38 LA09 Abstracts

IP1

Preconditioning for PDE-constrained Optimization

For many partial differential equations (PDEs) and sys-tems of PDEs efficient preconditioning strategies now existwhich enable effective iterative solution of the linear orlinearised equations which result from discretization. Fewwould now argue for the use of Krylov subspace methodsfor most such PDE problems without effective precondi-tioners. There are many situations, however, where it isnot the solution of a PDE which is actually required, butrather the PDE expresses a physical constraint in an Opti-mization problem: a typical example would be in the con-text of the design of aerodynamic structures where PDEsdescribe the external flow around a body whilst the goalis to select its shape to minimize drag. We will briefly de-scribe the mathematical structure of such PDE-constrainedOptimization problems and show how they lead to largescale saddle-point systems. The main focus of the talk willbe to present new approaches to preconditioning for suchOptimization problems. We will show numerical resultsin particular for problems where the PDE constraints areprovided by the Stokes equations for incompressible viscousflow.

Andy WathenOxford UniversityComputing Laboratoryandy.wathen@comlab.ox.ac.uk

IP2

Exploiting Sparsity in Matrix Functions, with Ap-plications to Electronic Structure Calculations

In this talk, some general results on analytic decay boundsfor the off-diagonal entries of functions of large, sparse(in particular, banded) Hermitian matrices will be pre-sented. These bounds will be specialized to various ap-proximations to density matrices arising in Hartree-Fockand Density Functional Theory (assuming localized basisfunctions are used), proving exponential decay (i.e., ‘near-sightedness’) for gapped systems at zero electronic tem-perature and thus providing a rigorous justification for thepossibility of linear scaling methods in electronic structurecalculations for non-metallic systems. The case of densitymatrices for disordered systems (Anderson model) and formetallic systems at finite temperature will also be consid-ered. Additional applications, possible extensions to thenon-Hermitian case, and some open problems will concludethe talk. This is joint work with Paola Boito and NaderRazouk.

Michele Benzi

DepartmentofMathematics and Computer ScienceEmory Universitybenzi@mathcs.emory.edu

IP3

Matrix Factoriazations and Total Positivity

The concept of a totally positive matrix grew out of threeseparate applications: Vibrating systems, interpolation,and probability. Since them, this class of matrices con-tinues to garner interest from both applied and theoreticalmathematicians. In 1953, a short paper appeared by AnneWhitney in which she established a ”reduction” result thatmay be one of the most powerful results on this topic. Myplan for this presentation is to survey the relevant historyfrom Anne’s paper to present day, and to discus the result-

ing impact of her reduction result. Particular attentionwill be paid to bidiagonal factorizations of totally positivematrices, along with a number of related applications.

Shaun M. FallatUniversity of ReginaCanadasfallat@math.uregina.ca

IP4

Model Reduction for Uncertainty Quantificationand Optimization of Large-scale Systems

For many engineering systems, decision-making underuncertainty is an essential component of achieving im-proved system performance; however, uncertainty quan-tification and stochastic optimization approaches are gen-erally computationally intractable for large-scale systems,such as those resulting from discretization of partial dif-ferential equations. This talk presents recent advances inmodel reduction methods — which aim to generate low-dimensional, efficient models that retain predictive fidelityof high-resolution simulations — that make possible thecharacterization, quantification and control of uncertaintyin the large-scale setting.

Karen WillcoxMassachusetts Institute of Technologykwillcox@mit.edu

IP5

Recent Advances in the Numerical Solution ofQuadratic Eigenvalue Problems

A wide variety of applications require the solution of aquadratic eigenvalue problem (QEP). The most commonway of solving the QEP is to convert it into a linear problemof twice the dimension and solve the linear problem by theQZ algorithm or a Krylov method. Much work has beendone to understand the linearization process and the effectsof scaling. Structure preserving linearizations have beenstudied, along with algorithms that preserve the spectralproperties in finite precision arithmetic. We present a newclass of transformations that preserves the block structureof widely used linearizations and hence does not requirecomputations with matrices of twice the original size. Weshow how to use these structure preserving transformationsto decouple symmetric and nonsymmetric quadratic matrixpolynomials.

Francoise TisseurUniversity of ManchesterDepartment of Mathematicsftisseur@maths.manchester.ac.uk

IP6

Definite versus Indefinite Linear Algebra

Matrices with symmetries with respect to indefinite innerproducts arise in many applications, e.g., Hamiltonian andsymplectic matrices in control theory and/or the theoryof algebraic Riccati equations. In this talk, we give anoverview on latest developments in the theory of matricesin indefinite inner product spaces and focus on the sim-ilarities and differences to the theory of Euclidean innerproduct spaces. Topics include normality, polar decom-positions, and singular value decompositions in indefinite

LA09 Abstracts 39

inner product spaces.

Christian MehlTechnische Universitaet BerlinInstitut fur Mathematikmehl@math.tu-berlin.de

IP7

Beyond the Link-Graph: Analyzing the Content ofPages to Better Understand their Meaning

In recent years the Web link graph has attracted much at-tention. However once we start analyzing the content Webpages many other fascinating graphs reveal themselves. Inmy talk I will discuss some of the information structuresthat we can construct from different Web content analysissources, such as: anchor text, query and click logs, socialtagging, natural language processing of text, and user pro-vided semantic data.

Hugo ZaragozaYahoo! Research Barcelonahugoz@yahoo-inc.com

IP8

Numerical Linear Algebra, Data Mining and ImageProcessing

In this talk, I will present two applications in data miningand image processing. The first application is related tothe problem where an instance can be assigned to multi-ple classes. For example, in text categorization tasks, eachdocument may be assigned to multiple predefined topics,such as sports and entertainment; in automatic image orvideo annotation tasks, each image or video may belongto several semantic classes, such as urban, building, road,etc; in bioinformatics, each gene may be associated with aset of functional classes. The second application is relatedto multiple class image segmentation problem. For ex-ample, foreground-background segmentation has wide ap-plications in computer vision (scene analysis), computergraphics (image editing) and medical imaging (organ seg-mentation). Both applications share the concept of semi-supervised learning and involve several numerical linear al-gebra issues. Finally, I will present some matrix compu-tation methods for solving these numerical issues. Exper-imental results are also given to demonstrate the applica-tions and computation methods.

Michael NgHong Kong Baptist UniversityDepartment of Mathematicsmng@math.hkbu.edu.hk

IP9

Confidence and Misplaced Confidence in Image Re-construction

Forming the image from a CAT scan and taking the blurout of vacation pictures are problems that are ill-posed. Bydefinition, small changes in the data to an ill-posed problemmake arbitrarily large changes in the solution. How can wehope to solve such problems using noisy data and inexactcomputer arithmetic? In this talk we discuss the use of sideconditions and bias constraints to improve the quality ofsolutions. We discuss their impact on solution algorithmsand show their effect on our confidence in the results.

Dianne P. O’Leary

University of Maryland, College ParkDepartment of Computer Scienceoleary@cs.umd.edu

IP10

Sparse Optimization: Algorithms and Applications

Many signal- and image-processing applications seek sparsesolutions to large underdetermined linear systems. Thebasis-pursuit approach minimizes the 1-norm of the solu-tion, and the basis-pursuit denoising approach balancesit against the least-squares fit. The resulting problemscan be recast as conventional linear and quadratic pro-grams. Useful generalizations of 1-norm regularization,however, involve optimization over matrices, and often leadto considerably more difficult problems. Two recent exam-ples: problems with multiple right-hand sides require joint-sparsity patterns among the solutions and involve mini-mizing the sum of row norms; matrix-completion problemsrequire low-rank matrix approximations, and involve min-imizing the sum of singular values. I will survey some ofthe most successful and promising approaches for sparseoptimization, and will show examples from a variety of ap-plications.

Michael P. FriedlanderDept of Computer ScienceUniversity of British Columbiampf@cs.ubc.ca

IP11

Using Combinatorics to Solve Systems in M-matrices in Nearly-linear Time

Symmetric M-matrices arise in areas as diverse as Sci-entific Computing, Combinatorial Optimization, and Ma-chine Learning. In this talk, we survey recently developedcombinatorial algorithms for preconditioning these matri-ces. These algorithms enable one to solve linear equa-tions in these matrices in asymptotically nearly-linear time.The problem of solving equations in symmetric M-matricesleads to questions in graph theory such as: What is thebest approximation of a graph by a sparser graph? Whatabout by a tree? And, what is a good decomposition of agraph? We will provide answers and explain their conse-quences. Finally, we will survey a randomized combinato-rial algorithm for scaling a symmetric M-matrix to makeit diagonally dominant.

Dan SpielmanDepartment of MathematicsMITspielman@math.mit.edu

CP1

Preconditioning Systems Arising from ElectronStructure Calculations Using the Korringa-Kohn-Rostoker Green Function Method

Electron structure calculations are known to be computa-tionally expensive and arise in different fields of physics.Recently a linear scaling method for metallic systems hasbeen developed by Zeller at Forschungszentrum Jlich. Thismethod uses QMR to solve a linear system with multipleright-hand sides arising from the discretization of the prob-lem. The convergence rate depends on the temperature ofthe underlying physical problem. We will present first re-

40 LA09 Abstracts

sults on preconditioning these systems.

Matthias BoltenUniversity of Wuppertalbolten@math.uni-wuppertal.de

CP1

On Preconditioners for Saddle Point ProblemsArising in Implicit Time Stepping of Maxwell’sEquations

Time integration of space-discretized Maxwell’s equationsis often carried out by second-order time integrationschemes, treating the curl terms explicitly and obeying theCFL stability restriction. In different applications, for in-stance, when locally refined unstructured meshes are em-ployed, the CFL restriction can be restrictive, so that un-conditionally stable implicit time integration becomes at-tractive. In implicit time integration of space discretizedMaxwell’s equations large-scale saddle-point a linear sys-tem has to be solved at every time step. As our recentresearch shows, an powerful preconditioner is then a must.We report on recent results on preconditioning for this typeof saddle-point problems.

Mike A. BotchevUniversity of Twentem.a.botchev@math.utwente.nl

Jan VerwerCWIThe Netherlandsjan.verwer@cwi.nl

CP1

A Two-Level Ilu Preconditioner for Electromag-netism Applications

In this work, we consider the solution of the linear systemsarising from electromagentism applications by precondi-tioned Krylov subspace methods. Sparse approximate in-verse preconditioners based on Frobenius norm minimiza-tion are quite effective for this type of problems. We ex-periment with a two-level ILU preconditioner with graphpartioning techniques, and discarding small entries whichcorrespond to far field interactions. Preliminary resultsshow that this two-level preconditioner is competitive.

Rafael BruD. Matematica Aplicada, ETSEAUniv. Politecnica de Valenciarbru@mat.upv.es

Natalia MallaInstituto Matematica MultidisciplinarUniversidad Politecnica de Valencianamalmar@doctor.upv.es

JOSE MarinInstitut de Matematica MultidisciplinarUniversitat Politecnica de Valenciajmarinma@imm.upv.es

CP1

Preconditioning GMRES(m) for Finite VolumeDiscretization of Incompressible Navier-StokesEquation by Hermitian/Skew-Hermitian Sepera-

tion

Existing papers show that, after finite difference and fi-nite element discretization of incompressible Naiver-Stokesequation, we will get special linear system with struc-ture of saddle point problem. GMRES(m) with newly de-veloped preconditioner Hermitian/Skew-Hermitian Sepa-ration (GMRES(m)-HSS) provides an efficient way to solvethe saddle point system. In this paper, firstly we discusssaddle point system from finite volume discretization of in-compressible Naiver-Stokes equation. Then, we solve thesystem by GMRES(m)-HSS. Furthermore, if the size of theoriginal PDE model is large, the solution speed of the linearsystem is still slow. Since large portion of computationaleffort of GMRES(m)-HSS is spent on building the orthog-onalization basis, in this paper, by newly developed high-level parallel programming language, we develop a methodto accelerate convergence of GMRES(m)-HSS by parallelGram-Schmidt (pGS) process. Theoretical analysis showsthat pGS can decrease time complexity of GMRES(m)-HSS from O(m) to O(m). Computational experimentsshow that pGS can significantly increase solution speed ofGMRES(m)-HSS.

Di ZhaoLouisiana Tech Universitydzh002@latech.edu

CP2

On a stable variant of Simpler GMRES and GCR

We propose and analyze a stable variant of Simpler GM-RES and GCR, which is based on an adaptive choice ofthe Krylov subspace basis at a given iteration step usingan intermediate residual norm decrease criterion. The newdirection vector is chosen as in the original implementationof Simpler GMRES or it is equal the normalized residualvector as in the GCR method. Such an adaptive strategyleads to a well-conditioned basis of the Krylov subspace,which results in a numerically stable and more robust vari-ant of Simpler GMRES or GCR.

Pavel JiranekCERFACSjiranek@cerfacs.fr

CP2

Progressive Gmres: A Minimum Residual KrylovMethod Which Is Equivalent to Minres in the Sym-metric Case

We study Beckermann and Reichel’s Progressive GM-RES (ProGMRES) algorithm, a minimum residual Krylovmethod for approximating solutions to n × n nearly sym-metric linear systems, with skew symmetric part of ranks << n. Generating the Krylov subspace basis requires thestorage of only s+2 vectors. We show that for symmetric,possibly indefinite systems, the algorithm is equivalent inexact arithmetic to MINRES. Experiments imply that, inmany cases, ProGMRES is computationally equivalent toMINRES for symmetric systems. However, in some exper-iments, ProGMRES appears to be less stable.

Kirk M. Soodhalter, Daniel B. SzyldTemple UniversityDepartment of Mathematicsksoodha@temple.edu, szyld@temple.edu

LA09 Abstracts 41

CP2

GIDR(s,l): IDR(s) with Higher-Degree Stabiliza-tion Polynomials

Recently Sleijpen and van Gijzen have proposed IDR(s)with stabilization polynomials of higher degree, namedIDRstab, which is an efficient method for solving real largenonsymetric systems of equations. Their derivation ofIDRstab is based on a complex version of the so-calledIDR principle, although they consider real systems of equa-tions. In the present talk we introduce a generalized realIDR principle, which enables us to incorporate stabilizationpolynomials of higher degree into IDR(s).

Masaaki Sugihara, Masaaki TanioGraduate School of Information Science and Technology,the University of Tokyom sugihara@mist.i.u-tokyo.ac.jp, masaaki tanio@mist.i.u-tokyo.ac.jp

CP2

Notes on the Convergence of the Restarted Gmres.

Our first result concerns the convergence of restarted GM-RES for normal matrices. We prove that the cycle-convergence of the restarted GMRES applied to systemswith a normal coefficient matrix is sublinear. The secondresult concerns the convergence of the restarted GMRESin the general case. In the general case, we prove thatany admissible cycle-convergence curve is possible for therestarted GMRES at a certain number of cycles. This re-sult holds for any eigenvalue distribution, therefore eigen-values alone do not determine the convergence of restartedGMRES.

Eugene VecharynskiUniversity of Colorado Denveryaugen.vecharynski@ucdenver.edu

Julien LangouDepartment of Mathematical & Statistical SciencesUniversity of Colorado Denverjulien.langou@ucdenver.edu

CP3

Interpolation of Reduced-Order Linear Operatorson Matrix Manifolds

A new method is presented for interpolating reduced-ordermodels on matrix manifolds. This method relies on thelogarithm and exponential mappings to move the interpo-lation data to a tangent space where standard multivariateinterpolation algorithms can be applied and bring it backto the manifold of interest, respectively. The proposedmethod is illustrated for several engineering applicationsassociated with the orthogonal manifold and non-compactStiefel manifold.

David Amsallem, Charbel FarhatStanford Universityamsallem@stanford.edu, CFarhat@stanford.edu

CP3

A New Algorithm for Computing a Darboux Trans-formation with Large Shifts

A monic Jacobi matrix is a tridiagonal matrix which con-tains the parameters of the three-term recurrence relation

satisfied by the sequence of monic polynomials orthogonalwith respect to a measure. The basic Geronimus trans-formation with shift α transforms the monic Jacobi matrixassociated with a measure dμ into the monic Jacobi matrixassociated with dμ/(x−α)+Cδ(x−α), for some constantC. In this paper we examine the algorithms available tocompute this transformation and we propose a more ac-curate algorithm, estimate its forward errors, and provethat it is forward stable. In particular, we show that forC = 0 the problem is very ill-conditioned, and present anew algorithm that uses extended precision.

Maria Isabel Bueno CachadinaThe University of California, Santa Barbarambueno@math.ucsb.edu

Alfredo DeanoUniversity of Cambridge, UKad495@cam.ac.uk

Edward TavernettiUniversity of California, Davisetavernetti@math.ucdavis.edu

CP3

Spectral Methods for Parameterized Matrix Equa-tions

We apply polynomial approximation techniques to thevector-valued function that satisfies a linear system ofequations where the matrix and the right hand side de-pend on a parameter. We derive both an interpolatorypseudospectral method and a residual-minimizing Galerkinmethod, and we show how each can be interpreted as solv-ing a truncated infinite system of equations; the differencebetween the two methods lies in where the truncation oc-curs.

Paul ConstantineStanford UniversityInstitute for Computational and MathematicalEngineeringpaul.constantine@stanford.edu

David F. GleichStanforddgleich@stanford.edu

CP3

Riemannian Optimization Algorithms for MatrixCompletion

We present new algorithms for the problem of recovering alarge low-rank matrix from a subset of its entries, known asmatrix completion. The methods minimize the least-squareerror of the projection of a matrix directly on the Rieman-nian manifold of matrices of fixed rank. The optimizationalgorithms are the generalization of steepest descent, non-linear cg and Newton’s algorithm to this Riemannian man-ifold. We will demonstrate the numerical performance incomparison with some recent algorithms.

Bart VandereyckenKatholieke Universiteit Leuvenbart.vandereycken@cs.kuleuven.be

Stefan VandewalleScientific Computing Research GroupKULeuven

42 LA09 Abstracts

stefan.vandewalle@cs.kuleuven.be

CP4

Generalized M-Matrices

Various classes of generalized M -matrices are discussed in-cluding the new class of GM -matrices. The matrices con-sidered are of the form sI − B where B has either thePerron-Frobenius property or the strong Perron-Frobeniusproperty but it is not necessarily nonnegative, I is the iden-tity matrix, and s is a nonnegative scalar not exceeding thespectral radius of B. Results that are analogous to thoseknown for M -matrices are shown for these classes of gener-alized M -matrices and various splittings of a GM -matrixare studied along with conditions for their convergence.

Abed ElhashashDrexel UniversityDepartment of Mathematicsaae36@drexel.edu

Daniel B. SzyldTemple UniversityDepartment of Mathematicsszyld@temple.edu

CP4

Mixed Class of H-Matrices and Schur Comple-ments

It is well-known that there exist nonsingular H-matricessuch that their comparison matrix M(A) is singular. Allequimodular matrices related to this case form the mixedclass of H-matrices, HM . Here, we study their Schur com-plements. In particular, it is determined that the Schurcomplement of a nonsingular matrix in HM exists and isan H-matrix, but it can belong to HM or to HI (M(S) isnonsingular). Conditions to distinguish these two cases aregiven.

Isabel Gimenez, Rafel Bru, Cristina CorralInstituto de Matematica MultidisciplinarUniversidad Politecnica de Valenciaigimenez@imm.upv.es, rbru@imm.upv.es,ccorral@imm.upv.es

Jose MasInstitut de Matematica MultidisciplinarUniversitat Politecnica de Valenciajmasm@imm.upv.es

CP4

Numerical Approximation of the MinimalGersgorin Set

One possibility to find a good numerical approximationto the minimal Gersgorin set (MGS) is presented in[Varga, Cvetkovic, Kostic, Approximation of the minimalGersgorin set of a square complex matrix, ETNA 30 (2008),398-405]. Here we will show some improvements of this al-giruthm, plotting an area which contains MGS itself, andfor which a limited number of boundary points are actualboundary points of MGS.

Richard VargaDepartment of Mathematical Sciences, Kent StateUniversityvarga@math.kent.edu

Ljiljana D. Cvetkovic, Vladimir KosticUniversity of Novi Sad, Faculty of ScienceDepartment of Mathematics and Informaticslila@im.ns.ac.yu, rikica@im.ns.ac.yu

CP4

QR Decomposition of H-Matrices

The hierarchical (H-) matrix format allows storing a vari-ety of dense matrices from certain applications in a specialdata-sparse way with linear-polylogarithmic complexity. Anew algorithm to compute the QR decomposition of an H-matrix will be presented. Like other H-arithmetic opera-tions the HQR decomposition is of linear-polylogarithmiccomplexity. We will compare our new algorithm with twoolder ones by using two series of test examples and discussbenefits and drawbacks of the new approach.

Peter Benner, Thomas MachTU ChemnitzGermanybenner@mathematik.tu-chemnitz.de,thomas.mach@mathematik.tu-chemnitz.de

CP5

Eigenspaces of Third-Order Tensors

Building upon a recently described tensor-tensor multipli-cation [Kilmer, Martin, and Perrone; submitted, October2008], we show that, on a particular module of n-by-n ma-trices, every linear transformation can be represented bytensor-matrix multiplication by a third-order tensor. Inthis context we investigate invariant sub-modules and de-fine tensor eigenvectors and their corresponding eigenma-trices. In addition, we examine an Arnoldi-like method forcomputation of tensor eigenvectors and eigenmatrices.

Karen S. BramanSouth Dakota School of MinesKaren.Braman@sdsmt.edu

CP5

Combinatorial Applications ofMultidimensional Matrix Theory

Recently the multidimensional matrix theory have foundseries of applications mostly in all areas of mathematics.In our talk we will be constrained only by our own findingsconcerning some new approaches in solving problems bymeans of multidimensional matrices, matrix constructionsand functions. Moreover, due to time restrictions, we willreport only on combinatorial applications explaining ourMatrix Network Method and its applications in combina-torial enumeration, in graph and hypergraph coloring, insymmetric function theory, and in substructural combina-torics.

Armenak S. GasparyanProgram Systems Institute of RASarmen@armen.pereslavl.ru

CP5

Fourier Basis Regression Against Multicollinearity

Multicollinearity is a long-standing problem in Statistics.This paper presents an alternative solution to deal withmulticollinear observations. In this paper, we propose touse orthogonal Fourier bases F to obtain an estimate of β

LA09 Abstracts 43

from the subspace spanned by the orthogonalized columnsof the X matrix. We also demonstrate that the orthogonal-ized independent variables have research-related physicalinterpretations which arise not just because of mathemat-ical convenience but because of the inherent informationcontained in these new bases. Finally, we show the mathe-matical elegance of using Fourier bases in regression analy-sis as opposed to the often messy mathematics of ordinaryregression. It is demonstrated that the two definitions ofmulticollinearity, namely, that based on variance inflationfactors and condition numbers are in fact equivalent defi-nitions.

Manuel Beronilla, Eduardo LacapUniversity of Rizal Systemmanuelberonilla@yahoo.com,lacap eduardo27@yahoo.com

CP5

Scalable Implicit Symmetric Tensor Approxima-tion

We describe some algorithms for producing a best multilin-ear rank r approximation of a symmetric tensor, useful e.g.for working with higher moments and cumulants of a largep-dimensional random vector. Since the approximated d-way tensor has pd entries, for large p it is critical never toexplicitly represent the entire tensor. Working implicitlyand approximately where appropriate, this approach en-ables the use of higher order factor models on large-scaledata.

Jason MortonStanford Universityjason@math.stanford.edu

CP6

Combinative Preconditioning Based on RelaxedNested Factorization and Tangential Filtering Pre-conditioner

The problem of solving block tridiagonal linear systemsarising from the discretization of PDE on structured gridis considered. Nested Factorization preconditioner intro-duced by Appleyard et. al. is an effective preconditionerfor certain class of problems and a similar method is imple-mented in Schlumerger’s widely used Eclipse oil reservoirsimulator. In this work, a relaxed version of Nested Fac-torization preconditioner is proposed. Effective multiplica-tive/additive preconditioning is achieved in combinationwith Tangential filtering preconditioner.

Pawan KumarINRIA Futursmisterpawan@gmail.com

Laura GrigoriINRIAFranceLaura.Grigori@inria.fr

Frederic NatafLaboratoire J.L. Lionsnataf@ann.jussieu.fr

Qiang NiuSchool of Mathematical Sciences, Xiamen Universitykangniu@gmail.com

CP6

Inverse Sherman-Morrison Factorization and ItsRelated Preconditioners

We analyze the Inverse Sherman-Morrison factorization ofan invertible matrix, in particular its relations with theLDU factorization. Since the ISM factorization is highlyparametrizable we analyze also some relations of the fac-tors for different parameter choices. Also the block versionof the factorization is studied. The main motivation forintroducing the ISM factorization is to use it as a precondi-tioner. Some existence properties of these preconditionersand some numerical results are shown.

Jose Mas, J. Cerdan, JOSE MarinInstitut de Matematica MultidisciplinarUniversitat Politecnica de Valenciajmasm@imm.upv.es, jcerdan@imm.upv.es,jmarinma@imm.upv.es

Miroslav TumaAcademy of Sciences of the Czech RepublicInstitute of Computer Sciencetuma@cs.cas.cz

CP6

The Importance of Structure in Algebraic Precon-ditioners

In recent years, algebraic preconditioners have achieved aconsiderable degree of efficiency and robustness. Incom-plete factorizations represent an important class of suchpreconditioners. Despite their success, for many problemsfurther tools are required to enhance performance and im-prove reliability. We propose an approach that is based onforcing the original matrix structure, or an appropriatelymodified structure of powers of the system matrix, into avalue-based incomplete Cholesky decomposition. Our im-plementation will be included within the HSL mathemati-cal software library.

Jennifer ScottRutherford Appleton LaboratoryComputational Science and Engineering Departmentjennifer.scott@stfc.ac.uk

Miroslav TumaAcademy of Sciences of the Czech RepublicInstitute of Computer Sciencetuma@cs.cas.cz

CP6

Optimal Block Method and Preconditioner forBanded Matrices

Optimized Schwarz methods (OSMs) for PDEs use Robin(or higher order) transmission conditions in the artificialinterfaces between subdomains, and the Robin parametercan be optimized. We present a completely algebraic ver-sion of the OSM, including an algebraic approach to findthe optimal operator or a sparse approximation thereof.We can thus apply this method to any banded or blockbanded linear system of equations, and in particular to dis-cretizations of PDEs on irregular domains. With the com-putable optimal operator, we prove that both the OSM andGMRES with OSM preconditioning converge in no morethan two iterations for the case of two subdomains. Veryfast convergence is attained even when the optimal opera-tor is approximated by a sparse transmission matrix. Nu-

44 LA09 Abstracts

merical examples illustrating these results are presented.

Daniel B. SzyldTemple UniversityDepartment of Mathematicsszyld@temple.edu

Martin J GanderUnivesity of GenevaSection de Math’ematiquesmartin.gander@unige.ch

Sebastien LoiselTemple UniversityDepartment of Mathematicsloisel@temple.edu

CP7

Sublinearly Fast Solvers for Finite Difference Op-erators on Mostly Structured Grids

For boundary value problems associated with constant co-efficient finite difference operators on structured grids with-out body loads, we will demonstrate there exists solverswhose complexity is lower than O(N), where N denotes thenumber of grid points. We analytically construct the fun-damental solution for the difference operator, then use fasttechniques for computing convolutions involving the funda-mental solution. The resulting methods scale on the orderof the number of grid points on the boundary.

Adrianna GillmanUniversity of Colorado at BoulderDepartment of Applied Mathematicsadrianna.gillman@colorado.edu

Per-Gunnar MartinssonUniversity of Colorado at Boulderper-gunnar.martinsson@colorado.edu

CP7

Linear Systems with Large Off-Diagonal Elementsand Discontinuous Coefficients

Previous work showed that “geometric” scaling of the equa-tions is useful for handling nonsymmetric systems withdiscontinuous coefficients. It is shown that after suchscaling, AAT contains relatively large diagonal elements.These two operations are inherent in the sequential CGMNand the block-parallel CARP-CG algorithms, making themvery effective for systems with large off-diagonal elementsand discontinuous coefficients. Examples involving hetero-geneous media include convection-diffusion equations withlarge convection and the Helmholtz equation with largewave numbers.

Dan GordonDept. of Computer ScienceUniversity of Haifagordon@cs.haifa.ac.il

Rachel GordonDept. of Aerospace EngineeringThe Technion - Israel Inst. of Technologyrgordon@tx.haifa.ac.il

CP7

Fast Iterative Solver for Incompressible Navier-Stokes Equations with Spectral Elements

We introduce a block preconditioning technique for spec-tral element discretization of the steady Navier-Stokesequations. This method is built from subsidiary DomainDecomposition solvers which use the Fast DiagonalizationMethod to eliminate degrees of freedom on subdomain in-teriors. We demonstrate the effectiveness of this precondi-tioner in numerical simulations of several benchmark ?owsincluding Kovasznay ?ow, ?ow in a lid-driven cavity and?ow over a backward-facing step.

P. Aaron LottNational Institute of Standards and TechnologyAaron.Lott@nist.gov

Howard C. ElmanUniversity of Maryland, College Parkelman@cs.umd.edu

CP7

Numerical Linear Algebra Applied to the Simula-tion of the Dynamic Interaction between a RailwayTrack and a High Speed Train

In the simulation of the dynamic interaction between arailway track and a high speed train, many useful linearalgebra techniques have to be applied effectively in orderto execute it as efficiently and accurately as possible. Atfirst, in the eigenvalue analysis of the large model consist-ing of two subsystems of the vehicles and the track, it is aninteresting problen how we should deal with the interface ofcontact between a wheel and a rail. Contact points of theinteraction move along the track and therefore the charac-teristic of the system depends on the time. An appropriatemethodology of eigenvalue analysis of these dynamicallyinteracting system is needed. Secondly, for railway engi-neers it is very important to know the influence of the sys-tem structural components on a variation of contact forcebetween a wheel and a rail. Because a large variation ofcontact force causes the early deterioration of the track ge-ometry. In this study, including the investigation of theabove problems, an application of numerical linear algebraand related techniques such as singular value decomposi-tion, inverse analysis method are described.

Akiyoshi YoshimuraTokyo University of TechnologyEmeritus Professoraki-yoshimura@kyi.biglobe.ne.jp

CP8

Cyclic Reduction and Multigrid

Given a discretized PDE on a uniform mesh, one stepof cyclic reduction amounts to ordering the gridpoints inred/black fashion and eliminating the ones that correspondto one of the colors. The resulting operator is spectrallybetter than its original (unreduced) counterpart, but adrawback of this approach is that implementation is noteasy, in particular in three dimensions, due to the irregularstructure of the associated mesh. In this talk we examinecyclic reduction in the context of multigrid, analyze thesmoothing properties of cyclically reduced operators, anddiscuss ways of efficiently implementing solvers based on

LA09 Abstracts 45

this approach.

Chen GreifUniversity of British ColumbiaCanadagreif@cs.ubc.ca

CP8

Adaptive Algebraic Block Iterative Methods

This talk will present a new method of adaptively con-structing block iterative methods based on Local Sensitiv-ity Analysis (LSA). The main application is the develop-ment of adaptive smoothers for multigrid methods. Givena linear system, Ax = b, LSA is used to identify blocksof the matrix, A, which are strongly connected in a sensesimilar to the traditional measure of strength in algebraicmultigrid methods. Block iterative Gauss-Seidel or Jacobimethods can then be constructed based on the identifiedblocks. The size of the blocks can be varied adaptively al-lowing for the construction of a class of block iterative alge-braic methods. Results will be presented for both constantand variable coefficient elliptic problems, systems arisingfrom both scalar and coupled system PDEs, as well as lin-ear systems not arising from PDEs. The simplicity of themethod will allow it to be easily incorporated into existingmultigrid codes while providing a powerful tool for adap-tively constructing smoothers tuned to the problem.

Bobby PhilipLos Alamos National Laboratorybphilip.kondekeril@gmail.com

Timothy P. ChartierDavidson Collegetichartier@davidson.edu

CP8

Halley and Newton Are One Step Apart

In this talk we onsider numerical solution of a nonlinearsystem of equations F (x) = 0 where the function is suffi-ciently smooth. While Newton’s method which has secondorder rate of convergence method, is usually used for solv-ing such systems, third order methods like Halley’s methodwill in general use fewer iterations than a second ordermethod to reach the same accuracy. However, the numberof arithmetic operations per iteration is higher for thirdorder methods than second order methods. We will showthat for a large class of problems the ratio of the num-ber of arithmetic operations of a third order method andNewton’s method is constant per iteration using a directmethod to solve the linear systems including the cost ofcomputing the function and the derivatives. We also dis-cuss replacing the direct method using an iterative method.

Trond SteihaugUniversity of BergenDepartment of Informaticstrond@ii.uib.no

Geir Gundersen, Sara SuleimanUniversity of Bergengeirg@ii.uib.no, sara.suleiman@student.uib.no

CP8

On the Parameterized Uzawa Methods for Saddle

Point Problems

Bai, Parlett and Wang recently proposed an efficient pa-rameterized Uzawa method for solving the nonsingularsaddle point problems (see Numer. Math. 102(2005)1-38).In this paper we investigates the W− norm convergence ofthe parameterized Uzawa methods for saddle point prob-lems under the a suitable weighted norm of the iterativematrix. Based on the estimates of the weighted norm ofthe iterative matrix, an new inexact parameterized Uzawamethods for saddle point problems is presented.

Bing ZhengSchool of Mathematics and StatisticsLanzhou University, Lanzhou, P.R. Chinabzheng@lzu.edu.cn

CP9

Implementing the Retraction Algorithm for Fac-toring Symmetric Banded Matrices

Problems arising when modeling structures subject to vi-brations or in designing optical fibers wrapped around aspool give rise to eigenvalue problems and linear systemswith symmetric, banded matrices. We present various vari-ations of the retraction algorithm which factors a matrixinto a product of matrices while preserving symmetry, thebandwidth, and element growth even when the original ma-trix is indefinite. The factorization may be used to solvea system or compute the inertia of a matrix. It is basedon the algorithm of Bunch and Kaufman for symmetricnonbanded systems.

Linda KaufmanWilliam Paterson Universitykaufmanl@wpunj.edu

CP9

Compact Fourier Analysis for Deriving MultigridAs Direct Solver

A Compact Fourier Analysis (CFA) for designing new effi-cient multigrid (MG) methods is considered. The principalidea of CFA is to model MG by means of generating func-tions and block symbols. We utilize the CFA for derivingMG as a direct solver and give necessary conditions thathave to be fulfilled by the MG components for this pur-pose. General practical smoothers and transfer operatorsthat lead to powerful MG algorithms are introduced.

Christos KravvaritisTechnical University of MunichDepartment of Informaticskravvarc@in.tum.de

Thomas K. HuckleInstitut fuer InformatikTechnische Universitaet Muenchenhuckle@in.tum.de

CP9

Diagonal Pivoting Methods for Solving Unsymmet-ric Tridiagonal Systems

This talk concerns the LBMT factorization of unsymmet-ric tridiagonal matrices, where L and M are unit lowertriangular matrices and B is block diagonal with 1× 1 and2 × 2 blocks. In some applications, it is necessary to form

46 LA09 Abstracts

this factorization without row or column interchanges whilethe tridiagonal matrix is formed. We propose methodsfor forming such a factorization that both exhibit boundedgrowth factors and are normwise backward stable.

Jennifer ErwayWake Forest Universityerwayjb@wfu.edu

Roummel F. MarciaDepartment of Electrical and Computer EngineeringDuke Universityroummel@ee.duke.edu

CP9

Orthosymmetric Block Reflectors and Qr Factor-izations

Reflectors and block reflectors are widely used tools in nu-merical linear algebra. Both types of reflectors are neededfor the construction of non-Euclidean variants of the QRfactorization, for example, in hyperbolic and symplecticscalar products. We shall present implementation details,perturbation and error analysis of the hyperbolic and thesymplectic QR factorization.

Sanja SingerFaculty of Mechanical Engineering and NavalArchitecture, University of Zagrebssinger@fsb.hr

Sasa SingerDepartment of Mathematics, University of Zagrebsinger@math.hr

Vedran NovakovicFaculty of Mechanical Engineering and NavalArchitecture, University of Zagrebvenovako@fsb.hr

CP10

Solutions of Systems of Algebraic Equations froma Physical model of Induction Machine

This work deals with the study of a system of algebraicequations arising in the mathematical modeling of an elec-tric machine. The machine considered may be an Inductionmachine, a synchronous machine, or a transformer. Ourmodel is primarily based on a transformer. - The trans-former may be used to transfer large amount of power,voltage, or current. In the present case we consider themachine windings for homogeneous or non-homogeneousmedium with Dirichlet and Mixed Boundary conditions.We study time independent equations first, and later com-pare it with time dependent equations. Many cases of timedependent as well as time independent Maxwell’s equationshave been studied before. A comparison of exact solutionsto numerical solutions is established for different layers oftransformer windings under dissimilar structures. We fur-ther look for extra eddy currents produced and the lossof power therein. A Power-current or skin depth plot andTorque-skin depth plot will be done to find out how to con-trol and minimize the power loss and increase amount oftorque required to start the machine or stop the machinein minimum time by using transient analysis and classicalNyquist and Bode plots. An example is considered thatdepicts how equations and solutions behave differently indifferent media/materials with respect to power and torquecontrol. The analysis is being done by using a combination

of software tools, which include FEMLAB, Mat-Lab andSimulink, and SCILAB.

Kumud S. AltmayerUniversity of Arkansaskumud altmayer@yahoo.com

Kamran IqbalUniversity of ArkansasLittle Rockkxiqbal@ualr.edu

CP10

Eigenvalue Localization for Matrices with ConstantRow Or Column Sum

Motivated by some research on synchronization on net-works of mobile agents, we are interested in finding a goodeigenvalue localization for a square, real-valued, non sym-metric matrix with the number k on the main diagonal andexactly k ”−1” on off-diagonal places (in random position)in each row. According to Gersgorin theorem, all the eigen-values of such a matrix are located within the circle withcenter in k and radius equak to k, but this localization isnot good enough. We are interested in tighter localizationof all eigenvalues, different from zero. More generalizedresuts for matrices with constant row or column sum areobtained.

Ljiljana D. CvetkovicUniversity of Novi Sad, Faculty of ScienceDepartment of Mathematics and Informaticslila@im.ns.ac.yu

Juan Manuel PenaDepartamento de Matematica Aplicada, Universidad deZaragozajmpena@unizar.es

Vladimir KosticUniversity of Novi Sad, Faculty of ScienceDepartment of Mathematics and Informaticsrikica@im.ns.ac.yu

CP10

A Hadamard Product Involving Inverse-PositiveMatrices with Checkerboard Pattern

A nonsingular real matrix A is said to be inverse-positiveif all the elements of its inverse are nonnegative. Thisclass contains the M -matrices. The Hadamard prod-uct involving M -matrices has studied for several authors,Markham, Johnson, Neumann and other. Recently Chenin 2006 prove a conjeture concerning the Hadamard pow-ers of inverse M -matrices. In this work we study theHadamard product of certain classes of inverse-positive ma-trices. And we prove that for any pair A , B of triangu-lar inverse-positive matrices with the checkerboard pattern,the Hadamard (entry-wise) product A◦B−1 is again trian-gular inverse-positive matrices with the checkerboard pat-tern. Some interesting inequalities about of the minors ofthis class of matrices are presented in this work.

Maria T. GassoInstituto Matematica MultidisciplinarUniversidad Politecnica Valenciamgasso@mat.upv.es

Juan R. Torregrosa, Manuel F. Abad

LA09 Abstracts 47

Instituto de Matematica MultidisciplinarUniversidad Politecnica Valenciajrtorre@mat.upv.es, maabrod@mat.upv.es,

CP10

Spectral Inclusion Sets for Structured Matrices

We derive eigenvalue inclusion sets for centrohermitian,skew-centrohermitian, perhermitian, skew-perhermitian,and persymmetric matrices, that can be considered struc-tured analogs of the Gersgorin sets. We also present thenonsingularity conditions associated with these spectral in-clusion sets. In addition, we derive sets that can be consid-ered the structured analogs of the Brauer sets. Our tech-niques are general enough to allow for other symmetries aswell.

Aaron MelmanSanta Clara UniversityDepartment of Applied Mathematicsamelman@scu.edu

CP11

Parallel Preconditioning for Block-Structured CFDProblems

At the Institute for Propulsion Technology of the GermanAerospace Center (DLR), the parallel simulation systemTRACE (Turbo-machinery Research Aerodynamic Com-putational Environment) has been developed specificallyfor the calculation of internal turbo-machinery flows. Thefinite volume approach with block-structured grids requiresthe parallel, iterative solution of large, sparse systems oflinear equations. For convergence acceleration of the itera-tion, parallel preconditioners based on multi-level imcom-plete factorizations have been investigated. Numerical andperformance results of these methods will be presented fortypical TRACE problems on parallel computer systems,including multi-core architectures.

Achim Basermann, Hans-Peter KerskenGerman Aerospace Center (DLR)Simulation and Software Technology (SISTEC)Achim.Basermann@dlr.de, hans-peter.kersken@dlr.de

CP11

Extending Algorithm Based Fault Tolerance in Or-der to Support Error on the Fly

In a previous paper, we showed that Algorithm BasedFault Tolerance can be used to perform high performantand fault tolerant matrix-matrix multiplication. We ex-tend these ideas to most of linear algebra operations. Webelieve our approach is systematic and simply relies on aABFT BLAS. We will show result of a prototype ABFTLAPACK.

Julien LangouDepartment of Mathematical & Statistical SciencesUniversity of Colorado Denverjulien.langou@ucdenver.edu

CP11

On the use of Cholesky in the CholeskyQR algo-rithm

In practice, the CholeskyQR algorithm is the fastest knownalgorithm for the QR factorization of a tall and skinny ma-

trix. Unfortunately, this algorithm is unstable and, for ill-conditioned matrices, the resulting vectors do not form anorthonormal basis. Several variants have been proposed,all requiring the eigenvalue decomposition of the symmet-ric matrix ATA. We propose shifted Cholesky to cure theproblem of breakdown in Cholesky, and a subsequent re-orthogonalization step.

Matthew W. Nabity, Julien LangouDepartment of Mathematical & Statistical SciencesUniversity of Colorado DenverMatthew.Nabity@ucdenver.edu,julien.langou@ucdenver.edu

CP11

Resolution of Large Sparse Non-Hermitian Com-plex Linear Systems on a Nationwide Cluster ofClusters

To solve a large sparse non-Hermitian complex linearsystem Ax=b, we asynchronously compute in parallelthe dominant eigenelements of A and a preconditionedrestarted GMRES method on GRID5000, a Nationwidecluster of heterogeneous distributed clusters. We ex-ploit the computed eigenvalues to accelerate the GMRESmethod using a LS method. This preconditioned methodis known to not be mathematically correct when the ma-trix elements are complexes. Then, we propose a solutionto use this method using the same storage space on eachnodes that other classical GMRES methods. We proposenumerical experimentations using several GRID5000 sitesand we compare the obtained performance, time and itera-tion, with other methods for large matrices from industrialapplications.

Serge G. PetitonINRIA and LIFL, Universite de Lilleserge.petiton@inria.fr

Guy BergereLIFL, CNRSguy.bergere@lifl.fr

Ye ZhangCNRS/LIFLye.zhang@lifl.fr

Pierre-Yves AquilantiLIFL; CNRSpierre-yves.aquilanti@lifl.fr

CP12

Numerical Approaches for Nonlinear Ill-Posed In-verse Problems

Difficult nonlinear inverse problems arise in a variety of sci-entific applications, and iterative optimization algorithmsare often used to efficiently compute solutions to large-scale problems. In this work, we investigate Krylov-basedsolvers and appropriate preconditioners for linear systemsthat arise within nonlinear optimization schemes. Further-more, regularization methods for nonlinear ill-posed prob-lems are required for computing stable solution approxi-mations. Applications from medical image processing areconsidered.

Julianne ChungUniversity of Marylandjmchung@cs.umd.edu

48 LA09 Abstracts

CP12

Calculating the Numerical Rank of Sparse Matrices

Algorithms for calculating the numerical rank of sparsematrices are compared. The algorithms include (1) rankcalculation using Sylvester’s inertia theorem, (2) calcu-lating the rank using the sparse QR factorization spqr(http://www.cise.ufl.edu/research/sparse/SPQR/) and(3) post-processing the spqrresults. The comparison will be based on matrices fromthe San Jose State University Singular Matrix Database(http://www.math.sjsu.edu/singular/matrices/).

Leslie V. FosterSan Jose State UniversityDepartment of Mathematicsfoster@math.sjsu.edu

CP12

Limited Approximation of Numerical Range ofNormal Matrix

Let A be an n× n normal matrix, whose numerical rangeNR[A] is a k–polygon. Adam and Maroulas (On compres-sions of normal matrices, Linear Algebra Appl., 341(2002),403-418) established that if a unit vector v ∈ W ⊆ Cn, withdimW = k and the point v∗Av ∈ IntNR[A], then NR[A]is circumscribed to NR[P ∗AP ], where P is an n×(k−−1)isometry of {span{v}}⊥W → Cn. In this paper, we investi-gate an internal approximation of NR[A] by an increasingsequence of NR[Cs] of compressed matrices Cs = R∗

sARs,with R∗

sRs = Ik+s−−1, s = 1, 2, · · · , n − −k and addi-tionally NR[A] is expressed as limit of numerical ranges ofk–compressions of A.

John B. MaroulasNational Technical UniversityDepartment of Mathematicsmaroulas@math.ntua.gr

Maria AdamUniversity of Central GreeceDepartment of Computer Science and BiomedicalInformaticsmadam@ucg.gr

CP12

Accurate Solutions of Structured Linear Systemsfrom Rank-Revealing Decompositions.

For many structured linear systems (among them, Cauchyand Vandermonde, either totally positive, or non totallypositive) for which there are not accurate solvers, we showhow to compute accurate solutions by using rank-revealingdecompositions (RRD). In this way we take advantage ofthe great effort done in the last decades to compute accu-rately the SVD using RRD’s. The algorithms used involveO(n3) flops.

Juan M. Molera, Froilan M. DopicoDepartment of MathematicsUniversidad Carlos III de Madridmolera@math.uc3m.es, dopico@math.uc3m.es

CP13

Title of the Paper Here. Don’t Use All Caps.

In this paper the positive realness problem in circuit and

control theory is studied. A numerical method is developedfor verifying the positive realness of a given proper ratio-nal matrix H(s) for which H(s)+HT (−s) may have purelyimaginary zeros. The proposed method is only based onorthogonal transformations, it is structure-preserving andhas a complexity which is cubic in the state dimension ofH(s). Some examples are given to illustrate the perfor-mance of the proposed method.

Delin ChuDepartment of Mathematics, National University ofSingapore2 Science Drive 2, Singapore 117543matchudl@nus.edu.sg

Peter BennerTU ChemnitzGermanybenner@mathematik.tu-chemnitz.de

Xinmin LiuDepartment of Electrical and Computer EngineeringUniversity of Virginia,USAxmliu@virginia.edu

CP13

Dimension of Orbits of Singular Systems underProportional and Derivative Feedback

We consider triples of matrices (E,A,B), representing sin-gular linear time invariant systems in the form Ex(t) =Ax(t) + Bu(t), with E,A ∈ Mp×n(C) (not necessarilysquares), and B ∈ Mn×m(C), under proportional andderivative feedback. After to obtain a canonical reducedform preserving the structure of the systems where thestandard subsystem is maximal and using geometric tech-niques, we obtain the tangent and normal space to theorbits of equivalent triples. This description permit us todeduce the dimension of the orbits and analyze structuralstability of the triples.

Maria Isabel Garcia-PlanasUniversitat Politecnica de Catalunyamaria.isabel.garcia@upc.edu

CP13

Analyzing Controllability and Observability by Ex-ploiting the Stratification of Matrix Pairs

The qualitative information about nearby linear time-invariant (LTI) systems is revealed by the theory of strat-ification. Utilizing the software tool StratiGraph, we showhow the stratification theory is used to analyze the control-lability and observability characteristics. We demonstratewith two examples, a mechanical system consisting of athin uniform platform supported at both ends by springs,and a linearized Boeing 747 model. For both examples,nearby uncontrollable systems are identified as subsets ofthe complete closure hierarchy for the associated systempencils. Some quantitative results are also reported.

Stefan JohanssonUmea UniversityComputing Sciencestefanj@cs.umu.se

Pedher JohanssonUmea UniversityComputing Science and HPC2N

LA09 Abstracts 49

pedher@cs.umu.se

Erik ElmrothDepartment of Computing ScienceUmea Universityelmroth@cs.umu.se

Bo T. KagstromUmea UniversityComputing Science and HPC2Nbokg@cs.umu.se

CP13

Exposing Structure Algorithms for a Nonsymmet-ric Algebraic Riccati Equation Arising in TransportTheory.

The nonsymmetric algebraic Riccati equation arising intransport theory is a special algebraic Riccati equationwhose coefficient matrices having some special structures.By reformulating the Riccati equation, they were foundthat its arbitrary solution matrix is of a Cauchy-like formand the solution matrix can be computed from a vectorform Riccati equation instead of the matrix form that. Inthis talk, we review some classical-type and Newton-typeiterative methods for solving the vector form Riccati equa-tion and present some work under investigation .

Linzhang LuSchool of Mathematical ScienceXiamen University, Xiamen, P.R. Chinalzlu@xmu.edu.cn llz@gznu.edu.cn

CP13

A Parameter Free ADI-like Method for the Numer-ical Solution of Large Scale Lyapunov Equations

A parameter free ADI-like method is presented for con-structing an approximate numerical solution to a largescale Lyapunov equation. The algorithm uses an approxi-mate power method iteration to obtain a basis update andthen constructs a re-weighting of this basis update to pro-vide a factorization update that satisfies ADI-like conver-gence properties.

Hung NongSandia National Laboratoriesryannong@gmail.com

Danny C. SorensenRice Universitysorensen@rice.edu

CP13

Nonnegativity of the Group-Projectors Used toObtain the Nonnegativity of Control Singular Sys-tems

In POSTA06 we studied the nonnegativity of control sin-gular systems via state-feedbacks by using a characteriza-tion of the nonnegativity of the group-projector of a squarematrix E assuming the nonnegativity of the matrix E. InPOSTA09 we will present a characterization of the nonneg-ativity of the group-projector, slightly simplified. In thiswork, we use this last characterization to improve the studymentioned at the beginning. This work has been partially

supported by Grant DGI MTM2007-64477.

ALICIA HerreroInstituto de Matematica MultidisciplinarUniversidad Politecnica de Valencia, Spainaherrero@mat.upv.es

Francisco J. RamirezInstituto Tecnologico de Santo DomingoDominican Republicframires@hotmail.com

Nestor ThomeInstituto de Matematica MultidisciplinarUniversidad Politecnica de Valencia, Valencia, Spainnjthome@mat.upv.es

CP14

A Note on the dqds Algorithm with Rutishauser’sShift

We are concerned with the behaviour of the dqds algo-rithm for computing sigular values, when incorporatedwith Rutishauser’s shift. It is proved that in the itera-tive process Rutishauser’s shift is selected finally and thedqds converges cubically. Furthermore the phenomenon isobserved that once Rutishauser’s shift is selected ever af-ter it is selected. In this talk we will give an theoreticalexplanation for the phenomenon.

Kensuke AishimaGraduate School of Information Science and Technology,University of TokyoKensuke Aishima@mist.i.u-tokyo.ac.jp

Takayasu MatsuoGraduate School of Information Science and Technology,the University of Tokyomatsuo@mist.i.u-tokyo.ac.jp

Kazuo MurotaGraduate School of Information Science and Technology,University of Tokyomurota@mist.i.u-tokyo.ac.jp

Masaaki SugiharaGraduate School of Information Science and Technology,the University of Tokyom sugihara@mist.i.u-tokyo.ac.jp

CP14

Extraction Strategies for Nonsymmetric Eigen-value Problems

We consider the extraction phase of subspace methods likethe Jacobi-Davidson method for solving large eigenvalueproblems. In the nonsymmetric case, known approaches donot satisfy any optimality property and, in practice, theirbehavior may be unsatisfactory, especially when searchingfor the eigenvalues with largest real part. In this talk, weshall review and compare existing methods, and also pro-pose some new strategies providing enhanced robustness.

Julien CallantService de Metrologie NucleaireUniversite Libre de Bruxelles (ULB)jcallant@ulb.ac.be

50 LA09 Abstracts

Yvan notayUniversite Libre de Bruxellesynotay@ulb.ac.be

CP14

On Singular Two-Parameter Eigenvalue Problem

In the 1960s Atkinson introduced an abstract algebraic set-ting for multiparameter eigenvalue problems. He showedthat a nonsingular multiparameter eigenvalue problem isequivalent to the associated system of generalized eigen-value problems. We extend his result to singular two-parameter eigenvalue problems and show that under verymild conditions the finite regular eigenvalues of both prob-lems agree. This enables one to solve such problems bycomputing the common regular eigenvalues of two singulargeneralized eigenvalue problems.

Andrej MuhicInstitute of Mathematics, Physics and MechanicsJadranska 19, SI-1000 Ljubljana, Sloveniaandrej.muhic@fmf.uni-lj.si

Bor PlestenjakDepartment of MathematicsUniversity of Ljubljanabor.plestenjak@fmf.uni-lj.si

CP14

Tropical Scaling of Polynomial Eigenvalue Problem

We introduce a general scaling technique, based on tropicalalgebra,to compute the eigenvalues of a matrix polynomial.We show that under nondegeneracy conditions, the ordersof magnitude of the different eigenvalues are given by cer-tain “tropical roots’. We use them to construct a scaling,and we show by experiments that this improves the accu-racy of the computations, particularly when the data havevarious orders of magnitude.

Meisam Sharify, Stephane GaubertINRIA Saclay Ile-de-France &Centre de Mathematiques appliquees, EcolePolytechniquemeisam.sharify@inria.fr, stephane.gaubert@inria.fr

CP14

On MRRR-Type Algorithms for the TridiagonalSymmetric Eigenproblem and the Bidiagonal SVD

The focus of our research is extending Dhillon & Parlett’salgorithm MR3 to compute accurate singular vectors for abidiagonal matrix. Doing this without spoiling accuracy isa challenge. Among other aspects we had to evaluate howkey parts of MR3 could be varied to improve reliability andefficiency. As main theoretical contribution we regard ourinvestigation of using block factorizations, allowing 2 × 2-pivots, in the representation tree. Numerical results willdetail the effectiveness of our techniques.

Paul R. WillemsUniversity of Wuppertal, GermanyApplied Computer Science Groupwillems@math.uni-wuppertal.de

CP14

Numerical Solutions of Generalized Eigenvalue

Problems with Spectral Transformations

This talk concerns algorithms for solving generalized eigen-value problems, with emphasis on efficient iterative solu-tion of the sequence of linear systems that arise when in-exact subspace iteration and implicitly restarted Arnoldimethod are applied to detect interior eigenvalues. We pro-vide new insights into the tuning of preconditioning pro-posed by Spence and his collaborators, and show that thecost of solving these linear systems can be further reducedby the combined use of several other strategies.

Fei XueUniversity of Marylandpetrinet@math.umd.edu

Howard C. ElmanUniversity of Maryland, College Parkelman@cs.umd.edu

CP15

An Algorithm for Constructing a Canonical Basisand the Jordan Normal Form of Arbitrary LinearOperators over Any Field.

It is well known that the Jordan normal form for a linearoperator A is defined and is constructed together with thecorresponding basis in case when the basic field k is al-gebraically closed or, more generally, if the characteristic(equivalently, the minimal) polynomial of A is decomposedinto a product of linear polynomials. We define the (gener-alized) Jordan normal form (of the second kind) for a linearoperator A of a linear space L over an arbitrary field k andwith an arbitrary characteristic polynomial. For such anoperator we give an algorithm which is used to constructthis form by finding a corresponding canonical basis. Asapplication some other results are obtained, too.

Samuel H. Dalalyandalalyan@ysu.amdalalyan@ysu.am

CP15

A General Subspace Structure Parameterization ofa Pre-Compesator for a Non-Interacting Controllerin Robotic Manipulation Systems and Mechanisms

This paper deals with a total general geometric approachto the study of robotics manipulators and general mecha-nisms dynamics. In particular a parametrization of a pre-compensator through structures of subspaces for a nonin-teracting controller is presented. Due to the technologicaldevelopment, robotics applications are growing in manyindustrial sectors and even in such applications as medicalone (micro manipulation of internal tissues or laparoscopy).In such a framework some typical problems in robotics aremathematically formalized and analyzed. The outcomesare so general that it is possible to speak of structural prop-erties in robotic manipulation and mechanisms. The prob-lem of designing of force/motion controller is investigated.A generalized linear model is used and a careful analysisis made. The presented results consist of proposing a gen-eral geometric structure to the study of mechanisms. Inparticular, a Lemma and a Theorem are shown which offera parametrization of a pre-compesator for a task–orientedchoice of input subspaces. The existence of these inputsubspaces are a necessary condition for a structural nonin-

LA09 Abstracts 51

teraction property.

Paolo MercorelliUniversity of Applied Sciences Wolfsburgp.mercorelli@fh-wolfsburg.de

CP15

A Sinkhorn-Knopp Fixed Point Problem

We consider the fixedpoint problem �x = (AT (A�x)(−1) )(−1) (where (−1) is theentry-wise inverse), which arises from the Sinkhorn-KnoppAlgorithm for transforming a positive matrix into a uniquedoubly stochastic matrix. We discuss basic properties ofsolutions, including for non-positive and complex matrices,and we consider some special cases, including some inter-esting results involving patterned matrices. This work wasperformed in part by four undergraduates as part of anNSF-sponsored undergraduate research project.

David M. StrongDepartment of MathematicsPepperdine UniversityDavid.Strong@pepperdine.edu

CP15

Some Properties of the K and (s+1) - Potent Ma-trices

The situations where a square matrix A coincides with anyof its powers have been studied as well as another class ofmatrices which involves an permutation matrix. So, theconcept of idempotency can be generalized. This conceptis widely used in several applications. We introduce a newkind of matrices called K and (s+1)-potent matrices as anextension of both situations mentioned above. A study ofthese matrices is developed with some examples.

Leila LebtahiUniversidad Politecnica de Valencia, SpainDepartamento de Matematica Aplicadanjthome@mat.upv.es

Nestor ThomeInstituto de Matematica MultidisciplinarUniversidad Politecnica de Valencia, Valencia, Spainnjthome@mat.upv.es

CP15

On Efficient Numerical Approximation of the Scat-tering Amplitude

This talk presents results on efficient and numerically well-behaved estimation of the scalar value cΛx, where cΛ de-notes the conjugate transpose of c and x solves the lin-ear system Ax = b, A 2 CNN is a nonsingular complexmatrix and b and c are complex vectors of length N. Inother words, we wish to estimate the scattering amplitudecΛA-1b . In our understanding, various approaches for nu-merical approximation of the scattering amplitude can beviewed as applications of the general mathematical con-cept of matching moments model reduction, formulatedand used in applied mathematics by Vorobyev in his re-markable book [3]. Using the Vorobyev moment problem,matching moments properties of Krylov subspace methodscan be described in a very natural and straightforward way,see [1]. This talk further develops the ideas from [1] intoefficient estimates of cΛA-1b, see [2]. We briefly outline

the matching moment property of the Lanczos, CG andArnoldi methods, and specify techniques for estimation ofcΛA-1b with A non- Hermitian, including a new algorithmbased on the BiCG method. We show its mathematicalequivalence to the existing estimates which use a complexgeneralization of Gauss quadrature, and discuss its numer-ical properties. The proposed estimate will be comparedwith existing approaches using analytic arguments and nu-merical experiments on a practically important problemthat arises from the computation of diffraction of light onmedia with periodic structure.

Petr TichyCzech Academy of Sciencestichy@cs.cas.cz

Zdenek StrakosAcademy of Sciences of theCzech Republic, Praguestrakos@cs.cas.cz

CP15

Jordan Forms For Two Matrices Which Commute

This article gives a method of obtaining the Jordan form oftwo commuting matrices R and A when R is diagonalizable.An nxn complex matrix R with the property that a positivepower of R gives the identity is examined and examples aregiven to demonstrate the methods developed in the paper.

James R. WeaverUniversity of West Floridajweaver@uwf.edu

CP16

Matching Figures with Sentences in Technical Ab-stracts

Figures are key content in technical literature, so the abil-ity to quickly access images relevant to a sentence in adocument’s abstract would provide a valuable resource toscientists. We use two main ideas to realize this goal: amixture language model, which can be viewed as a non-negative matrix factorization, and a Markov model. Weintroduce a utility metric and report results of applyingboth models to 100 documents from the biomedical litera-ture.

John M. ConroyIDA Center for Computing Sciencesconroyjohnm@gmail.com

Joseph BockhorstUniversity of Wisconsin - MilwaukeeDept. of Electrical Eng. and Comp. Sci.joebock@uwm.edu

Dianne P. O’LearyUniversity of Maryland, College ParkDepartment of Computer Scienceoleary@cs.umd.edu

Hong YuUniversity of Wisconsin - MilwaukeeDepts. of Health Sciences and Computer Sci.hongyubennett@gmail.com

52 LA09 Abstracts

CP16

Perturbation Theory for the LDU Factorization ofDiagonally Dominant Matrices and Its Applicationto Accurate Computation of Singular Values

We present a new structured perturbation theory for theLDU factorization of diagonally dominant matrices thatallows us to prove that the relative errors in the factorscomputed by using an algorithm developed by Q. Ye in2008 are bounded by O(n4ε), where n is the dimension ofthe matrix and ε the machine precision. The previous er-ror bounds proved for this algorithm were O(8nε), that areuseless for matrices with dimension as small as n = 20 indouble precision. This new error bound proves rigurouslythat the computed LDU factorization can be used to com-pute accurately the SVD of this type of matrices.

Froilan M. DopicoDepartment of MathematicsUniversidad Carlos III de Madriddopico@math.uc3m.es

Plamen KoevSan Jose State Universitykoev@math.sjsu.edu

CP16

PageRank As a Function of a Matrix Instead of aScalar

The PageRank vector for a stochastic matrix is a rationalfunction of a scalar parameter. Replacing the scalar witha matrix yields a PageRank function of a matrix parame-ter. The trivial relationship between the Markov chain andlinear system for PageRank then disappears. We thereforeinvestigate a few other analogies. These ideas are prelimi-nary and one goal is to solicit feedback on other potentialrelationships. We show a few numerical examples.

David F. GleichStanforddgleich@stanford.edu

Paul ConstantineStanford UniversityInstitute for Computational and MathematicalEngineeringpaul.constantine@stanford.edu

CP16

Models for Beta-Wishart Random Matrices

We present our recent progress on developing empiricalmodels for Beta-Wishart random matrices. These mod-els bridge the gap between the classical real, complex, andquaternion Wishart random matrices and provide a contin-uous treatment of the theory for any positive beta. Thiswork generalizes the identity-covariance (Beta-Laguerre)models of Dumitriu and Edelman to arbitrary covariance.

Plamen KoevSan Jose State Universitykoev@math.sjsu.edu

CP16

A Simplified Newton’s Method for a Rational Ma-

trix Problem

Motivated by the classical Newton-Schulz method for find-ing the inverse of a nonsingular matrix, we develop a newinversion-free method for obtaining the minimal Hermitianpositive semidefitive solution X− of the matrix rationalequation X + A∗X−1A = I, where I is the identity matrixand A is a given matrix. This equation appears in manyapplied areas including control theory, dynamical program-ming, ladder networks, stochastic filtering and statistics.We present convergence results and discuss stability prop-erties when the method starts from the available matrixAA∗. We also present numerical results to compare ourproposal with some recently developed inversion-free tech-niques for solving the same rational matrix equation.

Marlliny MonsalveDepartamento de Computaci’{o}n. Facultad de CienciasUniversidad Central de Venezuelamarlliny,monsalve@ciens.ucv.ve

Marcos RaydanDepartamento de Calculo Cientifico y Estadistica.Universidad Simon Bolivar. Caracas-Venezuelamarcos.raydan@ciens.ucv.ve

CP16

Markov Chain Small-World Model with Asymmet-ric Transition Probabilities

Motivated by existing results concerning the case of sym-metric transition probabilities, we study the effects of avarying degree of asymmetry in transition probabilities onthe behavior of the Markov chain small-world model ofD. Higham. Asymptotic results in this regard are devel-oped following a matrix-theoretic approach. An interest-ing outcome is that the behavior of the model is stronglyinfluenced by the intensity of asymmetry, which may findapplications in real-world networks involving imbalancedinteractions.

Jianhong XuSouthern Illinois University CarbondaleDepartment of Mathematicsjxu@math.siu.edu

CP17

A Numerical Method for Quadratic ConstrainedMaximization

The problem of maximizing a quadratic function subjectto an ellipsoidal constraint is considered. The algorithmproduces a nearly optimal solution in a finite number of it-erations. In particular, the method can be used to solve theill-conditioned problems in which the solution consists oftwo parts from two orthogonal subspaces. Without restric-tive assumptions, the solution generated by the methodsatisfies the first and second order necessary conditions fora maximizer of the objective function. Numerical experi-ments clearly demonstrate that the method is successful.

Alireza Esna AshariINRIA- Rocquencourtalireza.esna ashari@inria.fr

Ramine NikoukhahINRIA - Rocquencourtramine.nikoukhah@inria.fr

LA09 Abstracts 53

Stephen L. CampbellNorth Carolina State UnivDepartment of Mathematicsslc@math.ncsu.edu

CP17

Joint Spectral Characteristics of Matrices: a ConicProgramming Approach

The joint spectral radius and subradius are a research topicof growing impact in hybrid systems, combinatorics, nu-merical analysis, etc. We propose a new approach to com-pute them. To our knowledge this constitutes the firstmethod to compute the joint spectral subradius. We showhow previous results can be interpreted as particular casesof ours. We show the efficiency of our algorithms by ap-plying them to several problems of combinatorics, numbertheory and discrete mathematics.

Raphael JungersMassachusetts Institute of TechnologyLaboratory for Information and Decision Systemsraphael.jungers@uclouvain.be

Vladimir Y. ProtasovMoscow State UniversityDepartment of Mechanics and Mathematicsv-protassov@yandex.ru

Vincent BlondelUniversite catholique de Louvain.vincent.blondel@uclouvain.be

CP17

Minimization of the Kohn-Sham Energy with a Lo-calized, Projected Search Direction

With the constantly increasing power of computers, therealm of experimental chemistry is increasingly beingbrought into the field of computational mathematics. Thiswork integrates non-linear programming methods with aprojected search direction to directly minimize the Kohn-Sham energy. By ensuring that the search direction re-mains sparse, we guarantee that the computationally-intensive energy function is only evaluated at sparse it-erates. Maintaing sparsity, while avoiding local minima,is a critical first-step in developing a fully linear-scalingalgorithm to minimize the Kohn-Sham energy.

Marc MillstoneCourant Institute of Mathematical SciencesNew York Universitymarcm@courant.nyu.edu

Juan C. MezaLawrence Berkeley National Laboratorymeza@hpcrd.lbl.gov

Michael L. OvertonNew York UniversityCourant Instit. of Math. Sci.overton@cs.nyu.edu

Chao YangLawrence Berkeley National LabCYang@lbl.gov

CP17

Variable Projection Methods for Separable Nonlin-ear Least Squares Learning

We describe variable projection methods (VPM) for neural-network learning. We explain why classical VPM may notwork excellently, and describe how to modify VPM to in-clude the second-order information of the residual Hessianmatrix unlike the standard VPM that uses the Levenberg-Marquardt Hessian. The modification, supported by effi-cient stage-wise evaluation of the Hessian that requires es-sentially no more computation than for the Gauss-NewtonHessian, allows us to exploit the negative curvature if itexists.

Eiji MizutaniDept. of Industrial Management, National TaiwanUniversityof Science and Technologyeiji@mail.ntust.edu.tw

James W. DemmelUniversity of CaliforniaDivision of Computer Sciencedemmel@cs.berkeley.edu

CP17

Solving Large-Scale Eigenvalue Problems from Op-timization

We describe parameterized eigenvalue problems arising inmethods for large-scale trust-region subproblems and regu-larization. We present comparisons of matrix-free methodsfor solving the eigenproblems.

Marielba RojasTechnical University of Denmarkmr@imm.dtu.dk

CP17

Preconditioning for PDE-Constrained Optimiza-tion

Advances in algorithms and hardware have enabled moreresearch on the optimization of functions with constraintsgiven by partial differential equations. Problems of thistype arise in a variety of applications and pose significantchallenges to optimization algorithms and numerical meth-ods. In this talk we present preconditioners for problemsin a general setup and also when additional box constraintsare introduced for the control. These constraints add anextra layer of complexity to the optimization method andthe efficient solution of the linear system is very important.Block-diagonal preconditioners present one possibility tosolve the arising saddle point problems efficiently. In addi-tion, we discuss the use of block-triangular preconditionersthat can be used in a non-standard inner product iterativemethod. We show that the drawbacks of this method canbe easily overcome by choosing appropriate precondition-ers for the blocks of the saddle point system. We presentan eigenvalue analysis for the preconditioners and illustratetheir competitiveness on some examples.

Martin StollUniversity of Oxfordmartin.stoll80@gmail.com

Tyrone ReesOxford University

54 LA09 Abstracts

tyrone.rees@comlab.ox.ac.uk

Andy WathenOxford UniversityComputing Laboratoryandy.wathen@comlab.ox.ac.uk

CP18

Generalized Laplacians and First Transit Times forDirected Graphs

We extend results on average commute-times for undi-rected graphs to fully-connected directed graphs, corre-sponding to irreducible Markov chains. We show how thepseudo-inverse of an unsymmetrized generalized Laplacianmatrix yields the first-transit times and commute timeswith formulas, almost matching the undirected graph case.The matrices are positive semi-definite, leading to a naturalembedding in Euclidean space preserving commute times.These results are illustrated with an application to an anal-ysis of ad-hoc asymmetric wireless networks.

Daniel L. BoleyUniversity of MinnesotaDepartment of Computer Scienceboley@cs.umn.edu

Gyan Ranjan, Zhi-Li ZhangUniversity of MinnesotaComputer Sciencegranjan@cs.umn.edu, zhzhang@cs.umn.edu

CP18

Graph Laplacians and Logarithmic Forest Dis-tances

A new family of distances for graph vertices is proposed.They are a kind of logarithmically transformed forest dis-tances. They reduce to the shortest path distance and tothe resistance distance at the extreme values of the familyparameter. Additionally, they satisfy bottleneck additivity :d(i, j) + d(j, k) = d(i, k) if and only if every path from ito k contains j. The construction of the distances is basedon the matrix forest theorem and the graph bottleneck in-equality.

Pavel ChebotarevInstitute of Control Sciencesof the Russian Academy of Sciencespavel4e@gmail.com

CP18

Parallel Unsymmetric Nested Dissection Orderingfor Sparse Direct Solvers

Fill-reducing ordering algorithms for unsymmetric prob-lems are often inherently sequential (e.g., COLAMD). Forlarge problems, a common approach is to apply nested dis-section on the symmetrized matrix. The HUND algorithm,based on hypergraph partitioning, was recently proposedto overcome these drawbacks. We present the first parallelimplementation of the HUND algorithm, soon available inthe Zoltan toolkit. We show preliminary results for someapplication matrices.

Erik G. BomanSandia National Labs, NMScalable Algorithms Dept.

egboman@sandia.gov

Cedric ChevalierSandia National LaboratoriesScalable Algorithms Departmentccheval@sandia.gov

CP18

Multivariable Toeplitz Matrices

We are interested in formulas for inverse of positive definitemultivariable Toeplitz matrices T = (tk−l)k,l∈Λ with Λ ⊂Nd. Attempts to provide a direct generalization of theGohberg-Semencul formula have failed so far. In this paperwe use a multivariable Gohberg-Semencul type expressionwhich serves as an upper bound for the inverse.

Selcuk Koyuncu, Hugo WoerdemanDrexel Universitysk476@drexel.edu, hugo @ math.drexel.edu

CP18

Structured Distance to Normality in the BandedToeplitz Case

In this talk we are concerned with the structured and un-structured distances to normality of banded Toeplitz ma-trices in the Frobenius norm. We present formulas for thedistance of banded Toeplitz matrices of suitable restrictedbandwidth to the algebraic variety of similarly structurednormal matrices. We then investigate the distance to gen-eralized circularity, formulating a conjecture on the dis-tance to the variety of normal matrices. The talk is basedon joint work with Lothar Reichel.

Silvia NoscheseDipartimento di Matematica ”Guido Castelnuovo”SAPIENZA Universita‘ di Romanoschese@mat.uniroma1.it

CP18

Factorization of Block Toeplitz Matrices. Applica-tions to Wavelets and Functional Equations.

Block (or slanted) Toeplitz matrices have found a lot of ap-plications in functional equations, wavelets theory, proba-bility, combinatorics, etc. In the simplest case, 2-blockToeplitz matrices are N ×N -matrices (Ts)ij = p2i−j+s−1,s = 0, 1, and i, j ∈ {1, . . . , N} for a sequence of complex co-efficients p0, . . . , pN . We obtain a complete spectral factor-ization of those matrices depending on the sequence pk, de-scribe the structure of their kernels, root subspaces, and allcommon invariant subspaces. These results allow us to sim-plify a formula for the exponent of regularity of wavelets, toobtain a factorization theorem for refinable functions, andto characterize the manifold of smooth refinable functions.This also solves the problem of continuity of solutions ofthe refinement equations with respect to their coefficientsand gives a criterion of convergence of the correspondingcascade algorithms and subdivision schemes.

Vladimir Y. ProtasovMoscow State UniversityDepartment of Mechanics and Mathematicsv-protassov@yandex.ru

LA09 Abstracts 55

MS1

Title Not Available at Time of Publication

Abstract not available at time of publication.

Ernesto EstradaUniversity of Strathclydeee@maths.strath.ac.uk

MS1

Rational Approximation to Trigonometric Opera-tors

We will discuss the approximation of trigonometric opera-tor functions that arise in the numerical solution of waveequations by trigonometric integrators. It is well knownthat Krylov subspace methods for matrix functions with-out exponential decay show superlinear convergence be-havior if the number of steps is larger than the norm ofthe operator. Thus, Krylov approximations may fail toconverge for unbounded operators. In this talk, a rationalKrylov subspace method is proposed which converges notonly for finite element or finite difference approximationsto differential operators but even for abstract, unboundedoperators. In contrast to standard Krylov methods, theconvergence will be independent of the norm of the oper-ator and thus of its spatial discretization. We will discussefficient implementations for finite element discretizationsand illustrate our analysis with numerical experiments.

Volker GrimmFriedrich-Alexander Universitaet Erlangen-Nuernberggrimm@am.uni-duesseldorf.de

Marlis HochbruckHeinrich-Heine-University Duesseldorfmarlis@am.uni-duesseldorf.de

MS1

A New Scaling and Squaring Algorithm for the Ma-trix Exponential

The scaling and squaring method for the matrix exponen-tial is based on the approximation eA ≈ (rm(2−sA))2

s

,where rm(x) is the [m/m] Pade approximant to ex and theintegers m and s are to be chosen. Several authors haveidentified a weakness of existing scaling and squaring algo-rithms termed overscaling, in which a value of s much largerthan necessary is chosen, causing a loss of accuracy in float-ing point arithmetic. Building on the scaling and squaringalgorithm of Higham [SIAM J. Matrix Anal. Appl., 26(4):1179–1193, 2005], which is used by MATLAB’s expm,we derive a new algorithm that alleviates the overscalingproblem.

Nicholas J. HighamUniversity of ManchesterSchool of Mathematicshigham@ma.man.ac.uk

Awad H. Al-MohySchool of MathematicsThe University of Manchesteralmohy@maths.manchester.ac.uk

MS1

Computing the Phi-function of a Matrix

The phi-functions are certain functions related to the expo-nential: the zeroth phi-function is the exponential itself andthe first one is defined by φ1(x) = (ex − 1)/x. They playan important role in exponential integrators. We presentan algorithm to evaluate the action of φk(A) on a vector.The key ingredients in the algorithm are a Krylov-subspacemethod to reduce the size of the matrix and a time-steppingmethod akin to the scaling-and-squaring method for ma-trix exponentials.

Jitse NiesenUniversity of Leedsjitse@maths.leeds.ac.uk

Will M. WrightLa Trobe Universityw.wright@latrobe.edu.au

MS2

Ritz Value Localization and Convergence of Non-Hermitian Eigensolvers

Rayleigh-Ritz eigenvalue estimates for Hermitian matricesobey the well-known interlacing property, which plays acrucial role in explaining convergence of the restarted Lanc-zos method with exact shifts. Much less is understoodabout Ritz values of non-Hermitian matrices, and conse-quently a satisfactory convergence theory for the restartedArnoldi method has proved elusive. We shall describerecent progress on the localization of Ritz values, whichprovide sufficient (though strict) conditions for restartedArnoldi convergence.

Russell Carden, Mark EmbreeComputational and Applied MathematicsRice Universityrussell.l.carden@rice.edu, embree@rice.edu

MS2

Low Rank Tensor Approximation

We present a black-box type algorithm for the approxima-tion of tensors in high dimension d. The algorithm deter-mines adaptively the positions of entries of the tensor thathave to be computed or read, and using these few entriesit constructs a low rank tensor approximation. For effi-ciency reasons the positions are located on fibre-crosses.The minimization problem is solved by Newton’s methodwhich requires the Hessian. It can be evaluated in data-sparse form in O(d).

Lars GrasedyckMax-Planck-InstituteMath in Scienceslgr@mis.mpg.de

MS2

The Concept of Augmented Backward Stability forSome Iterative Algorithms, with Emphasis on theLanczos Process

Jim Wilkinson developed the ideas of backward stable nu-merical algorithms. Some crucial algorithms are clearlynot backward stable, but provide useful computed results.Several of these are based on orthogonality, but fail miser-

56 LA09 Abstracts

ably to provide this orthogonality. However there exists atruly orthogonal higher dimensional matrix for each com-puted one. By using this fact, some of these algorithmsare shown to give correct answers for “nearby’ higher di-mensional problems. We call this “augmented backwardstability’.

Christopher C. PaigeMcGill University, Montrealchris@cs.mcgill.ca

MS2

Tensor Network Computations and Analysis

A tensor network is a way to build high-order tensorsthrough structured contractions of many low-order tensors.The manipulation of such objects can lead to sequences ofQR factorizations and SVDs. Product versions of thesewell known matrix tools turn out to be important for nu-merical reasons.

Charles Van LoanCornell UniversityDepartment of Computer Sciencecv@cs.cornell.edu

MS3

Algebraic Multigrid

In this talk, we will give an overview of the algebraic multi-grid (AMG) method and some of the research issues asso-ciated with it. The talk will primarily serve as backgroundand motivation for the other presentations in the minisym-posium, filling in on topic areas that will not otherwise becovered. In particular, we will discuss in more detail de-velopments in AMG theory and on coarsening techniquesbased on compatible relaxation.

Robert FalgoutCenter for Applied Scientific ComputingLawrence Livermore National Laboratoryrfalgout@llnl.gov

MS3

Bootstrap AMG in Lattice QCD

Operators arising in lattice quantum-chromodynamics(LQCD) simulations pose various challenges to currentsolvers. For physically interesting configurations the per-formance of Krylov-subspace methods deteriorates quicklywhich fueled the search for preconditioners. Due to the na-ture of the background SU(n) gauge field classical algebraicmultigrid (AMG) fails that task, but by introduction ofadaptive techniques and generalization of AMG principleswe were able to overcome some of the immanent challengeswith our Bootstrap approach.

Karsten KahlUniversity of WuppertalDepartment of Mathematicskkahl@math.uni-wuppertal.de

Achi BrandtUCLAabrandt@math.ucla.edu

James BrannickDepartment of Mathematics

Pennsylvania State Universitybrannick@psu.edu

Ira LivshitsBall State Universityilivshits@bsu.edu

MS3

Towards Nonsymmetric Adaptive Smoothed Ag-gregation for Systems of Pdes

Smoothed aggregation (SA) applied to nonsymmetric lin-ear systems, Ax = b, lacks a minimization principle incoarse-grid correction. Consider applying SA to symmet-ric positive definite (SPD) matrices,

√AtA or

√AAt, for

which minimization principles exist. It is not computation-ally efficient to use either matrix directly in coarse-grid cor-rection as they are typically full. The proposed approachefficiently approximates these corrections using SA adap-tively to approximate right and left singular vectors of A.Linear systems arising from discretizations of nonsymmet-ric systems of PDEs are considered.

Geoffrey D. SandersUniversity of Colorado, Department of AppliedMathematicsgeoffrey.sanders@Colorado.edu

Marian BrezinaU. of Colorado, BoulderApplied Math Departmentmarian.brezina@colorado.edu

Tom ManteuffelUniversity of Colorado, Department of AppliedMathematicstmanteuf@colorado.edu

Steve McCormickDepartment of Applied MathCU-Boulderstevem@colorado.edu

John RugeUniversity of Coloradojruge@colorado.edu

MS3

Algebraic Multigrid for Massively Parallel Multi-Core Architectures

Algebraic multigrid has shown to be extremely efficient ondistributed-memory architectures. However, the new gen-eration of high-performance computers with an increasingnumber of nodes, which include multiple cores sharing thebus and several caches and an increased core-to-main mem-ory performance gap, presents new challenges. Algorithmsneed to have good data locality, few synchronization con-flicts, and increased small grain parallelism while at thesame time improving scalability to deal effectively withmillions of cores. This presentation discusses the designand implementation of algebraic multigrid on such archi-tectures.

Ulrike YangLawrence Livermore National Laboratoryumyang@llnl.gov

LA09 Abstracts 57

MS4

Computing the Distance to Bounded Realness viathe Solution of Structured Eigenvalue Problems

We study control systems x = Ax + Bu, y = Cx + Duthat arise via discretization and model reduction in elec-tric and magnetic field computation. While the originalsystem is typically strictly passive (in simple words it doesnot generate energy) the approximate system often haslost its passivity. The system is then usually made pas-sive by small perturbations to the system matrices. Thecomputation of small or smallest perturbations that per-form this task leads to the task of finding smallest pertur-bations of Hamiltonian matrices that move eigenvalues offthe imaginary axis. We give explicit formulas for the min-imal perturbations to move purely imaginary eigenvaluesof different sign characteristics together and also to movemultiple purely imaginary eigenvalues to specific values inthe complex plane. We also present a numerical methodto compute upper bounds for the minimal perturbationsand discuss how to employ these ideas in the context ofpassivation.

Rafikul AlamAssistant Professor, Department of Mathematics,Indian Institute of Technology Guwahati, INDIArafik@iitg.ernet.in

Shreemayee BoraDepartment of Mathematics, IIT Guwahati, Indiashbora@iitg.ernet.in

Michael KarowTechnical University of BerlinGermanykarow@math.tu-berlin.de

Volker MehrmannInst. f.Mathematik, TU Berlinmehrmann@math.tu-berlin.de

Julio MoroUniversidad Carlos III de MadridDepartamento de Matematicasjmoro@math.uc3m.es

MS4

Structured Distance to Normality and EigenvalueSensitivity in the Banded Toeplitz Case

Matrix nearness problems have been the focus of much re-search in linear algebra. In particular, characterizationsof the algebraic variety of normal matrices and distancemeasures to this variety have received considerable atten-tion. Banded Toeplitz matrices arise in many applicationsin signal processing, time-series analysis, and numericalmethods for the solution of partial differential equations.In this talk we first describe spectral properties of nor-mal (2k + 1)-banded Toeplitz matrices of order n, withk ≤ n/2�. We then present formulas for the distanceof (2k + 1)-banded Toeplitz matrices to the algebraic va-riety of structured normal matrices of same bandwidth.We focus on the perturbation analysis in connection withthe distance to normality, devoting special attention to thesensitivity of the spectrum to structure-preserving pertur-bations in the tridiagonal Toeplitz case, and describing aninteresting example involving a Jordan block. Finally, wereport on an application regarding the determination of aset containing the field of values of a (2k + 1)-banded ma-

trix. The talk is based on joint work with Lionello Pasquiniand Lothar Reichel.

Silvia NoscheseDipartimento di Matematica ”Guido Castelnuovo”SAPIENZA Universita‘ di Romanoschese@mat.uniroma1.it

MS4

Structured Quadratic Matrix Polynomials: Sol-vents and Inverse Problems

A monic Hermitian matrix polynomial Q(λ) of degree 2 canbe factorized into a product of two linear matrix polynomi-als, say Q(λ) = (Iλ−S)(Iλ−A). For the inverse problem offinding a quadratic matrix polynomial with given spectraldata it is natural to prescribe a right divisor A and thendetermine compatible left divisors S. In applications, Q(λ)may be Hermitian, real symmetric, hyperbolic, elliptic, orhave mixed real/non-real spectrum, and these structuresadd extra constraints to the inverse problem. These issuesare explored in this talk.

Francoise TisseurUniversity of ManchesterDepartment of Mathematicsftisseur@maths.manchester.ac.uk

MS4

Computing the Hamiltonian Schur Form by Blocks

We discuss block methods for computing the HamiltonianSchur form of a Hamiltonian matrix. By placing each clus-ter of eigenvalues together in a block, we obtain a more ro-bust algorithm. This is joint work with Christian Schroederand Volker Mehrmann.

David S. WatkinsWashington State UniversityDepartment of Mathematicswatkins@math.wsu.edu

MS5

High Performance Algorithms for Materials fromNanocrystals to Crystals

A challenging problem in materials physics is to developalgorithms that permit prediction of materials propertiesat disparate length scales. Most algorithms are specializedfor a periodic system such as a crystal, or a localized sys-tem such as a nanocrystal. I will discuss new algorithms tohandle both regimes. Our algorithms are based on imple-menting the Kohn-Sham problem in real space and solvingthe corresponding eigenvalue problem without explicit re-sort to standard diagonalization methods.

Jim ChelikowskyUniversity of Texas at Austinjrc@ices.utexas.edu

MS5

The Quantum Monte Carlo Method and Linear Al-gebra Problems

In the Quantum Monte Carlo method we must evaluatedeterminant ratios for successive matrices in a very longsequence. One approach is to solve a sequence of linearsystems, another to solve generalized eigenvalue problems,

58 LA09 Abstracts

and yet another to approximate the determinants or ratiosdirectly estimating bilinear forms of matrix functions. Wewill give a brief overview and then focus on linear solvers,particularly on a multilevel preconditioner that can be up-dated very cheaply.

Eric De SturlerVirginia Techsturler@vt.edu

MS5

Electronic Structure: Basic Theory and PracticalMethods

Recent advances in quantitative theory of electronic prop-erties of materials have occurred through the combinationof new theoretical developments, clever algorithms, and in-creased computational power. This talk will focus on as-pects of the basic theory, with emphasis upon the struc-ture of the methods. In particular, I will describe currentwork to combine density functional theory with many-bodymethods in order to create robust methods to treat broadclasses of materials.

Richard M. MartinDepartment of Applied PhysicsStanford Universityrmartin@uiuc.edu

MS5

On the Convergence of the Self Consistent FieldIteration for a Class of Nonlinear Eigenvalue Prob-lems

Methods based on Density Functional Theory (DFT) re-quire solving a nonlinear eigenvalue problem derived fromthe Schrodinger equation. At the heart of these codes,one typically finds a Self Consistent Field (SCF) iteration,which is the predominant method for solving this under-lying nonlinear eigenvalue problem. Despite its popularityhowever, there are few convergence results available. I willdiscuss some new and interesting results showing the con-vergence behavior for the SCF iteration.

Juan C. MezaLawrence Berkeley National Laboratorymeza@hpcrd.lbl.gov

Chao YangLawrence Berkeley National LabCYang@lbl.gov

Weiguo GaoFudan University, Chinawggao@fudan.edu.cn

MS6

CPU-GPU Hybrid Eigensolvers for SymmetricEigenproblems

We examine the use of CPU-GPU hybrid systems to ac-celerate a few symmetric eigensolvers. We investigate thebisection and inverse iteration methods for tridiagonal ma-trices by performing redundant computation, optimizingfor higher resource utilization and reduced memory traf-fic. We then examine a Lanczos method for a particu-lar spectral partitioning problem, finding that using theCullum-Willoughby test without reorthogonalization leads

to a 180x speedup on a CPU-GPU system. Our approachesare generally applicable to hybrid systems with heteroge-neous parallel processors.

Bryan CatanzaroEECS DepartmentUniversity of California; Berkeleycatanzar@eecs.berkeley.edu

Vasily Volkov, Bor-Yiing Su, Narayanan SundaramEECS DepartmentUniversity of California, Berkeleyvolkov@eecs.berkeley.edu, subrian@eecs.berkeley.edu,narayans@eecs.berkeley.edu

Jim DemmelUC Berkeleydemmel@eecs.berkeley.edu

Kurt KeutzerEECS DepartmentUniversity of California, Berkeleykeutzer@eecs.berkeley.edu

MS6

Parallel Distributed Reduction to CondensedForms - Scheduling Sequences of Givens Rotations

Some algorithms for two-sided reduction of matrices andmatrix pairs to condensed forms generate sequences ofGivens rotations acting on rows or columns i and i+ 1 forcontiguous indices i. This raises some interesting schedul-ing problems when using 2D block-cyclic distribution of thematrices. We present static scheduling techniques basedon partial wavefronts for both rectangular and triangularmatrices. The techniques also apply to the application ofaccumulated Givens rotations using level 3 BLAS.

Lars KarlssonUmea UniversityComputing Science and HPC2Nlarsk@cs.umu.se

Bo T. KagstromUmea UniversityComputing Science and HPC2Nbokg@cs.umu.se

MS6

Iterative Refinement of Schur Decompositions forMixed Precision Computation

Assume that the Schur or spectral decomposition of a ma-trix A is available at a considerably low machine precisionbut the application demands for higher accuracy. Exam-ples for this setting of current interest include mixed sin-gle/double precision computations on Cell processors andGPUs as well as mixed double/quad precision on standardCPUs. In both cases, the operations performed in high pre-cision are significantly more costly and should therefore belimited to the bare minimum. This talk will discuss refine-ment procedures for entire decompositions that allow us inmany cases to have a highly accurate Schur decompositionessentially at the cost of the low precision computation.Particular attention will be paid to situations where someof the eigenvalues of A are highly ill-conditioned.

Daniel KressnerETH Zurich

LA09 Abstracts 59

daniel.kressner@sam.math.ethz.ch

Christian SchroderTU Berlinschroed@math.tu-berlin.de

MS6

A Novel Parallel QR Algorithm for Hybrid Dis-tributed Memory HPC Systems

Key techniques used in our novel parallel QR algorithm in-clude multi-window bulge chain chasing and distributed ag-gressive early deflation, which enable level 3 chasing oper-ations and improved eigenvalue convergence. Mixed MPI-OpenMP coding techniques are utilized for DM platformswith multithreaded nodes, such as multicore processors.Application and test benchmarks confirm the superb per-formance of our parallel QR algorithm. In comparison withthe existing ScaLAPACK code, the new parallel software isone to two orders of magnitude faster for sufficiently largeproblems.

Robert GranatDept. of Computing Science and HPC2NUmea University, Swedengranat@cs.umu.se

Bo T. KagstromUmea UniversityComputing Science and HPC2Nbokg@cs.umu.se

Daniel KressnerETH Zurichdaniel.kressner@sam.math.ethz.ch

MS6

Accelerating the Reduction to Upper HessenbergForm through Hybrid GPU-based Computing

We present a Hessenberg reduction algorithm for hybridmulticore+GPU systems that gets 16x performance im-provement over the current LAPACK algorithm runningjust on multicores (in double precision). This enormous ac-celeration is due to proper matching of algorithmic require-ments to architectural strengths of the hybrid components.The reduction is an important linear algebra problem, es-pecially with its relevance to eigenvalue solvers. The re-sults are significant because Hessenberg reduction has notyet been accelerated on multicore architectures.

Stanimire TomovInnovative Computing LaboratoryUniversity of Tennessee; Knoxvilletomov@eecs.utk.edu

Jack DongarraUniversity of Tennessee Knoxvilledongarra@eecs.utk.edu

MS7

Computing Enclosures for the Matrix Square Root

Given an approximation X to the square root of a matrixA, we present methods based on interval arithmetic whichcompute an entrywise enclosure for the ‘true’ square root.Our approach is based on a modification of Krawczyk’smethod which reduces the complexity from O(\�) to O(\�).

Numerical results using the Matlab toolbox Intlab will bereported.

Andreas J. FrommerBergische Universitaet WuppertalFachbereich Mathematik und Naturwissenschaftenfrommer@math.uni-wuppertal.de

Behnam HashemiAmirkabir University of Technology, Tehranhashemi am@aut.ac.ir

MS7

Green’s Function Calculations in Nano Research

The non-equilibrium Green’s function formalism providesa powerful conceptual and computational framework fortreating quantum transport in nanodevices. The Green’sfunction that we are interested in is a function of a bi-infinite block tridiagonal Hermitian matrix, and only onespecial diagonal block of the Green’s function is requiredin applications. Computing this special matrix has beena challenging problem for nano-scientists. In this talk wepresent a satisfactory solution to this problem.

Chun-Hua GuoUniversity of ReginaDept of Math & Statisticschguo@math.uregina.ca

Wen-Wei LinNational Chiao Tung Universitywwlin@math.nctu.edu.tw

MS7

Computing Matrix Means

The definition of geometric mean of two positive definitematrices is not straightforward. However, imposing someexpected properties leads to the following matrix

A(A−1B)1/2,

which is defined as the geometric mean of A and B anddenoted by A#B. For more than two matrices the situa-tion is more complicated. Filling a list of properties thata geometric mean should verify does not lead to a uniquematrix. In fact, there is no complete agreement on the def-inition of the geometric mean of more than two matrices.We briefly survey some applications and problems relatedto the matrix geometric means and present some numericalmethods to compute them.

Bruno IannazzoUniversita‘ di Perugia, Italybruno.iannazzo@dipmat.unipg.it

MS7

An Effective Matrix Geometric Mean Satisfyingthe Ando-Li-Mathias Properties

There are in literature several proposals to define a propergeneralization of the geometric mean to k ≥ 3 positive def-inite matrices. Ando, Li and Mathias listed ten propertiesthat a ”good” mean should fulfill, and proposed a definitionsatisfying them, based on the limit of an iteration process.We propose another definition, as the limit of a cubicallyconvergent iteration, having all the desired properties but

60 LA09 Abstracts

much easier to compute.

Dario Bini, Beatrice MeiniDept. of MathematicsUniversity of Pisa, Italybini@dm.unipi.it, meini@dm.unipi.it

Federico PoloniScuola Normale Superioref.poloni@sns.it

MS8

Missing Value Estimation Using Matrix Factoriza-tions

Many predictive modeling tasks in data analysis may beposed as missing value estimation problems. Recently, suchproblems have sparked enormous interest in the contextof recommender systems, specifically due to the Netflixmillion-dollar challenge problem. In this talk, I will sur-vey and discuss methods for such missing value estimationproblems that rely on low-rank matrix factorizations of in-complete data matrices.

Inderjit S. DhillonUniversity of Texas at Austininderjit@cs.utexas.edu

MS8

Krylov Subspace-Based Techniques for Order Re-duction of Large-Scale RCL Networks

The development of many of the most powerful algorithmsin linear algebra was driven by the need to solve challeng-ing problems arising in new application areas. In this talk,we describe how the need for order reduction of large-scaleRCL networks arising in VLSI circuit simulation led to thedevelopment of Krylov subspace-based order-reduction oflinear dynamical systems. In particular, we focus on re-cent advances in reduction techniques that preserve certainstructures of RCL networks.

Roland W. FreundUniversity of California, DavisDepartment of Mathematicsfreund@math.ucdavis.edu

MS8

Polynomial Approximation Problems and Conver-gence of Krylov Subspace Methods

The convergence of Krylov subspace methods for solvinglinear algebraic systems or eigenvalue problems can be de-scribed or at least bounded by the value of certain polyno-mial approximation problems involving the given matrix.Unfortunately, such problems are notoriously difficult toanalyze, particularly for nonnormal matrices. In this talkI will review some of the results in this area, and discussrecent work on the so-called Chebyshev polynomials of ma-trices.

Jorg LiesenInstitute of MathematicsTU Berlinliesen@mail.math.tu-berlin.de

MS8

Do We Understand Gaussian Elimination?

Despite the enormous progress in improving efficiency ofalgorithms based on Gaussian elimination for solving largesparse systems of linear equations, there is still a strongneed for analyzing and improvements of the resulting pro-cedures, in particular, in the field of incomplete decompo-sitions. This talk will review some results in this area anddiscuss the possibilities for further developments. A spe-cial attention will be given to the role of the matrix inversewhich is implicitly present in the considered algorithms.

Miroslav TumaAcademy of Sciences of the Czech RepublicInstitute of Computer Sciencetuma@cs.cas.cz

MS9

A Week in the Life of an Industrial Linear Alge-braist

We will present a survey of all of the linear algebra prob-lems that we have been working on during our years atLSTC including - KKT systems for nonlinear solution forimplicit mechanics - Block matrix methods including lowrank approximations for subblocks for applications in elec-tromagnetics and acoustics - Iterative methods for CFDand thermal mechanics - Consistent constraint explicitwhere we are applying the consistent constraint technol-ogy of implicit to the explicit formulation.

Cleve AshcraftLivermore Software TechnologyCorporationcleve@lstc.com

Roger GrimesLivermore Software Technology Corporationgrimes@lstc.com

MS9

The Role of Numerical Linear Algebra in Large-Scale Electromagnetic Modeling

We will present a survey of all of the linear algebra prob-lems that we have been working on during our years atLSTC including KKT systems for nonlinear solution forimplicit mechanics, block matrix methods including lowrank approximations for subblocks for applications in elec-tromagnetics and acoustics, iterative methods for CFDand thermal mechanics, and consistent constraint explicitwhere we are applying the consistent constraint technologyof implicit to the explicit formulation.

Lie-Quan LeeSLAC National Accelerator Laboratoryliequan@slac.stanford.edu

MS9

Role of Numerical Linear Algebra in DOE

The U.S. Department of Energy is the owner of one of thelargest research enterprises in the United States. It hasmore than ten research laboratories scattered throughoutthe country. These laboratories are well known for theirwork in physical and biological sciences. Numerical linearalgebra plays a crucial role in many of the scientific appli-

LA09 Abstracts 61

cations. In this talk the speaker will sample some of theseapplications and highlight the impact of numerical linearalgebra.

Esmond G. NgLawrence Berkeley National LaboratoryLawrence Berkeley National Laboratoryegng@lbl.gov

MS9

Linear Algebra Meets the Long Tail

In this presentation we discuss large data problems for amarketplace like eBay. eBay is a Long tail market placewhere buyers and sellers come together to trade for, typ-ically, one-of-a-kind items. Given this long tail nature ofproducts for sale building applications like Search, Rec-ommender systems and Classification solutions pose hugechallenges. Data dimensions are represented by users,keywords, products, items, item properties, or categories.Most of these dimensions extend to tens to hundreds of mil-lions. Matrices representing combinations of these dimen-sions tend to be sparse and are candidates for dimension-ality reduction. Given the scale of the data these compu-tations need to be highly parallelized. Other sets of prob-lems include large volume text data short text (feedbackcomments) or long text (product description and productreviews). We will discuss the problem space and the chal-lenges and also approaches to address these challenges.

Neel SundaresaneBay Research Labstba

MS10

Maximum Entropy Principle for Composition Vec-tor Method in Phylogeny

The composition vector (CV) method is an alignment-freemethod for phylogeny. Since the phylogenetic signals inthe biological data are often obscured by noise and bias,denoising is necessary when using the CV method. Re-cently a number of denoising formulas have been proposedand found to be successful in phylogenetic analysis of bac-teria, viruses etc. By using the maximum entropy principlefor denoising and utilizing the structure of the constraintmatrix to simplify the optimization, we derive several newformulas. With these formulas, we obtain a phylogenetictree which identifies correct relationships between differ-ent gerera of Archaea strains in family Halobacteriaceaeby using the 16S rRNA dataset; and also a phylogenetictree which correctly groups birds and reptiles together, andthen Mammals and Amphibians successively by using the18S rRNA dataset.

Raymond ChanThe Chinese University of Hong Kongtba

MS10

Alternative Minimization Method with AutomaticSelection of Regularization Parameter for TotalVariation Image Restoration

Total variation (TV) based image restoration has been in-tensively studied recent years for its nice property of edgepreservation. In this talk, we present an alternative min-imization method with automatic selection of regulariza-tion parameter for TV based image restoration problem.

Under the framework of alternative minimization method,we solve each sub-problem and update the regularizationparameter iteratively. Experimental results show that theperformance of the proposed method is quite promising.

Haiyong LiaoHong Kong Baptist University, Hong Kong06459358@hkbu.edu.hk

Michael K. NgDepartment of Mathematics, Hong Kong BaptistUniversitymng@math.hkbu.edu.hk

MS10

A Fast Algorithm for Updating and Tracking theDominant Kernel Principal Components

Many important kernel methods in the machine learningarea, i.e., kernel Principal Component Analysis, featureapproximation, denoising, compression, prediction requirethe computation of the dominant set of eigenvectors of thesymmetrical kernel Gram matrix. Recently, an algorithmfor tracking the dominant kernel eigenspace has been pro-posed. In this talk,a new algorithm to incrementally com-pute the dominant kernel eigenbasis, allowing to track thekernel eigenspace dynamically, will be presented. This ap-proach exploits the properties of some structured matrices,like arrowhead, Cauchy and Householder ones.

Nicola MastronardiIstituto per le Applicazioni del CalcoloNational Research Council of Italy, BariN.Mastronardi@ba.iac.cnr.it

MS10

Tensor Decomposition Methods in HyperspectralData Analysis

Nonnegativity constraints on solutions, or approximate so-lutions, to numerical problems are pervasive throughoutcomputational science and engineering [?]. An importantresearch area in hyperspectral image data analysis is touse nonnegativity constraints to identify materials presentin the object or scene being imaged and to quantify theirabundance in the mixture. Since spectral unmixing is anill-posed inverse problem, the use of prior information andregularization constraints are essential tools. In this work,we develop analysis methods based on nonnegative ma-trix and tensor decompositions that focus on the follow-ing goals: material identification, material abundance esti-mation, data compression, segmentation, and compressedsensing. Applications include satellite and debris trackingand analysis for space activities [?].

Bob PlemmonsWake Forest UniversityDept. Mathematics, Box 7388plemmons@wfu.edu

MS11

Full Multigrid and Least Squares for Incompress-ible Resistive Magnetohydrodynamics

Magnetohydrodynamics (MHD) is a fluid theory that de-scribes Plasma Physics by treating the plasma as a fluid ofcharged particles. Hence, the equations that describe theplasma form a nonlinear system that couples Navier-Stokes

62 LA09 Abstracts

with Maxwells equations. This talk develops a nested-iteration-Newton-FOSLS-AMG approach to solve this typeof system. Most of the work is done on the coarse grid, in-cluding most of the linearizations. We show that at mostone Newton step and a few V-cycles are all that is neededon the finest grid. Here, we describe how the FOSLSmethod can be applied to incompressible resistive MHDand how it can be used to solve these MHD problems ef-ficiently in a full multigrid approach. An algorithm is de-veloped which uses the a posterior error estimates of theFOSLS formulation to determine how well the system isbeing solved and what needs to be done to get the mostaccuracy per computational cost. In addition, various as-pects of the algorithm are analyzed, including a timestep-ping analysis to confirm stability of the numerical schemeas well as the benefits of an efficiency based adaptive meshrefinement method. A 3D steady state and a reduced 2Dtime-dependent test problem are studied. The latter equa-tions can simulate a large aspect-ratio tokamak. The goalis to resolve as much physics from the test problems withthe least amount of computational work. This talk showsthat this is achieved in a few dozen work units. (A workunit equals one fine grid residual evaluation.)

James H. AdlerUniversity of Colorado at Boulderjames.adler@colorado.edu

MS11

Algebraic Multigrid for Linear Elasticity

We are interested in the efficient solution of large systemsof PDEs arising from elasticity applications. We proposeextending the AMG interpolation operator to exactly in-terpolate the rotational rigid body modes by adding addi-tional degrees of freedom at each node. Our approach isan unknown-based approach that builds upon any existingAMG interpolation strategy and requires nodal coarsening.The approach fits easily into the AMG framework and doesnot require any matrix inversions. We demonstrate the ef-fectiveness on 2D and 3D elasticity problems.

Allison H. BakerCenter For Applied Scientific ComputingLawrence Livermore National Laboratoryabaker@llnl.gov

Tzanio V. Kolev, Ulrike Meier YangCenter for Applied Scientific ComputingLawrence Livermore National Laboratorytzanio@llnl.gov, umyang@llnl.gov

MS11

AMG for Definite and Semi-definite Maxwell Prob-lems

Recently, there has been a significant interest in auxiliary-space methods for linear systems arising in electromagneticdiffusion simulations. Based on a novel stable decomposi-tion of the Nedelec finite element space due to Hiptmairand Xu, we implemented a scalable solver for second order(semi-)definite Maxwell problems, utilizing internal AMGV-cycles for scalar and vector Poisson-like matrices. Inthis talk we present the linear algebra theory, numericalperformance, and discuss some recent developments in thealgorithm. in its implementation.

Tzanio V. Kolev, Panayot S. VassilevskiCenter for Applied Scientific Computing

Lawrence Livermore National Laboratorytzanio@llnl.gov, panayot@llnl.gov

MS11

Local Fourier Analysis for Systems of PDEs

For many systems of PDEs, multigrid methods are amongstthe most efficient techniques for solving the resulting ma-trix systems. This efficiency results from achieving appro-priate complementarity in the smoothing and coarse-gridcorrection processes. In this talk, we discuss recent work inlocal Fourier analysis of multigrid methods for systems ofPDEs, including staggered discretizations and overlappingmultiplicative smoothers. The resulting tools are demon-strated for the Stokes, curl-curl, and grad-div systems.

Scott MaclachlanDepartment of MathematicsTufts Universityscott.maclachlan@tufts.edu

Kees OosterleeDelft University of Technologyc.w.oosterlee@cwi.nl

MS12

The Use of Sylvester Equation Solvers for the Com-putation of Eigenvalues of Integral Operators

Sylvester equations are important in mathematical mod-els of many areas of applied mathematics, such as con-trol theory, stochastic dynamic general equilibrium models,etc. Here we consider its application to the computationof spectral elements of integral operators when there areeigenvalues of multiplicity greater than 1. In this case, amethod for the computation of eigenpairs of Fredholm in-tegral operators in a Banach space, which combines defectcorrection with power iteration method, requiring the so-lution of an intermediate Sylvester equation, will be used.Different solvers for Sylvester equations will be compared.

Filomena DIAS D’ALMEIDAFac. Engenharia Univ. PortoCMUP - Centro Matematica Univ. Portofalmeida@fe.up.pt

Paulo B. VASCONCELOSFac. Economia Univ. PortoCMUP - Centro Matematica Univ. Portopjv@fep.up.pt

MS12

Triple dqds Algorithm: Eigenvalues of Real Un-symmetric Tridiagonals

The progressive quotient difference algorithm with shifts(qds) was presented by Rutishauser as early as 1954. It isequivalent to the shifted LR algorithm written in a specialnotation for tridiagonal matrices. The much more recentdifferential qds (dqds) is a sophisticated variant of qds.We will present the 3dqds algorithm which performes im-plicitly three dqds steps and such that real arithmetic ismaintained in the presence of complex eigenvalues. Wewill show some interesting numerical experiments.

Carla FERREIRAMathematics DepartmentUniversity of Minho

LA09 Abstracts 63

caferrei@math.uminho.pt

Beresford PARLETComputer Science DepartmentUniversity of California, Berkeleyparlett@math.berkeley.edu

MS12

A Framework for Analyzing Nonlinear Eigenprob-lems and Parametrized Linear Systems

Associated with an n × n matrix polynomial of degree ,

P (λ) =∑�

j=0λjAj , are the eigenvalue problem P (λ)x = 0

and the linear system problem P (ω)x = b, where in thelatter case x is to be computed for many values of the pa-rameter ω. Both problems can be solved by conversion toan equivalent problem L(λ)z = 0 or L(ω)z = c that is lin-ear in the parameter λ or ω. This linearization process hasreceived much attention in recent years for the eigenvalueproblem, but it is less well understood for the linear systemproblem. We develop a framework in which more generalversions of both problems can be analyzed, based on one-sided factorizations connecting a general nonlinear matrixfunction N(λ) to a simpler function M(λ), typically a poly-nomial of degree 1 or 2. Our analysis relates the solutionsof the original and linearized problems and in the linear sys-tem case indicates how to choose c and recover x from z.For the eigenvalue problem this framework includes manyspecial cases studied in the literature, including the vec-tor spaces of pencils L1(P ) and L2(P ) recently introducedby Mackey, Mackey, Mehl, and Mehrmann and a class ofrational problems. We use the framework to investigatethe conditioning and stability of the parametrized linearsystem P (ω)x = b and thereby study the effect of scaling,both of the original polynomial and of the pencil L. Ourresults identify situations in which scaling can potentiallygreatly improve the conditioning and stability and our nu-merical results show that dramatic improvements can beachieved in practice.

Laurence GRAMMONTUniversite de Saint-etienne,Laurence.Grammont@univ-st-etienne.fr

Francoise TisseurUniversity of ManchesterDepartment of Mathematicsftisseur@maths.manchester.ac.uk

Nicholas HIGHAMSchool of MathematicsThe University of Manchesterhigham@maths.manchester.ac.uk

MS12

Newton-like Methods for Schur and Gauss Trian-gular Forms

Ill conditioned eigenproblems may lead to poor numericalapproximations of eigenvalues, for example when the datamatrix is far from being a normal one or of eigenvectors, forexample when the data matrix has close eigenvalues. How-ever, these numerical approximations may serve as initialpoints of iterative schemes which are able to refine themup to a desired precision. The goal of this paper is to showhow the classical Newton-Kantorovich method for differ-entiable nonlinear problems and its Fixed Slope variantcan be adapted in real arithmetic to refine eigenelements

of a square complex matrix. Numerical evidence is pro-vided with famous matrices of the Gallery collection. Wepropose a nonlinear formulation of Schur and Gauss trian-gularizations in such a way that Newton-like methods canbe used to refine the initial approximations to these uppertriangular forms.

Alain LARGILLIERLaboratoire de MathematiquesUniversite de Saint-Etiennelarg@univ-st-etienne.fr

Mario AHUESLaboratoire de Mathematiques del’Universite de Saint-Etienne (EA3989)mario.ahues@univ-st-etienne.fr

Muzafar HAMALaboratoire de MathematiquesUniversite de Saint-Etiennehamamuzafar@yahoo.com

MS13

On the Early History of Matrix Iterations: TheItalian Contribution

In this talk I will review some early work on iterative meth-ods for solving linear systems by Italian mathematiciansduring the 1930s, with particular attention to the contri-butions of Lamberto Cesari (1910–1990) and GianfrancoCimmino (1908–1989). I will also provide some backgroundinformation on Italian applied mathematics and especiallyon Mauro Picone’s Istituto Nazionale per le Applicazionidel Calcolo, where most of this early numerical work tookplace. Finally, I will illustrate the influence of Cimmino’swork on modern and current research.

Michele Benzi

DepartmentofMathematics and Computer ScienceEmory Universitybenzi@mathcs.emory.edu

MS13

Cayley, Sylvester, and Early Matrix Theory

Last year marked the 150th anniversary of “A Memoir onthe Theory of Matrices’ by Arthur Cayley (1821–1895)—the first paper on matrix algebra. The subject was devel-oped by Cayley and his colleague James Joseph Sylvester(1814–1897), who coined the term “matrix’, along withmany others, in particular the Berlin school of Weiertrass,Kronecker, and Frobenius. I will give an overview of theseearly developments, leading up the time when matrices be-came an accepted tool in applied mathematics.

Nicholas J. HighamUniversity of ManchesterSchool of Mathematicshigham@ma.man.ac.uk

MS13

Hermann Grassmann and the Foundations of Lin-ear Algebra

Hermann Grassmann (1809-1877), one of the mathemat-ical geniuses of the 19th century, laid the foundations ofLinear Algebra with his “Calculus of Extension” of 1844.Being notoriously difficult to read and far ahead of its time,

64 LA09 Abstracts

Grassmann’s work was accepted only gradually by othermathematicians. On the occasion of Grassmann’s 200thbirthday, I will discuss his main mathematical achieve-ments as well as some of the influences that led to hiscreation of Linear Algebra.

Jorg LiesenInsitute of MathematicsTU Berlinliesen@math.tu-berlin.de

MS13

Kron(A,B): A Product of the Times

A quick study of Zehfuss (1858) and subsequent develop-ments suggests that the Kronecker product (KP) shouldreally be referred to as the ”Zehfuss product.” But nevermind, we will fast forward to the 1960s and later and tracehow this all-important matrix operation has worked its wayinto mainstream numerical linear algebra. Connectionsto fast transforms, PDE discretization, high-dimensionalmodeling, and tensor computations will be stressed. Usinghistory as a guide, we will argue that KP-based problem-solving will become increasing prominent as N approachesinfinity.

Charles Van LoanCornell UniversityDepartment of Computer Sciencecv@cs.cornell.edu

MS14

Lyapunov Equations, Energy Functionals, andModel Order Reduction

We give an overview on the use of algebraic Gramians in thecontext of model order reduction (MOR). Particularly, weemphasize their relation to energy functionals, Lyapunov(-type) equations, and balancing transformations. New re-sults on using these concepts for balanced truncation MORmethods for stochastic and bilinear systems will be pre-sented. Using Carleman linearization together with bal-anced truncation for bilinear systems, nonlinear reduced-order models for nonlinear systems can be computed. Nu-merical examples illustrate our results.

Peter BennerTU ChemnitzGermanybenner@mathematik.tu-chemnitz.de

Tobias DammTU KaiserslauternGermanydamm@mathematik.uni-kl.de

MS14

Model Reduction in Computational Flight Testing

Brute-force use of usual CFD-methods for numerical sim-ulation in the context of flight testing is not feasible dueto large numbers of different simulation settings and highsimulation cost per setting. Models of reduced order forthe calculation of stationary and instationary aerodynam-ical data of the airplane are desired, which are applicablefor a wide range of flight envelopes. We will report on firststeps towards the use of model reduction methods in this

context.

Heike FassbenderTU Braunschweig, GermanyInstitut Computational Mathematicsh.fassbender@tu-bs.de

MS14

Discrete Empirical Interpolation for NonlinearModel Reduction

A dimension reduction method called Discrete Empir-ical Interpolation (DEIM) is proposed and shown todramatically reduce the computational complexity ofthe Proper Orthogonal Decomposition (POD) methodfor constructing reduced-order models for unsteadyand/or parametrized nonlinear partial differential equa-tions (PDE). DEIM is a modification of POD that reducesthe complexity as well as the dimension of general nonlin-ear systems of ordinary differential equations. In particularit applies to nonlinear ODE arising from discretization ofPDE.

Danny C. Sorensen, Saifon ChaturantabutRice Universitysorensen@rice.edu, sc3@rice.edu

MS14

Approaches to Nonlinear Model Order Reduction

In the design of electrical circuits, simulation has alwaysbeen a crucial factor, e.g., for circuit and system level veri-fication and for capturing parasitic effects. Shrinking struc-ture sizes and increasing packing densities require refinedmodels, causing very high dimensional linear and nonlin-ear dynamical systems to be treated numerically. Mea-sures have to be taken to enable simulation in adequatetime. Model order reduction for linear and nonlinear prob-lems is one, industry is interested in. We give an overviewof techniques and present questions and approaches fromsemiconductor industry.

Michael StriebelTechnische Universitat Chemnitz, Germanymichael.striebel@mathematik.tu-chemitz.de

Joost RommesNXP Semiconductors, The Netherlandsjoost.rommes@nxp.com

MS15

Nonlinear Eigenvalue Problems in AcceleratorCavity Modeling

Computational design and optimization of accelerator cav-ities involves solving large-scale numerical eigenvalue prob-lems. The electromagnetic power loss due to external cou-pling in the cavities leads to nonlinear terms which dependon the eigenvalues. Due to the complexity of cavity geome-tries, the problem size of the systems is often very large.We will present the formulation and algorithms we em-ployed to solve those large-scale nonlinear eigenvalue prob-lems from accelerator cavity modeling.

Lie-Quan LeeSLAC National Accelerator Laboratoryliequan@slac.stanford.edu

LA09 Abstracts 65

MS15

The Computation of Purely Imaginary Eigenvalueswith Application to the Detection of Hopf Bifurca-tions in Large Scale Problems

The computation of purely imaginary eigenvalues with ap-plication to the detection of Hopf Bifurcations in large scaleproblems The determination of Hopf bifurcations of a dy-namical system is often a challenging problem. The goalis to compute the system parameter values for which thesystem has purely imaginary eigenvalues. In this talk, wepresent a method that computes these parameter valueswithout computing the imaginary eigenvalues themselves.The method is based on residual inverse iteration for aproblem of squared dimension.

Karl MeerbergenK. U. Leuvenkarl.meerbergen@cs.kuleuven.be

MS15

On the Stationary Schrodinger Equation Describ-ing Electronic States of Quantum Dots in a Mag-netic Field

In this work we derive a pair/quadruple of nonlinear eigen-value problems corresponding to the one–band effectiveHamiltonian governing the electronic states of a quantumdot in a homogenous magnetic field. Exploiting the min-max characterization of its eigenvalues we devise an effi-cient iterative projection methods simultaneously handlingthe pair/quadruple of nonlinear problems and thereby sav-ing about 25%/40-50% of the computation time compar-ing to the Nonlinear Arnoldi method applied to each of theproblems separately.

Heinrich VossHamburg Univ of TechnologyInstitute of Numerical Simulationvoss@tu-harburg.de

Marta M. BetckeUniv of ManchesterSchool of Mathematicsm.betcke@maths.manchester.ac.uk

MS15

A Parallel Additive Schwarz PreconditionedJacobi-Davidson Algorithm for Polynomial Eigen-value Problems in Quantum Dot Simulation

Abstract not available at time of publication.

Weichung WangNational Taiwan UniversityDepartment of Mathematicswwang@math.ntu.edu.tw

MS16

Eigenvalue Problems in Sensing and Imaging

Nonuniqueness arises from typical sensing and imagingproblems, and can be resolved by seeking sparse solution tounderdetermined linear (integral) equations, usually witha Fourier kernel. We connect the task of sparse solutionto the design of Gaussian quadratures and to a class ofeigenvalue problems with their computational issues.

Yu Chen

Courant Institute, NYUUSAyuchen@cims.nyu.edu

MS16

Minimizing Communication in Linear Algebra

A lower bound on communication (words of data moved)needed to perform dense matrix multiplication is known tobe Omega(f/sqrt(M)) where f is the number of multiplica-tions (n3) and M is the fast memory size. We extend thisresult to all direct methods of linear algebra (BLAS, LU,QR, eig, svd, etc), to sequential or parallel algorithms, andto dense or sparse matrices. We show large speedups overalgorithms in LAPACK and ScaLAPACK.

James DemmelUC Berkeley, USAdemmel@eecs.berkeley.edu

Grey BallardUniversity of California, Berkeleyballard@eecs.berkeley.edu

Olga HoltzUniversity of California, BerkeleyTechnische Universitat Berlinoholtz@eecs.berkeley.edu

Oded SchwartzTechnische Universitat Berlinodedsc@math.tu-berlin.de

MS16

Structured Matrix Computations: Recent Ad-vances and Future Work

Fast algorithms for strucutred matrices, such as theToeplitz matrices and Cauchy matrices, have a long andrich history. Recent advances have made many of such al-gorithms both fast and numerically stable. In addition,strucutred matrix techniques have been successfully usedto solve new classes of problems that are not structured inthe classical sense. In this talk, we discuss some recent ad-vances and interesting open questions in structured matrixcomputations.

Ming GuUniversity of California, BerkeleyMathematics Departmentmgu@math.berkeley.edu

MS16

Minimum Sobolev Norm Schemes

Minimum Sobolev Norm (MSN) schemes are a higher ordermesh-free discretization schemes that suppress the RungePhenomenon. The main idea is to find interpolatory poly-nomials of order >> N -the number of nodes, with min-imum Sobolev Norm. The infinite order case of the in-terpolation operator, produces a Golumb Wienberger ker-nel, that has FMM structure. We present the ideas be-hind the MSN scheme, results from convergence, and fast-algorithms for their evaluation.

Karthik JayaramanDept. Elect. & Comp. Engr.U. Cal. Santa Barbara

66 LA09 Abstracts

jrk@ece.ucsb.edu

MS16

Fast Inversion of Integral Operators

The talk will describe a scheme for rapidly constructing anapproximate inverse to the dense matrices arising from thediscretization of 1D integral operators and boundary inte-gral operators in the plane. The scheme is easily derivedvia hierarchical application of the Woodbury formula formatrix inversion. Its computational cost typically scaleslinearly with problem size, with a small constant of pro-portionality. The use of randomized sampling to facilitatethe compression of matrices will be discussed.

Gunnar MartinssonDept. of Applied MathematicsU. of Colorado, Bouldermartinss@colorado.edu

MS17

The Extended Krylov Subspace Method and Or-thogonal Laurent Polynomials

An orthogonal basis for the extended Krylov subspaceK�,m(A) = span{A−�+1v, . . . , A−1v, v, Av, . . . , Am−1v}can be generated using short recursion formulas. Theseformulas are derived using properties of Laurent polynomi-als. The derivation is presented for the general case whenm = k where k is a positive non-zero integer. The methodis applied to the approximation of matrix functions of theform f(A)v.

Carl JagelsHanover Collegejagels@hanover.edu

MS17

A Newton-Smith-Block Arnoldi Method for LargeScale Algebraic Riccati Equations

In this talk, we present a Newton-Block Arnoldi method forsolving large continuous algebraic Riccati equations. Suchproblems appear in control theory, model reduction, cir-cuit simulation and others. At each step of the Newtonprocess, we solve a large Lyapunov matrix equation witha low rank right hand side. These equations are solved byusing the block Arnoldi process associated with a precon-ditioner based on the Alternating Direction Implicit (ADI)iteration method. We give some theoretical results andreport numerical tests to show the effectiveness of the pro-posed approach.

Khalid JbilouUniversity Littoral Cote dOpale, Calais, Francejbilou@lmpa.univ-littoral.fr

MS17

A New Implementation of the Block GMRESMethod

The Block GMRES is a block Krylov solver for solving nonsymmetric systems with multiple right hand side. Thismethod is classicaly implemented by first applying theArnoldi iteration as a block orthogonalization process tocreate a basis of the block Krylov space and then solving ablock least squares problems, which is equivalent to solv-ing several least squares problems but with the same Hes-

senberg matrix. These laters are solved by using a blockupdating procedure for the QR decomposition of the Hes-senberg matrix based on Givens rotations. A more effectivealternative was given in which uses the Householder reflec-tors.In this paper we will propose a new and a simple implemen-tation of the block least squares problem. Our approach isbased on a genralization of Ayachour’s method, given forthe GMRES with a single right hand side. Finally we willgive some numerical experiments to illustrate the perfor-mance of the new implementation.

Hassane SadokUniversit’e du LittoralCalais Cedex, Francehassane.sadok@lmpa.univ-littoral.fr

Lothar ReichelKent State Universityreichel@math.kent.edu

MS17

On a Structure-preserving Arnoldi-like Method

In this talk, an Arnoldi-like method is presented. Themethod preserves the structures of Hamiltonian or skew-Hamiltonian matrices. Different variants are also intro-duced. The obtained structure-preserving methods areneeded for reducing the size of large and sparse Hamil-tonian or skew-Hamiltonian matrices. Numerical experi-ments are given.

Ahmed SalamUniversite Lille-Nord de France, Calais, France.Ahmed.Salam@lmpa.univ-littoral.fr

MS18

Recent Developments on the Inertias of Patterns

A number of techniques have been introduced in the liter-ature to show a specific pattern is inertially arbitrary. Wewill focus on the problem of finding sets of inertias thata pattern can attain. To do this, we describe the inertiasrealizable by certain sets of polynomials, the effect the in-ertia of a matrix has on its characteristic polynomial, andprovide sufficient conditions for reducible patterns to beinertially arbitrary.

Michael CaversDept of Mathematics and StatisticsUniversity of Reginacavers2m@math.uregina.ca

MS18

On Nilpotence Indices of Sign Patterns

We provide classes of n by n sign patterns of nilpotenceindices at least 3, if they are potentially nilpotent. Fur-thermore it is shown that if a full sign pattern A of ordern has nilpotence index k with 2 ≤ k ≤ n − 1, then signpattern A has nilpotent realizations of nilpotence indicesk, k + 1, . . . , n.

In-Jae KimMinnesota State University, Mankatoin-jae.kim@mnsu.edu

LA09 Abstracts 67

MS18

Allow Problems Concerning Spectral Properties ofSign Patterns

A sign pattern has entries in {+,−, 0}. For a given prop-erty P, a sign pattern S allows P if there exists at leastone real matrix with the sign pattern of S that has prop-erty P. The following problems are surveyed: sign patternsthat allow all possible spectra (spectrally arbitrary); signpatterns that allow all inertias (inertially arbitrary); andsign patterns that allow nilpotency (potentially nilpotentpatterns). Proof techniques, constructions and exampleswill be discussed, and open problems identified.

Dale OleskyDepartment of Computer ScienceUniversity of Victoriadolesky@cs.uvic.ca

MS18

Potentially Nilpotent Zero-Nonzero Patterns overFinite Fields

The idea of exploring spectrally arbitrary patterns was ini-tially related to inverse eigenvalue and eigenvalue comple-tion problems. For various patterns, one seemingly criticalstep for determining whether a pattern is spectrally arbi-trary is to determine if the pattern is potentially nilpo-tent (PN). We introduce an investigation into which zero-nonzero patterns are PN over finite fields. Tools from al-gebraic geometry and commutative algebra are developed.We classify irreducible PN patterns of order three over Zp.

Kevin N. Vander MeulenDepartment of MathematicsRedeemer University Collegekvanderm@cs.redeemer.ca

Adam Van TuylDepartment of MathematicsLakehead Universityavantuyl@lakeheadu.ca

MS19

Quantum Monte Carlo Methods: The Use of Lin-ear Solvers and Evaluating Determinant Ratios

Quantum Monte Carlo is one of the most accurate methodsto calculate physical properties of large systems. Simula-tions are currently done on a number of electrons N be-tween 100 and 1000. These calculations involve evaluatinga series of determinants of N × N matrices that change(slowly) throughout the simulation. Asymptotically thisevaluation costs O(N3) and is one of the dominating prac-tical bottlenecks in simulating larger systems. In addi-tion, functions of these determinants such as gradients andlaplacians must be calculated. We will discuss the natureof this problem and explore a series of techniques that aredesigned to decrease the asymptotic cost of these evalua-tions in system of physical interest.

Bryan ClarkDepartment of PhysicsUniversity of Illinois at Urbana-Champaignbkclark@uiuc.edu

Kapil AhujaVirginia Tech

kahuja@vt.edu

MS19

Computing the Matrix Sign Function in the Over-lap Fermion Model of QCD

At non-zero chemical potential, the overlap Dirac operatoris obtained as the sign function of a non-hermitian ma-trix. We consider techniques to compute the action of thesign function on a vector, including the deflation of smalleigenvalues. One approach uses rational functions and amultishift version of deflated FOM, the other is based onthe Arnoldi-Lanczos recurrence with a new idea to evaluatethe matrix function on the projected system.

Andreas J. FrommerBergische Universitaet WuppertalFachbereich Mathematik und Naturwissenschaftenfrommer@math.uni-wuppertal.de

MS19

Title Not Available at Time of Publication

Abstract not available at time of publication.

Scott MaclachianTufts UniversityScott.MacLachlan@tufts.edu

MS19

Large-scale Eigenvalue Solvers in ComputationalMaterials

Eigenvalue problems continue to be a largely unavoidablebottleneck in materials science applications. We discusscurrent state of the art eigenmethods and revisit Lanc-zos type algorithms which have fallen out of favor becausethey cannot utilize multiple initial guesses and precondi-tioning. We show that our recent eigCG technique thatfinds Lanczos quality eigenpairs without keeping the wholeLanczos basis can be used by itself or together with othermethods such as Jacobi-Davidson in PRIMME to improveeigenvalue convergence.

Andreas StathopoulosCollege of William & MaryDepartment of Computer Scienceandreas@cs.wm.edu

MS20

PMOR, an Interpolation Approach

Recently, the Loewner matrix pencil framework to systemidentification from measured data has been developed. Inthis talk we will propose an extension to this frameworkwhich includes systems depending on parameters. Thisamounts to treating the transfer function of the systemas a rational function in several complex or real variables.Reduction of the order of the underlying system followsnaturally, as in the single parameter case.

Athanasios C. AntoulasDept. of Elec. and Comp. Eng.Rice Universityaca@rice.edu

68 LA09 Abstracts

MS20

H∈-Optimal Interpolation for Model Reduction ofParameterized Systems

We consider interpolatory model reduction of parameter-ized linear dynamical systems executed in such a way thatthe parametric dependence of the original system is repro-duced in the reduced order model. Interpolation is guar-anteed both respect to selected complex frequencies andselected parameter value choices. We develop optimal pa-rameter selection strategies that will produce H∈-optimalreduced order systems that also minimize a least squareserror measure taken over the parameter range.

Christopher A. BeattieVirginia Polytechnic Institute and State Universitybeattie@math.vt.edu

MS20

Interpolation Based Optimal Model Reduction forLTI-Systems

The input-output behaviour of a linear time invariant (LTI)system can be characterized by the transfer function H(s)of the system, a p × m dimensional rational matrix func-tion, where m is the number of inputs and p the numberof outputs of the system. To compute reduced order mod-els for a large-scale LTI system we may project the statespace such that the transfer functions of the reduce ordersystems are rational tangential Hermite interpolations ofthe original transfer function. The interpolation data canbe chosen in such a way that first order necessary condi-tions for an optimal approximation in the H2,α-, or h2,α-norm are satisfied, where α is a bound for the poles of thesystems. Properties as well as the computation of such re-duced order models are presented and relations to balancedtruncation methods are discussed.

Angelika Bunse-Gerstner, Dorota KubalinskaUniversitaet BremenZentrum fuer Technomathematik, Fachbereich 3Bunse-Gerstner@math.uni-bremen.de, dorota@math.uni-bremen.de

MS21

Large Scale Non-linear Eigenvalue Optimization inElectronic Structure Calculations

We will discuss very large scale non-linear eigenvalue prob-lems that arise in Density Functional Theory calculations.In particular, we are interested in simulations that involvemany thousands of atoms and we will illustrate the chal-lenges that arise in extreme scale-out of such problems. Wewill illustrate the success of powerful non-linear optimiza-tion methods and show their scalability to thousands ofprocessors.

Costas BekasIBM ResearchZurich Research LaboratoryBEK@zurich.ibm.com

MS21

Subspace Accelerated Inexact Newton Method forSolving the Kohn-Sham Equations in Large ScaleCondensed Matter Applications.

Discretizing the Kohn-Sham equations of Density Func-tional Theory by Finite Differences methods leads to large

sparse matrix eigenvalue problems. To calculate the eigen-vectors associated to the lowest eigenvalues we proposea Subspace Accelerated Inexact Newton (SAIN) methodcombined with a multigrid preconditioner. This efficientiterative solver takes into account the non-linear characterof the Hamiltonian operator and avoids solving a series ofsurrogate linearized problems. The method is illustratedwith examples of real applications with hundreds of eigen-vectors.

Jean-Luc FattebertLawrence Livermore National Lab.fattebert1@llnl.gov

MS21

Nonlinear Subspace Iteration

To compute the electronic structure of materials, the Den-sity Functional Theory (DFT) technique converts the orig-inal n-particle problem into an effective one-electron sys-tem, resulting in a coupled one-electron Schrodinger equa-tion and a Poisson equation. This coupling is nonlinear andrather complex. A recently proposed method [Y. Zhou etal., Phys. Rev. E. 74, pp.066704 (2006)] to solve this prob-lem is to resort to a nonlinear variant of subspace iteration.The usual (linear) subspace iteration [Bauer, 1960] builds abasis of the subspace associated with the p largest eigenval-ues of a matrix by a kind of polynomial iteration (filtering)on an initial basis. Convergence is usually enhanced byusing Chebyshev acceleration. The nonlinear form of thisapproach consists of only updating the Hamiltonian matrixat each restart. We will illustrate this approach and showtechniques to enhance its robustness by using Homotopy.

Youcef SaadUniversity of Minnesotasaad@cs.umn.edu

MS21

Solving Nonlinear Eigenvalue Problems in Elec-tronic Structure Calculations

One of the key tasks in electronic structure calculation isto solve a nonlinear eigenvalue in which the matrix Hamil-tonian depends on the eigenvectors to be computed. Thisproblem can be solved either as a system of nonlinear equa-tions or as a constrained nonlinear optimization problem.We will examine the numerical algorithms used in bothapproaches and compare their convergence properties. Wewill also discuss the use of Broyden updates and trust re-gion techniques in these algorithms.

Chao YangLawrence Berkeley National LabCYang@lbl.gov

MS22

Use of Semi-Separable Approximate Factorizationand Direction-Preserving for Constructing Effec-tive Preconsitioners

For a symmetric positive definite matrix A of size n-by-n, we developed a fast and backward stable algorithmto approximate A by a symmetric positive-definite semi-separable Cholesky factorization, accurate to any pre-scribed tolerance. Our algorithm ensures that the Schurcomplements during the Cholesky factorization all remainpositive definite after approximation. In addition, it pre-serves the product, AZ, for a given matrix Z of size n-by-d,

LA09 Abstracts 69

where d n. We present numerical results to demonstratethe effectiveness of the preconditioners, and discuss appli-cations in large-scale computations.

Xiaoye S. LiComputational Research DivisionLawrence Berkeley National Laboratoryxsli@lbl.gov

Ming GuUniversity of California, BerkeleyMathematics Departmentmgu@math.berkeley.edu

Panayot VassilevskiLawrence Livermore National Laboratoryvassilevski1@llnl.gov

MS22

On the Schur Complements of the DiscretizedLaplacian

We show that the Schur complements of the matrix ob-tained by finite difference discretization of the Laplacianon the unit square with Dirichlet and Neumann boundaryconditions exhibit off-diagonal blocks with low rank. Thisis also true of the finite difference matrix in three dimen-sions under suitable reordering of the grid. This in turnimplies that fast direct solvers could be used to efficientlysolve these problems.

Naveen SomasunderamDepartment of Electrical EngineeringUniversity of California, Santa Barbaranaveen@umail.ucsb.edu

Shiv ChandrasekaranU.C. Santa Barbarashiv@ece.ucsb.edu

Ming GuUniversity of California, BerkeleyMathematics Departmentmgu@math.berkeley.edu

Patrick DeWildeTU Delft, Netherlandsp.dewilde@ewi.tudelft.nl

MS22

Robust and Efficient Factorization and Precondi-tioning for Sparse Matrices

We present a robust approximate factorization and precon-ditioning method for sparse matrices such as those arisingfrom the discretization of some PDEs. Sparse matrix tech-inques are combined with structured factorizations of densematrices. Structural compressions of dense matrix blockslead to high efficiency. Robustness enhancement is auto-matically integrated into the factorization without extracost. The factor can work as an effective preconditionerwithout strict structural requirement. The cost and storagefor such a factorization are roughly linear in the matrix size.The efficiency, robustness, and effectiveness are discussedand demonstrated with examples such as Helmholtz, linearelasticity, and Maxwell equations. Numerical experimentsshow that this preconditioner is also relatively insensitive

to frequencies or wave numbers.

Jianlin XiaPurdue Universityxiaj@math.purdue.edu

Ming GuUniversity of California, BerkeleyMathematics Departmentmgu@math.berkeley.edu

MS22

Butterfly Algorithm and its Applications

We describe the Butterfly algorithm, which is a general ap-proach for the rapid evaluation of oscillatory interactions.We also discuss two applications: sparse Fourier transformand discrete Fourier integral operators. In each case, thebutterfly algorithm is combined with special interpolationtools to achieve optimal computational complexity.

Lexing YingUniversity of TexasDept. of Mathematicslexing@math.utexas.edu

MS23

FORTRAN 77 Implementations of SuperfastSolvers

We will describe the current status of FORTRAN 77 im-plementations of superfast algorithms for solving positivedefinite Toeplitz systems of equations. The algorithms in-clude the original O(n log2 n) method described in [SIMAX9(1988):61–76] as well as the O(n log3 n) procedure of Stew-art [SIMAX 25(2003):669–693], which has improved stabil-ity properties.

Greg AmmarNorthen Illinois Universityammar@math.niu.edu

MS23

An Implicit Multishift QR-algorithm for Symmet-ric Plus Low Rank Matrices

We present an efficient computation of all the eigenvalues ofthe Hessenberg form of a hermitian matrix perturbed by apossibly non-symmetric low-rank correction. We propose arepresentation based on Givens transformations that allowsto perform steps of the implicit multishift QR algorithm us-ing O(\) operations per iteration. A deflation technique isproposed as well. The representation by means of Givensrotations, makes the algorithm numerically stable as con-firmed by the extensive numerical tests.

Gianna M. Del CorsoUniversity of Pisa, Department of Computer Sciencedelcorso@di.unipi.it

Raf VandebrilKatholieke Universiteit LeuvenDepartment of Computer Scienceraf.vandebril@cs.kuleuven.be

MS23

Structured Regularization Operators for

70 LA09 Abstracts

Tikhonov’s Method

We consider the solution of linear discrete ill-posed prob-lems with the aid of Tikhonov regularization. The regu-larized problem is said to be in standard form if the reg-ularization operator is the identity. Tikhonov regulariza-tion problems in standard form are easier to solve thanregularization problems in general form. We discuss theconstruction of structured square regularization operatorsthat allow easy transformation of the regularized problemto standard form. This talk presents joint work with QiangYe.

Lothar ReichelKent State Universityreichel@math.kent.edu

MS23

On Tridiagonal Matrices Unitarily Equivalent toNormal Matrices

In this talk the unitary equivalence transformation of nor-mal matrices to tridiagonal form is studied. It is well-known that any matrix is unitarily equivalent to a tridi-agonal matrix. In case of a normal matrix the resultingtridiagonal inherits a strong relation between its super-and subdiagonal elements. The corresponding elements ofthe super- and subdiagonal will have the same absolutevalue. In this talk some basic facts about a unitary equiva-lence transformation of an arbitrary matrix to tridiagonalform are firstly studied. Both an iterative manner basedon Krylov sequences as a direct manner via Householdertransformations are deduced. This equivalence transforma-tion is then reconsidered for the normal case and equalityof the absolute value between the super- and subdiago-nals is proved. Self-adjointness of the resulting tridiago-nal matrix with regard to a specific scalar product will beproved. Flexibility in the reduction will then be exploitedand properties when applying the reduction on symmetric,skew-symmetric, Hermitian, skew-Hermitian and unitarymatrices and their relations with, e.g., complex-symmetricand pseudo-symmetric matrices are presented. It will beshown that the reduction can then be used to compute thesingular value decomposition of normal matrices makinguse of the Takagi factorization. Finally some extra prop-erties of the reduction as well as an efficient method forcomputing a unitary complex symmetric decomposition ofa normal matrix are given.

Raf VandebrilKatholieke Universiteit LeuvenDepartment of Computer Scienceraf.vandebril@cs.kuleuven.be

MS24

The Minimum Rank Problem for Planar Graphs

The minimum rank problem for a given simple, undirectedgraph G is to determine the minimum rank (or maximumnullity ) over all symmetric matrices whose off-diagonalnonzero pattern corresponds to G. Two graph parame-ters, the path cover number and zero forcing number, haveplayed an important role in the determination of the maxi-mum nullity for trees, unicyclic graphs, outerplanar graphs,and rectangular grids. This talk will survey several knownresults and report on recent progress.

Wayne BarrettBrigham Young Universitywayne@math.byu.edu

MS24

Acyclic and Unicyclic Graphs Whose MinimumSkew Rank is Equal to the Minimum Skew Rankof a Diametrical Path

The minimum skew rank of a simple graph G over the fieldof real numbers is the smallest possible rank among allskew-symmetric real matrices whose (i, j)-entry (for i �=j) is nonzero whenever {i, j} is an edge in G and is zerootherwise. In this paper we give necessary and sufficientconditions for a connected acyclic graph to have minimumskew rank equal to the minimum skew rank of a diametricalpath. We also characterize connected unicyclic graphs withthis property.

Luz M. DeAlbaDrake UniversityDepartment of Mathematicsluz.dealba@DRAKE.EDU

MS24

Minimum Rank Problems

A graph (directed or undirected, simple or allowing loops)can be used to describe a pattern of nonzero entries of amatrix (over the real numbers, or more generally over afield). A minimum rank problem is a problem of determin-ing the minimum of the ranks of the matrices describedby a graph, perhaps requiring additional matrix propertiessuch as symmetry or positive semidefiniteness. This talkwill survey recent developments in minimum rank prob-lems and applications, and serve as an introduction to themini-symposium.

Leslie HogbenIowa State Universitylhogben@iastate.edu

MS24

Diagonal Rank and Skew Symmetric Matrices

A beautiful result of Fiedler’s is that if A is an n× n sym-metric matrix such that the rank of D+A is at least n− 1for each diagonal matrix D, then A is permutationally sim-ilar to an irreducible, tridiagonal matrix. In this talk, wediscuss the analogous problem for skew-symmetric matri-ces.

Bryan L. Shader, Reshmi Nair, Colin Garnett, MaryAllisonDepartment of MathematicsUniversity of Wyomingbshader@uwyo.edu, rnair@uwyo9.edu,cgarnett@uwyo.edu, mallison@uwyo.edu

MS25

Application of the IDR Theorem to Stationary It-erative Methods and their Performance Evaluation

The appearance of IDR(s) methods based on the IDR The-orem strongly influenced some researchers in Krylov sub-space (KS) methods. Like some KS methods, classicalstationary iterative (SI) methods, e.g, Gauss-Seidel andSOR, may be seen in a new light too. This change of viewbuilds on the relation between the conventional residualrecursion rk+1 = Brk of the SI method and the recursionrk+1 = B(rk + γk(rk − rk−1)) of the IDR(s) method. Here

LA09 Abstracts 71

B is the iteration matrix, and γk is a scalar.

Seiji FujinoComputing and Communications CenterKyushu Universityfujino@cc.kyushu-u.ac.jp

MS25

IDR — A Brief Introduction

The Induced Dimension Reduction (IDR) method intro-duced first by Sonneveld in 1979 has been reconsidered andgeneralized by Sonneveld and van Gijzen recently. Whilethe original IDR became the forerunner of Bi-CGSTAB,the new IDR(s) is related to the ML(k)BiCGSTAB methodof Yeung and Chan and to the nonsymmetric band Lanczosalgorithm. It proved amazingly effective and offers someflexibility. This talk intends to give a brief overview ofthese approaches and their connections.

Martin H. GutknechtSeminar for Applied MathematicsETH Zurichmhg@math.ethz.ch

MS25

ML(n)BiCGStab: Reformulation, Analysis and Im-plementation

With the help of index functions, we rederive theML(n)BiCGStab algorithm in a more systematic way.There are n ways to define the ML(n)BiCGStab residualvector. Each different definition will lead to a differentML(n)BiCGStab algorithm. We demonstrate this by de-riving a second algorithm which requires less storage. Wealso analyze the breakdown situations. Implementation is-sues are also addressed. We discuss the choices of the pa-rameters in ML(n)BiCGStab and their effects on the per-formance.

Man-Chung YeungUniversity of Wyomingmyeung@uwyo.edu

MS25

An IDR-variant that Efficiently Exploits Bi-orthogonality Relations

The IDR theorem provides a way to construct a sequence ofnested subspaces, of which the smallest subspace is zero di-mensional. IDR-based algorithms construct residuals thatare in these nested subspaces. The translation from theo-rem to algorithm is not unique. In the talk we will presenta variant of IDR(s) that imposes bi-orthogonalization con-ditions on the iteration vectors. The resulting algorithm isaccurate and efficient.

Martin B. van Gijzen, Peter SonneveldDelft Inst. of Appl. MathematicsDelft University of Technologym.b.vangijzen@tudelft.nl, p.sonneveld@tudelft.nl

MS26

Linear Algebra and Stochastic Finite ElementComputations

Stochastic Finite Element methods for solving partial dif-ferential equations with random data have become an es-

tablished technique for uncertainty quantification (UQ). Itsimplementation involves a number of challenging linear al-gebra tasks, among these the solution of very large linearsystems of equations, long sequences of closely related lin-ear and nonlinear systems of equations and large denseeigenvalue problems. This talk will lead into the minisym-posium by describing the basic UQ problem setting, intro-ducing the key discretization techniques and also presentsome of our own recent work in addressing these tasks.

Oliver G. ErnstTU Bergakademie FreibergFakultaet Mathematik und Informatikernst@math.tu-freiberg.de

MS26

Parallelization and Preconditioning of IterativeSolvers for Linear SFEM Systems

We develop iterative solvers for the solution of linear sys-tems stemming from a stochastic Galerkin projection ofstochastic partial differential equations. We demonstratethe parallelism potential of these solvers and implementthem with the Trilinos high performance computing en-vironment. We show how these algorithms can mitigatethe curse of dimensionality while being very efficient andattuned to high performance resources. These solvers arevariations on solvers previously developed by one of the au-thors which have been adapted to current computationalresources.

Roger Ghanem, Ramakrishna TipireddyUniversity of Southern CaliforniaAerospace and Mechanical Engineering and CivilEngineeringghanem@usc.edu, tipiredd@usc.edu

Maarten ArnstUniversity of Southern Californiaarnst@usc.edu

MS26

Iterative Solvers for Stochastic Galerkin Finite El-ement Discretizations

A linear stochastic partial differential equation can be con-verted by the stochastic Galerkin method into a high di-mensional algebraic system. Such system, characterized bya tensor product structure, can be solved efficiently by amultigrid solver. This yields a convergence rate indepen-dent of the discretization parameters. Non-optimal meth-ods, based on block splitting approaches or on a hierarchyin stochastic space, may require however less computingtime. A comparison of several solvers will be presented.

Eveline RosseelDept. Computer ScienceK.U. LeuvenEveline.Rosseel@cs.kuleuven.be

Stefan VandewalleScientific Computing Research GroupKULeuvenstefan.vandewalle@cs.kuleuven.be

MS26

Iterative Solvers in Stochastic Galerkin Finite El-ement Computations: Preconditioners and Multi-

72 LA09 Abstracts

level Techniques

The discretization of linear stochastic partial differentialequations (SPDEs) by means of the stochastic Galerkin fi-nite element method [1] results in general in a very largeand coupled but structured linear system of equations.Our task is the design of efficient iterative solvers forthese Galerkin equations. The stochastic diffusion equa-tion serves as our model problem. For the primal formu-lation of this SPDE we review the recently proposed Kro-necker product preconditioner [2] for the Galerkin matrixwhich - in contrast to the popular mean-based precondi-tioner - makes use of the entire information contained in theGalerkin matrix. Furthermore, we extend the idea of Kro-necker product preconditioning to the discretized mixedformulation of the stochastic diffusion equation and presentnumerical results of test problems where the stochastic dif-fusion coefficient is a lognormal random field. Finally wereport on our attempts to utilize multilevel techniques forthe Galerkin equations. In contrast to previous works,where many researchers have applied multilevel methodsexclusively to the deterministic finite element spaces, wefocus on multilevel decompositions for the stochastic vari-ational space and combined deterministic/stochastic mul-tilevel approaches for the global Galerkin system.

Elisabeth UllmannTU Bergakademie Freibergullmann@math.tu-freiberg.de

MS27

Adaptive Least Squares Algebraic Multigrid forComplex-valued Systems

We present an algebraic multilevel method for solving HPDlinear systems. The proposed approach adaptively com-putes a set of testfunctions and then uses them in a leastsquares fit to define interpolation. In this way, the algo-rithm iteratively improves the multilevel hierarchy to bet-ter approximate the lowest modes of the MG relaxation(solver) until suitable performance is achieved. We testthe proposed approach for a variety of applications in boththe Classical AMG and Smoothed Aggregation settings,showing promising results.

James BrannickDepartment of MathematicsPennsylvania State Universitybrannick@psu.edu

MS27

Smoothed Aggregation Multigrid on Multicore Ar-chitectures

Multicore computer architectures present new challengesto linear solvers, including algebraic multigrid multigrid.The advent of many cores has effectively increased the im-balance between node compute power and off-node com-munication throughput. We report our experiences usingnonstandard smoothed aggregation multigrid on multicorearchitectures to address the computation/communicationimbalance. We consider multigrid variants with domaindecomposition smoothing, where domains are allowed tocover more than one core in a compute node and presentparallel numerical results.

Jonathan J. HuSandia National LaboratoriesLivermore, CA 94551jhu@sandia.gov

Chris SiefertSandia National Laboratoriescsiefer@sandia.gov

MS27

Smoothed Aggregation AMG Solvers for Least-Squares Finite Element Systems

Least-squares finite element methods are an attractive classof methods for the numerical solution of certain div-curlsystems. We consider a discretization where we approxi-mate H(curl,Ω) ∪ H(div,Ω) conformally with respect tothe divergence operator (e.g. using face elements) and usea discrete approximation of the curl operator. We derive amultigrid method from the previous work of Bochev, Hu,Siefert, Tuminaro, Xu and Zhu (2007), using specializedprolongators to transform the system into a nodal Laplace-like problem.

Chris SiefertSandia National Laboratoriescsiefer@sandia.gov

Pavel BochevSandia National LaboratoriesComputational Math and Algorithmspbboche@sandia.gov

Kara PetersonSandia National Laboratorieskjpeter@sandia.gov

MS27

A Generalization of Energy Minimization Ideas toAlgebraic Multigrid for Nonsymmetric Systems

We consider the generalization of energy minimization al-gebraic multigrid concepts to nonsymmetric systems. Anew method is proposed where construction of prolonga-tion (and restriction) is based on a few iterations of GM-RES. These iterations essentially minimize the sum of theATA norm of the grid transfer basis functions. This mini-mization occurs within some Krylov space while satisfyinga set of constraints. Numerical comparison are given on aset of highly convective nonsymmetric problems.

Ray S. TuminaroSandia National LaboratoriesComputational Mathematics and Algorithmsrstumin@sandia.gov

MS28

Teaching Introductory Linear Algebra Incorporat-ing MATLAB

The first wave of technology supporting mathematics in-struction surged during the 90s. We took the opportunityto offer a laboratory experience to accompany our generalintroductory linear algebra course. A fundamental objec-tive was to support linear algebra concepts with graphicsfor visualization, design experiments to engage students ona variety of levels, and explore connections. A discussion oftactics, examples, student reactions, changes encountered,and challenges involved with the current second wave oftechnologies that are emerging will be included.

David HillTemple University

LA09 Abstracts 73

dhill001@temple.edu

MS28

A Second Course in Linear Algebra for Science andEngineering

This talk will describe a second course in linear algebra thathas evolved over the past ten years. The course is aimed atstudents in engineering and in the physical and computersciences. Main new topics: (1) geometry in vector spaces,(2) matrix games and linear programming, and (3) finite-state Markov chains. Enrollment has grown over the pastfive years and now ranges around 200 students annually.

David LayUniversity of Marylandlay@math.umd.edu

MS28

When MATLAB Gives ”Wrong” Answers; How toGet Students in Numerical Linear Algebra ClassesExcited About Finite-precision Computations

One of the main differences between teaching the standardlinear algebra course and a course in numerical linear alge-bra is that in the latter course all computations are doneusing finite precision arithmetic. One way to illustrate theimportance of this difference is to look at examples wherecomputational software packages such as MATLAB appearto be giving wrong answers. In this talk we examine fouror five such scenarios. In each case we look at examplesand explain how and why MATLAB arrives at its answers.In our final example we examine a MATLAB program thatclearly produces an impossible answer. In this case, whenthe author of the program tried to debug it by printing outintermediate results, the value of the computed solutionchanged. What is going on? Is MATLAB exhibiting somesort of Heisenberg effect? All will be explained at the talk.

Steven J. LeonUniversity of Massachusetts Dartmouthleonste@gmail.com

MS28

Key Points in a Linear Algebra Course

I will discuss my experience in teaching and writing aboutlinear algebra. I will focus on three points in the course,one at the start and one at the heart and one at the veryend (often barely reached):1. I now give an early lecture to point the way – matrixmultiplication and words like independence are introducedby specific examples. I believe we learn language this way,by hearing words, making mistakes, and eventually gettingthem right.2. The ”Four Fundamental Subspaces” give a structureto the subject. Students catch on to the big picture –dimensions and orthogonality.3. A highlight at the end is the SVD. How to introducethis idea? It depends on A(A′A) = (AA′)A. It involvespositive definiteness. And it gives perfect bases for thefour subspaces.

Gil StrangMITgs@math.mit.edu

MS29

Interpolation with Semi-separable Matrices: theRatio-of-Polynomials Case

An interesting problem connected to semi-separable ma-trices and operators is the issue of interpolation. Givena general (upper) matrix or operator, one may wish tofind reduced approximate representations for it of the semi-separable type. This issue leads to all sort of generaliza-tions of classical interpolation problems, with very similarapproximation properties. An interesting case is when therepresentation is of the type ’ratio of generalized polyno-mials’. In the generalized case the polynomials are series ofshifted diagonals, their ratio are essentially ratios of (non-uniformly) banded matrices. The theory provides for ageneralization to matrices of the classical Lowner inter-polation theory as originally developed by Antoulas, Ball,Kang and Willems, has interesting connections with con-trol theory and is also computationally attractive.

Patrick DewildeInstitute for Advanced Study of TUMtba

MS29

Structured Matrix Methods for the Constructionof Interpolatory Subdivision Masks

In this talk we present a general approach for the construc-tion of interpolatory subdivision masks based on polyno-mials and structured matrix computations. This is a jointwork with Costanza Conti from the University of Florenceand Lucia Romani from the University of Milano-Bicocca.

Luca Gemignani

Univeristy of Pisa (Italy)Department of Mathematicsgemignan@dm.unipi.it

MS29

Efficient Computations with Tensor StructuredMatrices

In this talk we present a new method for basic matrix op-erations with multilevel tensor structured matrices, It isbased on a special low-parametric and stable representa-tion for tensors and keeps the complexity linear in thenumber of levels. Some of these results are presented inthe following papers:

I.V.Oseledets and E.E.Tyrtyshnikov, Breaking thecurse od dimensionality, or how to use SVDin many dimensions, Research Report 09-03,Kowloon Tong, Hong Kong: ICM HKBU, 2009(www.math.hkbu.edu.hk/ICM/pdf/09-03.pdf).

I.V.Oseledets,D.V.Savostyanov, and E.E.Tyrtyshnikov, Linear al-gebra for tensor problems, Research Report 08-02,Kowloon Tong, Hong Kong: ICM HKBU, 2008(www.math.hkbu.edu.hk/ICM/pdf/08-02.pdf).

Eugene TyrtyshnikovInstitute of Numerical Mathematics, Russian Academy ofSci.tee@inm.ras.ru

MS29

Multivariable Orthogonal Polynomials and Struc-

74 LA09 Abstracts

tured Matrix Computations

The recurrence coefficients of polynomials in one variablewith respect to a discrete inner product can be computedby solving a structured inverse eigenvalue problem. In thistalk we will investigate how this inverse eigenvalue problemis modified in case of multivariable orthogonal polynomials.We will also indicate how this inverse eigenvalue problemcan be solved only using orthogonal similarity transforma-tions. Some numerical experiments will show the validityof this approach.

Marc Van BarelKatholieke Universiteit LeuvenDepartment of Computer Sciencemarc.vanbarel@cs.kuleuven.be

MS30

Low-rank Matrix Recovery From Noisy Observa-tions Using Nuclear Norm Minimization

The field of Compressed Sensing has demonstrated thatsparse signals can be recovered robustly from incompletesets of measurements. It is natural to ask what other ob-jects or structures might also be recovered from incom-plete, limited, and noisy measurements. It has been shownrecently that a similar framework can be developed for re-covering low-rank matrices from limited observations, byminimizing the nuclear norm of the matrix. In this talk,we extend this framework to include cases where the obser-vations are noisy and the matrix is not perfectly low-rank,and provide bounds on the recovery error for several classesof measurements. Furthermore, we discuss conditions un-der which the matrix recovered from noisy measurementswill have the correct rank and singular spaces.

Maryam FazelUniversity of WashingtonElectrical Engineeringmfazel@ee.washington.edu

MS30

Nuclear Norm Minimization for the MaximumClique and Biclique Problems

We consider the problems of finding a maximum cliquein a graph and finding a maximum-edge biclique in a bi-partite graph. Both problems are NP-hard. We write bothproblems as matrix-rank minimization and then relax themusing the nuclear norm. This technique, which may beregarded as a generalization of compressive sensing, hasrecently been shown to be an effective way to solve rankoptimization problems. In the special cases that the inputgraph has a planted clique or biclique (i.e., a single largeclique or biclique plus diversionary edges), our algorithmsuccessfully provides an exact solution to the original in-stance. For each problem, we provide two analyses of whenour algorithm succeeds. In the first analysis, the diversion-ary edges are placed by an adversary. In the second, theyare placed at random. In the case of random edges for theplanted clique problem, we obtain the same bound as Alon,Krivelevich and Sudakov as well as Feige and Krauthgamer,but we use di erent techniques.

Brendan AmesUniversity of Waterloobpames@math.uwaterloo.ca

Stephen A. Vavasis

University of WaterlooDept of Combinatorics & Optimizationvavasis@math.uwaterloo.ca

MS30

Explicit Sensor Network Localiza-tion using Semidefinite Programming and CliqueReductions

The sensor network localization (SNL) problem in embed-ding dimension r, consists of locating the positions of adhoc wireless sensors, given only the distances between sen-sors that are within radio range and the positions of asubset of the sensors (called anchors). There are advan-tages for formulating this problem as a Euclidean distancematrix completion (EDMC) problem, and ignoring the dis-tinction between anchors and sensors. Current solutiontechniques relax this problem to a weighted, nearest, (posi-tive) semidefinite programming, SDP, completion problem,by using the linear mapping between EDMs and SDP ma-trices. The relaxation consists in ignoring the rank r for theSDP matrices. The resulting SDP is solved using primal-dual interior point solvers, yielding an expensive and inex-act solution. Moreover, the relaxation is ill-conditioned intwo ways. First, it is implicitly highly degenerate in thesense that the feasible set is restricted to a low dimensionalface of the SDP cone. This means that the Slater constraintqualification fails. Second, nonuniqueness of the optimalsolution results in large sensitivity to small perturbationsin the data. The degeneracy in the SDP arises from cliquesin the graph of the SNL problem. These cliques implicitlyrestrict the dimension of the face containing the feasibleSDP matrices. In this paper, we take advantage of theabsence of the Slater constraint qualification and derive atechnique for the SNL problem, with exact data, that ex-plicitly solves the corresponding rank restricted SDP prob-lem. No SDP solvers are used. We are able to efficientlysolve this NP-hard problem with high probability, by find-ing a representation of the minimal face of the SDP conethat contains the SDP matrix representation of the EDM.The main work of our algorithm consists in repeatedly find-ing the intersection of subspaces that represent the facesof the SDP cone that correspond to cliques of the SNLproblem.

Nathan KrislockUniversity of Waterloongbkrisl@uwaterloo.ca

Henry WolkowiczUniversity of WaterlooDept of Combinatorics and Optimizationhwolkowi@math.uwaterloo.ca

MS30

Fast and Near–Optimal Matrix Completion viaRandomized Basis Pursuit

Motivated by the philosophy and phenomenal success ofcompressed sensing, the problem of reconstructing a ma-trix from a sampling of its entries has attracted much at-tention recently. Such a problem can be viewed as aninformation–theoretic variant of the well–studied matrixcompletion problem, and the main objective is to designan efficient algorithm that can reconstruct a matrix by in-specting only a small number of its entries. Although thisis an impossible task in general, it has heen recently shownthat under a so–called incoherence assumption, a rank rn× n matrix can be reconstructed using semidefinite pro-

LA09 Abstracts 75

gramming (SDP) after one inspects O(nr log6 n) of its en-tries. In this paper we propose an alternative approachthat is much more efficient and can reconstruct a largerclass of matrices by inspecting a significantly smaller num-ber of the entries. Specifically, we first introduce a class ofso–called stable matrices and show that it includes all thosethat satisfy the incoherence assumption. Then, we proposea randomized basis pursuit (RBP) algorithm and show thatit can reconstruct a stable rank r n×n matrix after inspect-ing O(nr log n) of its entries. Our sampling bound is onlya logarithmic factor away from the information–theoreticlimit and is essentially optimal. Moreover, the runtime ofthe RBP algorithm is bounded by O(nr2 logn+n2r), whichcompares very favorably with the Ω(n4r2 log12 n) runtimeof the SDP–based algorithm. Perhaps more importantly,our algorithm will provide an exact reconstruction of theinput matrix in polynomial time. By contrast, the SDP–based algorithm can only provide an approximate one inpolynomial time.

Zhisu ZhuStanford Universityzhuzhisu@stanford.edu

Anthony Man-Cho SoThe Chinese University of Hong Kong, Hong Kongmanchoso@se.cuhk.edu.hk

Yinyu YeStanford Universityyinyu-ye@stanford.edu

MS31

BiCGStab2 and GPBiCG Variants of the IDR(s)Method

Thehybrid BiCG methods such as BiCGSTAB, BiCGStab2,GPBiCG, and BiCGstab() are well-known efficient solversfor linear system with nonsymmetric matrices. Recently,IDR(s) has been proposed, and it has been reported to bemore effective than the hybrid BiCG methods. Moreover,IDR(s) variants incorporating BiCGstab() strategies havebeen designed, which converge even faster than the orig-inal IDR(s) method. Here, we propose IDR(s) variantsincorporating strategies of BiCGStab2 and GPBiCG.

Kuniyoshi AbeFaculty of Economics and InformationGifu Shotoku Universityabe@gifu.shotoku.ac.jp

Gerard L.G. SleijpenMathematical InstituteUtrecht Universityg.l.g.sleijpen@uu.nl

MS31

Exploiting BiCGstab() Strategies to Induce Di-mension Reduction

IDR(s) and BiCGstab() are among the most efficient iter-ative methods for solving nonsymmetric linear systems ofequations. Here, we derive a new method called IDRstabcombining the strengths of IDR(s) and BiCGstab(). Toderive IDRstab we extend previous work, where we con-sidered Bi-CGSTAB as an IDR method. We analyze thestructure of IDR in detail and introduce the new conceptof a Sonneveld space. Numerical experiments show that

IDRstab can outperform both IDR(s) and BiCGstab().

Gerard L. SleijpenMathematical InstituteUtrecht UniversityG.L.G.Sleijpen@uu.nl

Martin B. van GijzenDelft University of TechnologyDelft, The Netherlandsm.b.vangijzen@tudelft.nl

MS31

On the Convergence Behaviour of IDR(s)

During the development of the prototype IDR(s) method[Sonneveld and van Gijzen, SIAM J. Sci. Comput. 31,1035-1062 (2008)], several ”intelligent” choices for the sauxiliary vectors were tried, with poor results. Only arandom choice for these vectors appears to be suitable.In numerical experiments the convergence plots show, forincreasing s, a kind of ”convergence” to full GMRES. Astatistical explanation of this behaviour will be given inrelation to the random choice of the auxiliary vectors.

Peter SonneveldDelft Inst. of Appl. MathematicsDelft University of Technologyp.sonneveld@tudelft.nl

MS31

Eigenvalue Perspectives of the IDR Family

We investigate the natural correspondence between Krylovmethods for the approximate solution of linear systems ofequations and the approximation of eigenvalues tailored tothe IDR family. This is work in progress.

Jens-Peter M. ZemkeInst. of Numerical SimulationTU Hamburg-Harburgzemke@tu-harburg.de

Martin GutknechtSeminar for Applied MathematicsETH Zurichgutknecht@math.ethz.ch

MS32

Model Reduction of Switched Dynamical Systems

A hybrid dynamical system is a system described byboth differential equations (continuous flows) and differ-ence equations (discrete transitions). It has the benefit ofallowing more flexible modeling of dynamic phenomena, in-cluding physical systems with impact such as the bouncingball, switched systems such as the thermostat, and eventhe internet congestion as examples. Hybrid dynamicalsystems pose a challenge since almost all reduction meth-ods cannot be directly applied. Here we show some recentdevelopments in the area of model reduction of switcheddynamical systems.

Younes ChahlaouiCICADAThe University of Manchester, UKYounes.Chahlaoui@manchester.ac.uk

76 LA09 Abstracts

MS32

A Structure-Preserving Method for Positive Real-ness Problem

In this talk the positive realness problem in circuit andcontrol theory is studied. A numerical method is developedfor verifying the positive realness of a given proper ratio-nal matrix H(s) for which H(s)+HT (−s) may have purelyimaginary zeros. The proposed method is only based onorthogonal transformations, it is structure-preserving andhas a complexity which is cubic in the state dimension ofH(s). Some examples are given to illustrate the perfor-mance of the proposed method.

Delin ChuDepartment of Mathematics, National University ofSingapore2 Science Drive 2, Singapore 117543matchudl@nus.edu.sg

Peter BennerTU ChemnitzGermanybenner@mathematik.tu-chemnitz.de

Xinmin LiuDepartment of Electrical and Computer EngineeringUniversity of Virginia,USAxmliu@virginia.edu

MS32

Thick-Restart Krylov Subspace Techniques for Or-der Reduction of Large-Scale Linear DynamicalSystems

In recent years, Krylov subspace techniques have provento be powerful tools for order reduction of large-scale lin-ear dynamical systems. The most widely-used algorithmsemploy explicit projection of the data matrices of the dy-namical systems, using orthogonal bases of the Krylov sub-spaces. For truly large-scale systems, the generation andstorage of such bases becomes prohibitive. In this talk, weexplore the use of thick-restart Krylov subspace techniquesto reduce the computational costs of explicit projection.

Roland W. FreundUniversity of California, DavisDepartment of Mathematicsfreund@math.ucdavis.edu

MS32

AMLS for an Unsymmetric Eigenproblem Govern-ing Free Vibrations of Fluid-Solid Structures

Automated Multi-Level Substructuring (AMLS) has beendeveloped to reduce the computational demands of fre-quency response analysis and is very efficient for huge sym-metric and definite eigenvalue problems. This contributionis concerned with an adapted version of AMLS for an un-symmetric eigenvalue problem governing free vibrations offluid solid structures. Although we take advantage of anequivalent Hermitian eigenproblem of doubled dimensionour method needs essentially the same computational workas the original AMLS algorithm.

Heinrich VossHamburg Univ of TechnologyInstitute of Numerical Simulationvoss@tu-harburg.de

Markus StammbergerHamburg Univ of TechnologyInst. of Numerical Simulationmarkus.stammberger@tuhh.de

MS33

Scalable Tensor Factorizations with Missing Data

Handling missing data is a major challenge in many dis-ciplines. Missing data problem has been addressed inthe context of matrix factorizations and these approacheshave been extended to tensor factorizations. We are inter-ested in fitting tensor models, in particular, the CANDE-COMP/PARAFAC (CP) model to data with missing en-tries. We propose the use of a gradient-based optimizationalgorithm (CP-WOPT) in order to have methods scalableto large tensors. Using numerical experiments we showthat the algorithm we propose is accurate and faster thanan alternative technique based on nonlinear least squaresapproach.

Evrim Acar, Tamara G. Kolda, Daniel M. DunlavySandia National Laboratorieseacarat@sandia.gov, tgkolda@sandia.gov,dmdunla@sandia.gov

MS33

Scalable Implicit Tensor Approximation

We describe some algorithms for producing a best multi-linear rank r approximation of a symmetric tensor, usefulfor working with higher moments and cumulants of a largep-dimensional random vector. Since the approximated d-way tensor has pd entries, for large p it is critical never toexplicitly represent the entire tensor. Working implicitlyand approximately where appropriate, this approach en-ables the use of higher order factor models on large-scaledata.

Jason MortonStanford Universityjason@math.stanford.edu

MS33

Decomposing a Third-Order Tensor in Rank-(L,L,1) Terms by Means of Simultaneous MatrixDiagonalization

The decompositions of a third-order tensor in block compo-nents are new multilinear algebra tools, that generalize theParallel Factor (PARAFAC) decomposition. We focus onone of these recently introduced decompositions, the Block-Component-Decomposition in rank-(L,L,1) terms, referredto as BCD-(L,L,1). In previous work, we have proposed anAlternating Least Squares (ALS) algorithm for the com-putation of the BCD-(L,L,1) and a sufficient bound forwhich uniqueness of this decomposition is guaranteed. Inthis talk, we show that the BCD-(L,L,1) can be equiva-lently reformulated as a problem of Simultaneous Diago-nalization (SD) of a set of matrices. This reformulationhas two major advantages. First, the resulting SD-basedalgorithm reveals to be more accurate and less sensitiveto ill-conditioned data than the standard ALS algorithm.Second, the SD-based reformulation of the BCD-(L,L,1) in-volves itself a new sufficient uniqueness bound. The latterbound is more relaxed than the one previously derived, inthe sense that the number of rank-(L,L,1) terms that canbe extracted from the tensor is significantly greater, under

LA09 Abstracts 77

a few conditions on the dimensions.

Dimitri NionK.U. Leuvendimitri.nion@kuleuven-kortrijk.be

Lieven De LathauwerKatholieke Universiteit LeuvenLieven.DeLathauwer@kuleuven-kortrijk.be

MS33

On the Convergence of Lower Rank Approxima-tions to Super-symmetric Tensors

We consider a type of rank-r approximation to super-symmetric tensors of even order (of size N ×N × · · ·×N),in which a set of r orthonormal basis vectors and anr × r × · · · × r core tensor are sought (with r < N) tominimize a Froebenius norm discrepancy. The problem isequivalent to finding basis vectors which optimally com-press a super-symmetric tensor into one of smaller dimen-sions via a standard multi-linear transformation. We pro-pose an iterative algorithm that may be understood as asymmetrized version of alternating least squares, with animportant distinction: whereas alternating least squarescan only be shown to converge to a stationary point (whichmay be a minimum or saddle point of the approximation),the symmetric variant under study is shown to convergemonotonically to a minimum of the approximation prob-lem. The proof exploits inequalities induced by convexityof a related functional, and extends our earlier results forthe rank-one super-symmetric approximation problem tothe rank-r case.

Phillip RegaliaCatholic Universityregalia@cua.edu

MS34

An Algebraic Multilevel Approach for Large Scale3D Helmholtz Equations in Heterogenous Media

An algebraic multilevel preconditioner is presented for theHelmholtz equation in heterogeneous media. It is basedon a multi-level block ILU that preserves the underlyingcomplex symmetry. The preconditioner uses an algebraiccoarsening strategy based on controlling ‖L−1‖ for numer-ical stability. In combination with shifting and correct-ing the systems, a relatively robust preconditioner is con-tructed. Our numerical results demonstrate the efficiencyof the preconditioner, even in the high-frequency regime.This is joint work with Marcus Grote and Olaf Schenk fromthe University of Basel and the ZUniversity of Castellon.

Matthias BollhoeferTU Braunschweigm.bollhoefer@tu-bs.de

MS34

Parallel Algebraic Hybrid Linear Solver for Fre-quency Domain Acoustic Wave Modeling

In this talk, we will present a parallel algebraic non-overlapping domain decomposition methods for the so-lution of the Helmholtz equations involved in frequency-domain full-waveform inversion. The preconditioning ap-proach is based on the shift Laplacian technique. The nu-merical behaviour and the parallel performance of the lin-

ear solver will be investigated on large 2D and 3D models.

Azzam HaidarINPT-ENSEEIHT-IRIT, Francehaidar@cerfacs.fr

Luc GiraudINRIAToulouse, Francegiraud@n7.fr

H. Benhadjali, S. OpertoUMR Geosciences Azurbenhadj@geoazur.unice.fr, stephane.operto@geoazur.obs-vlfr.fr

MS34

An AMG Preconditioner Based on Damped Oper-ators for Time-harmonic Wave Equations

An algebraic multigrid approximation of the inverse of anphysically damped operator is used as a preconditionerfor time-harmonic scattering problems in fluids and solids.The AMG uses a graph based coarsening and an under-relaxed Jacobi smoother. Numerical experiments demon-strate the behavior of the method in complicated domains.The number of GMRES iterations grows roughly linearlywith respect to the frequency. This approach leads to anefficient solution procedure for low and medium frequencyproblems.

Jari ToivanenStanford Universitytoivanen@stanford.edu

MS34

Spectral Analysis of the Helmholtz Operator Pre-conditioned with a Shifted Laplacian

We present a comprehensive spectral analysis of theHelmholtz operator preconditioned with a shifted Lapla-cian. By combining the results of this analysis with anupper bound on the GMRES-residual norm we are ableto provide an optimal value for the shift, and to explainthe mesh-depency of the convergence of GMRES precon-ditioned with a shifted Laplacian. We will illustrate ourresults with a seismic test problem.

Martin van GijzenNumerical Analysis GroupDelft University of TechnologyM.B.vanGijzen@tudelft.nl

Yogi ErlanggaDept. of Earch SciencesUniv. of British Columbiayerlangga@eos.ubc.ca

Kees VuikT. U. Delftc.vuik@math.tudelft.nl

MS35

Structures and Dimension of Linearizations of Sin-gular Matrix Polynomials

Recently, the authors have found strong linearizations

78 LA09 Abstracts

which are valid for both regular and singular square matrixpolynomials and that allow us to recover the whole eigen-structure of the polynomial. Nonetheless, the problem offinding structured linearizations of structured singular ma-trix polynomials remains open. In this talk, we study thisproblem for T -palindromic polynomials. In particular, weshow how to construct T -palindromic linearizations of T -palindromic polynomials with odd degree. In the case ofeven degree we provide necessary and sufficient conditionsfor the existence of T -palindromic linearizations, most ofwhich have lower dimension than the usual one.

Fernando De Teran, Froilan M. DopicoDepartamento de MatematicasUniversidad Carlos III de Madridfteran@math.uc3m.es, dopico@math.uc3m.es

D. Steven MackeyDepartment of MathematicsWestern Michigan Universitysteve.mackey@wmich.edu

MS35

Invariant Pairs for Polynomial and NonlinearEigenvalue Problems

We consider matrix eigenvalue problems that are polyno-mial or genuinely nonlinear in the eigenvalue parameter.One of the most fundamental differences to the linear caseis that distinct eigenvalues may have linearly dependenteigenvectors or even share the same eigenvector. This canbe a severe hindrance in the development of general numer-ical schemes for computing several eigenvalues of a poly-nomial or nonlinear eigenvalue problem, either simultane-ously or subsequently. The purpose of this talk is to showthat the concept of invariant pairs offers a way of repre-senting eigenvalues and eigenvectors that is insensitive tothis phenomenon. We will demonstrate the use of this con-cept with a number of numerical examples. This is partlyjoint work with Timo Betcke, University of Reading.

Daniel KressnerETH Zurichdaniel.kressner@sam.math.ethz.ch

MS35

Diagonalizing Quadratic Matrix Polynomials

A quadratic eigenvalue problem (Mλ2 + Dλ + K)x = 0with M nonsingular is said to be diagonalizable if its lin-earization

[D MM 0

]λ +

[−K 00 M

]

is diagonalizable by equivalence or congruence transfor-mations, as appropriate. We characterize all admissiblecanonical forms. Also, we identify isomorphisms betweenthe sets of transformations and subsets of the centralizerof the Jordan form. This is a report on collaborative workwith Ion Zaballa.

Peter LancasterDepartment of Mathematics and StatisticsUniversity of Calgary, Canadalancaste@ucalgary.ca

MS35

Smith Forms of Structured Matrix Polynomials

A much-used computational approach to polynomial eigen-problems starts with a linearization of the underlying ma-trix polynomial P , such as the companion form, and thenapplies a general-purpose method like the QZ algorithm tothis linearization. But when P is structured, it can be ad-vantageous to use a linearization with the “same’ structureas P , if one can be found. In this talk we discuss the scopeof this structured linearization strategy for the classes of“alternating’ and “palindromic’ matrix polynomials, usingthe Smith form as the central tool.

Steve MackeyWestern Michigan Universitysteve.mackey@wmich.edu

Niloufer MackeyWestern Michigan UniversityDept of Mathematicsnil.mackey@wmich.edu

Christian MehlTechnische Universitaet BerlinInstitut fur Mathematikmehl@math.tu-berlin.de

Volker MehrmannInst. f.Mathematik, TU Berlinmehrmann@math.tu-berlin.de

MS36

Fast and Accurate Computations with SomeClasses of Quasiseparable Matrices

In the last decade many algorithms have been developedto perform fast computations with quasiseparable matri-ces by working with the quasiseparable generators. In gen-eral, the accuracy and stability of these algorithms is notguaranteed. We present in this talk some subsets of qua-siseparable matrices that allow us to perform both fast andaccurate computations if a proper parametrization is used.These subsets include totally nonnegative quasiseparablematrices and zero diagonal symmetric and skew-symmetricquasiseparable matrices.

Froilan M. DopicoDepartment of MathematicsUniversidad Carlos III de Madriddopico@math.uc3m.es

Tom BellaUniversity of Rhode IslandDepartment of Mathematicstombella@math.uri.edu

Vadim OlshevskyDept. of MathematicsU. of Connecticut, Storrs.olshevsky@uconn.edu

MS36

A Survey of Some Recent Results for Quasisepara-

LA09 Abstracts 79

ble Matrices and their Applications

Abstract not available at time of publication.

Vadim OlshevskyDept. of MathematicsU. of Connecticut, Storrs.olshevsky@uconn.edu

MS36

Wavelet Matrices as Green’s Matrices with NewFactorizations

Olshevsky and Zhlobich have identified a special classof five-diagonal matrices with a remarkable factorization.These matrices are 2 by 2 block bidiagonal. They are prod-ucts of 2 by 2 block diagonal matrices in which the blocksof the second factor are shifted by a row and column. Weidentify these factors in the case of the famous Daubechies-4 wavelet filters. These are block Toeplitz and the newfactorization shows how easily they could be made time-varying (and finite length) without losing their orthogonal-ity.

Gilbert StrangMassachusetts Institute of Technologygs@math.mit.edu

MS36

A Quasiseparable Matrices Approach to CMV andFiedler Matrices

Several new classes of structured matrices have appearedrecently in the scientific literature. Among them there areso-called CMV and Fiedler matrices which are found to berelated to polynomials orthogonal on the unit circle andHorner polynomials respectively. Both matrices are fivediagonal and have a similar structure, although they haveappeared under completely different circumstances. We es-tablish a link between these matrices by showing that theyboth belong to the wider class of twisted Hessenberg-Order-One quasiseparable matrices. We also obtain a general de-scription of five-diagonal matrices in terms of recurrencerelations satisfied by polynomials they are related to.

Pavel Zhlobich, Vadim OlshevskyUniversity of ConnecticutDepartment of Mathematicszhlobich@math.uconn.edu, olshevsky@math.uconn.edu

Gilbert StrangMassachusetts Institute of Technologygs@math.mit.edu

MS37

Modified, Regularized Total Least Norm Approachto Signal Restoration

Total Least Norm (TLN) is a common choice for model-ing blind deconvolution problems. In this talk, we presenta modified, regularized TLN model for signal deblurringproblems where both the blurring operator and the blurredsignal contain noise. We introduce an alternating methodthat uses this model to obtain better restoration not onlyof the signal, but also of the blurring operator.

Malena I. EspanolGraduate Aeronautical LaboratoriesCalifornia Institute of Technology

mespanol@caltech.edu

Misha E. KilmerTufts Universitymisha.kilmer@tufts.edu

MS37

Non-smooth Solutions to Least Squares Problems

In an attempt to overcome the ill-posedness or ill-conditioning of inverse problems, regularization methodsare implemented by introducing assumptions on the solu-tion. Common regularization methods include total vari-ation, L-curve, Generalized Cross Validation (GCV), andthe discrepancy principle. It is generally accepted thatall of these approaches except total variation unnecessarilysmooth solutions, mainly because the regularization oper-ator is in an L2 norm. Alternatively, statistical approachesto ill-posed problems typically involve specifying a prioriinformation about the parameters in the form of Bayesianinference. These approaches can be more accurate thantypical regularization methods because the regularizationterm is weighted with a matrix rather than a constant.The drawback is that the matrix weight requires informa-tion that is typically not available or is expensive to cal-culate. Ø The χ2 method developed by the author andcolleagues can be viewed as a regularization method thatuses statistical information to find matrices to weight theregularization term. We will demonstrate that unique andsimple L2 solutions found by this method do not unec-essarily smooth solutions when the regularization term isaccurately weighted with a diagonal matrix.

Jodi MeadBoise State UniversityDepartment of Mathematicsjmead@boisestate.edu

MS37

Using Confidence Ellipsoids to Choose the Regu-larization Parameter for Ill-Posed Problems

Confidence ellipsoids for discretized ill-posed problems areelongated in directions corresponding to small singular val-ues. Intersecting a confidence ellipsoid with another ellip-soid guaranteed to contain the solution gives better local-ization of that solution with the same confidence level. Wepresent a regularization method with a one-to-one corre-spondence between values of the regularization parameterand convex combinations of the two ellipsoids. The pa-rameter value is optimized by minimizing the size of theconvex combination.

Bert W. RustNational Institute of Standards and Technnologybert.rust@nist.gov

MS37

Regularization in Polynomial Approximations,Resolution of the Runge Phenomenon

The polynomial interpolation based on a uniform gridyields the well-known Runge phenomenon. The maximumpoint-wise error is unbounded for functions with complexroots in the Runge zone. In this work, we first investigatethe Runge phenomenon with finite precision operations.Then a truncation method based on the truncated singu-lar value decomposition is proposed to resolve the Runge

80 LA09 Abstracts

phenomenon. The method consists of two stages: In thefirst stage a statistical filtering matrix is applied to the in-terpolation matrix as preconditioner. In the second stage apseudo inverse of the interpolation matrix is formed usinga truncated singular value decomposition. We investigatethe structure of the singular vectors and singular valuesand determine a truncation point based on the oscillatorybehavior of the signular vectors and decay behavior of thesingular values. We then show with numerical examplesthat exponential decay of the approximation error can beachieved if an appropriate smoothness parameter that de-pends on the interpolated function is chosen.

Wolfgang StefanUniversity at Buffalo, SUNYwstefan@buffalo.edu

Jae-Hun JungDepartment of MathematicsSUNY at Buffalojaehun@buffalo.edu

MS38

Block Preconditioners for Generalized Saddle-Point Matrices

Abstract not available at time of publication.

Zhong-Zhi BaiInstitute of Computational MathematicsChinese Academy of Sciences, Beijing, Chinabzz@lsec.cc.ac.cn

MS38

A Geometric View of Krylov Subspace Methods onSingular Systems

We show why one can apply CG to consistent systemsAx = b where A is symmetric and positive semi-definite,by decomposing CG into the R(A) and N (A) components,which shows that the method is essentially equivalent toCG applied to a positive diagonal system in R(A). Next,we analyze GMRES for systems where A is nonsymmetricand singular by decomposing GMRES into the R(A) andR(A)⊥ components, providing a geometric interpretationof the convergence conditions given by Brown and Walker.

Ken HayamiNational Institute of Informaticshayami@nii.ac.jp

Masaaki SugiharaGraduate School of Information Science and Technology,the University of Tokyom sugihara@mist.i.u-tokyo.ac.jp

MS38

On Restrictively Preconditioned HSS IterationMethods for Non-Hermitian Positive-Definite Lin-ear Systems

A restrictively preconditioned Hermitian/skew-Hermitiansplitting (RPHSS) iteration method is presented for thesolution of the non-Hermitian and positive-definite systemof linear equations. Theoretical analyses on the conver-gence condition for RPHSS method , the optimal param-eters and the choice of preconditioners are given. From

practical point of view, the implementation and conver-gence of the inexact RPHSS method are also discussed indetail. Finally, a number of numerical experiments are usedto show that RPHSS method is efficient and comparableto standard HSS iteration method.

Jun-Feng YinDepartment of MathematicsTongji University, Shanghai, P.R. Chinayinjf@tongji.edu.cn

MS38

On Parameterized Uzawa Methods for Saddle-Point Problems

Abstract not available at time of publication.

Bing ZhengSchool of Mathematics and StatisticsLanzhou University, Lanzhou, P.R. Chinabzheng@lzu.edu.cn

MS39

Coupled Tensor and Matrix Factorizations

Matrix and tensor factorizations have proved to be pow-erful tools in data analysis. There are limited, though, todata that can be represented as a single object. We proposeto consider problems where the data has multiple aspects,represented by multiple matrices and tensors with sharedmodes. Our goal is to analyze these data simultaneouslyvia coupled decompositions. Specifically, we propose all-at-once optimization techniques, which are more effective inour experiments than alternating methods. In this talk, wedescribe the problem, present several examples, and showcomputational results.

Evrim Acar, Tamara G. KoldaSandia National Laboratorieseacarat@sandia.gov, tgkolda@sandia.gov

Danny DunlavyComputer Science and Informatics DepartmentSandia National Laboratoriesdmdunla@sandia.gov

MS39

A Fast and Efficient Algorithm for Low-rank Ap-proximation of a Matrix and Generalization to Ten-sor

Abstract not available at time of publication.

Nam H. NguyenDepartment of Electrical and Computer EngineeringJohns Hopkins Universitynam@jhu.edu

MS39

Commuting Birth-and-Death Processes

We use methods from combinatorics and algebraic statis-tics to study analogues of birth-and-death processes thathave as their state space a finite subset of the m-dimensional lattice and for which the m matrices thatrecord the transition probabilities in each of the lattice di-rections commute pairwise. One reason such processes areof interest is that the transition matrix is straightforward

LA09 Abstracts 81

to diagonalize, and hence it is easy to compute n step tran-sition probabilities. The set of commuting birth-and-deathprocesses decomposes as a union of toric varieties, with themain component being the closure of all processes whosenearest neighbor transition probabilities are positive. Weexhibit an explicit monomial parametrization for this maincomponent, and we explore the boundary components us-ing primary decomposition.

Bernd SturmfelsUniversity of California , Berkeleybernd@Math.Berkeley.EDU

Steve Evans, Caroline UhlerDepartment of StatisticsUC Berkeleyevans@stat.berkeley.edu, cuhler@stat.berkeley.edu

MS39

Finitely Generated Cumulants: Developments andApplications

This continues work on algebraic aspects of cumulants de-veloped in Pistone and Wynn (1999, Statistica Sinica) and(2000, J. Symb. Comp). The definition of the finitely gen-erated cumulant (FGC) property is that, in the univariateor multivariate case, if K(s) is the cumulant generatingfunction for a distribution then the vector of first partialderivatives with respect to s, namely K’, and the matrixof second order (partial) derivatives, K’, satisfy an implicitpolynomial equation f(K’,K’) = 0. This generalizes theMorris class, in which K’ is a quadratic function of K’. TheFGC property is a way of introducing polynomial algebra,and symbolic methods, into continuous distribution theoryand a surprisingly large class of distributions satisfy theFGC property. Applications include the asymptotic theoryof maximum likelihood, saddle-point approximations, mul-tivariate dependency for non-Gaussian random variables,mixture models and information geometry.

Henry WynnLondon School of Economicsh.wynn@lse.ac.uk

MS40

A Key for the Choice of the Subspaces in RestartedKrylov Methods

The restarted Krylov subspace methods allow computingsome eigenpairs of large sparse matrices. The size of thesubspace in these methods is chosen empirically. A poorchoice of this size could lead to the non-convergence of themethods. We propose a technique, based on the projectionof the problem on several subspaces instead of a single one,to remedy to this problem. Our approach is validated byits application on IRA method.

Nahid EmadUniversity of Versailles, PRiSM LaboratoryNahid.Emad@prism.uvsq.fr

MS40

A Numerical Method Based on the Residue The-orem for Computing Eigenvalues in Multiply Con-nected Region

In this talk, we consider the solution of eigenvalues in a fi-nite and multiply connected region. Such problems arise in

the application of photonic crystal waveguides and there isa strong need for the fast solution of the problems. For solv-ing the problems efficiently, we extend the Sakurai-Sugiuramethod which has been proposed for simply connected re-gion. We also analyze the error of the computed eigenval-ues.

Takafumi Miyata, Lei Du, Tomohiro Sogabe, YusakuYamamoto, Shao-Liang ZhangNagoya University, Japanmiyata@na.cse.nagoya-u.ac.jp, lei-du@na.cse.nagoya-u.ac.jp, sogabe@na.cse.nagoya-u.ac.jp,yamamoto@na.cse.nagoya-u.ac.jp, zhang@na.cse.nagoya-u.ac.jp

MS40

Efficient Subspace Alignment Technique for theSelf-Consistent Field Iterations

The subspace alignment is a crucial preprocessing step forthe self-consistent field iteration to solve nonlinear eigen-value problems arising from electronic structure calcula-tions. Its computational kernel involves the matrix po-lar decomposition. The popular scaled Newton methodrequires explicit matrix inversion, which is dominated bycommunication in distributed and multi-core computing.In this talk, we present a new polar decomposition algo-rithm based on the QR-decomposition (without pivoting)and a communication optimal implementation of the sub-space alignment.

Yuji NakatsukasaUniversity of California, DavisApplied Mathynakatsukasa@ucdavis.edu

Zhaojun BaiUniversity of Californiabai@cs.ucdavis.edu

Francois GygiUniversity of California, DavisLawrence Livermore National Laboratoryfgygi@ucdavis.edu

MS40

A Hierarchical Parallel Method for Solving Nonlin-ear Eigenvalue Problems

In this talk, we present a parallel method for finding eigen-values in a given domain and corresponding eigenvectors ofnonlinear eigenvalue problems. In our method, the origi-nal problem is converted to a smaller generalized eigenvalueproblem, which is obtained numerically by solving a set oflinear equations. These linear equations are independentand can be solved in parallel.

Tetsuya SakuraiDepartment of Computer ScienceUniversity of Tsukubasakurai@cs.tsukuba.ac.jp

Junko AsakuraResearch and Development DivisionSquare Enix Co. Ltd.asakura@mma.cs.tsukuba.ac.jp

Hiroto TadanoDepartment of Computer Science

82 LA09 Abstracts

University of Tsukubatadano@cs.tsukuba.ac.jp

Tsutomu IkegamiInformation Technology Research InstituteAISTt-ikegami@aist.go.jp

Kinji KimuraDepartment of Applied MathematicsKyoto Universitykkimur@amp.i.kyoto-u.ac.jp

MS41

Fast Algorithms for Approximating the Pseu-dospectral Abscissa and Radius

The ε-pseudospectral abscissa and radius of an n×n matrixare respectively the maximum real part and the maximalmodulus of points in its ε-pseudospectrum. Existing tech-niques compute these quantities accurately but the cost ismultiple SVDs of order n. We present a novel approachbased on computing only the spectral abscissa or radius ora sequence of matrices, generating a monotonic sequence oflower bounds which, in many but not all cases, convergesto the pseudospectral abscissa or radius.

Nicola GuglielmiUniversity of L’AquilaItalyguglielm@univaq.it

Michael L. OvertonNew York UniversityCourant Instit. of Math. Sci.overton@cs.nyu.edu

MS41

Structured Pseudospectra of Hamiltonian Matrices

We consider the variation of the spectrum of Hamiltonianmatrices under Hamiltonian perturbations. The first partof the talk deals with the associated structured pseudospec-tra. We show how to compute these sets and give someexamples. In the second part we discuss the robustnessof linear stability. In particular we determine the smallestnorm of a perturbation that makes the perturbed Hamil-tonian matrix unstable.

Michael KarowTechnical University of BerlinGermanykarow@math.tu-berlin.de

MS41

Multiple Eigenvalues with Prespecified Multiplici-ties and Pseudospectra

Suppose z is a point in the complex plane where two com-ponents of the epsln-pseudospectrum of A coalesce. Alamand Bora deduced the existence of matrices within epsln-neighborhood of A with z as a multiple eigenvalue. There-fore smallest epsln such that two components of the epsln-pseudospectrum coalesce is the distance to the nearest ma-trix with a multiple eigenvalue. Here we establish the con-nection between the pseudospectra and nearest matrices

with eigenvalues with prespecified algebraic multiplicity.

Emre MengiDept. of Mathematics, UC San Diegoemengi@math.ucsd.edu

MS41

Characterization and Construction of the NearestDefective Matrix via Coalescence of Pseudospectra

Let w(A) be the distance from A to the set of defectivematrices. and let c(A) be the supremum of all ε for whichthe pseudospectrum of A has n distinct components. Itis known that w(A) ≥ c(A), with equality holding for the2-norm. We show that w(A) = c(A) for the Frobeniusnorm too, and that the minimal distance is attained by adefective matrix in all cases. The results depend on thegeometry of the pseudospectrum near points where coales-cence of the components occurs.

Michael L. OvertonNew York UniversityCourant Instit. of Math. Sci.overton@cs.nyu.edu

MS42

Hessenberg Quasiseparable Matrices and Polyno-mials

The classification of tridiagonal matrices in terms of poly-nomials orthogonal on a real interval, as well as the classi-fication of unitary Hessenberg matrices in terms of Szegopolynomials (orthogonal on the unit circle) are well–knownresults, and form the basis for many fast algorithms involv-ing these classes. In this talk, we give complete classifica-tions of matrices having Hessenberg–quasiseparable struc-ture in terms of the polynomials related in the same way asthe above examples. Several subclasses are also classified,including Hessenberg–semiseparable matrices, and the mo-tivating special cases of tridiagonal and unitary Hessenbergmatrices.

Tom BellaUniversity of Rhode IslandDepartment of Mathematicstombella@math.uri.edu

MS42

Characteristic Polynomials, Eigenvaluesand Eigenspaces of Quasiseparable of Order OneMatrices

We discuss spectral properties of quasiseparable of orderone matrices. This class of matrices contains at least threewell-known classes: diagonal plus semiseparable matrices,tridiagonal matrices, unitary Hessenberg matrices. We de-rive different recurrence relations for characteristic polyno-mials of principal leading submatrices of a quasiseparablematrix. Some basic algorithms to compute eigenvalues arepresented. We obtain conditions when an eigenvalue of aquasiseparable matrix is simple. For the case of a multipleeigenvalue the structure of the corresponding eigenspace isdescribed. Illustrative examples are presented.

Yuli EidelmanTel Aviv UniversityDepartment of Mathematicseideyu@post.tau.ac.il

LA09 Abstracts 83

Israel Gohberg, Iulian HaimoviciDepartment of MathematicsTel-Aviv Universitygohberg@post.tau.ac.il, iulianh@zahav.net.il

Vadim OlshevskyUniversity of ConnecticutDepartment of Mathematicsolshevsky@math.uconn.edu

MS42

A Fast Euclidean Algorithm for QuasiseparablePolynomials via Non-casual Factorizations

We have elaborated that there exist a one to one corre-spondence of Classical Euclid algorithm with GKO, HOand OS algorithms when we express the Euclid algorithmin one way of the matrix form. It is noticeable that Be-zoutian plays a major role in here by preserving differentdisplacement structures according to the above three algo-rithms. All these result drive us to design a fast Euclidalgorithm for polynomial represent in general bases andthe result can be specialized for cases of Quasiseparable,Orthogonal and Szego polynomials.

Sirani PereraUniversity of ConnecticutDepartment of Mathematicsmperera@math.uconn.edu

Vadim Olshevskyco-authorvadim.olshevsky@uconn.edu

MS42

Parameterization and Stability of Methods forQuasiseparable Matrices

For the standard parameterization of a quasiseparable ma-trix, a small perturbation of the parameters ( i.e. smallstructured error) do not correspond to a small perturba-tion of the matrix (i.e. small normwise error). As a conse-quence, when working with a quasiseparable matrix, someadditional stability assumption on the parameters is re-quired to guarantee normwise backward stability. This talkdescribes alternate parameterizations based on plane rota-tions for which small structured error implies small norm-wise error.

Michael StewartGeorgia State UniversityDepartment of Mathematicsmastewart@gsu.edu

MS43

A More Efficient Version of the Lucas-KanadeTemplate Tracking Algorithm Using the Ulv De-composition

Template tracking refers to the problem of tracking anobject through a video sequence using a template. Thetemplate is the image of the tracked object, usually ex-tracted from the first frame. The aspect of template track-ing that we are interested in, is the problem of updatingthe template. Template updates are necessary, especiallyin long video sequences, because the appearance of the ob-ject changes significantly and the initial template is soonobsolete. In this work we suggest improvements, in a com-

putational sense, to the template update algorithm pro-posed by Matthew et. al (The Template Update Problem).The update strategy proposed in Matthew et. al. requirescomputation of the principal components of the augmentedimage matrix at every iteration, since the PCs correspondto the left singular vectors of image matrix, we suggest us-ing URV updates as an alternative to computing PCs abinitio at every iteration.

Jesse L. BarlowPenn State UniversityDept of Computer Science & Engbarlow@cse.psu.edu

Anupama ChandrasekharPennsylvania State Universityanupama.psu@gmail.com

MS43

Analysis of Model Uncertainty Using OptimizationIterates

Throughout the course of an optimization run, multiplepoints are generated and evaluated. In this talk, we willdescribe how this data can be used to evaluate parametersensitivities and uncertainties while providing additionalinsight into the model. We will also describe how the opti-mization iterate set can be supplemented to provide morestatistically significant results and how these results mightbe computed during the course of the optimization to im-prove the search results.

Genetha GraySandia National Laboratoriesgagray@sandia.gov

Katie FowlerClarkson Universitykfowler@clarkson.edu

Matthew GraceSandia National Labsmgrace@sandia.gov

MS43

Multisplitting for Regularized Least Squares: AMultiple Right Hand Side Problem for ImageRestoration and Reconstruction

Least squares problems are one of the most often used nu-merical formulationsin engineering. Many such problemslead to ill-posed systems of equations for which a solutionmay be found by introducing regularization. Here we ex-tend the use of multisplitting least squares, as originallyintroduced by Renaut (1998) for well-posed least squaresproblems, to Tikhonov regularized large scale least squaresproblems. Regularization at both the global and subprob-lem level is considered, hence providing a means for mul-tiple parameter regularization of large scale problems. Wefind out that in solving the local splitting, each local prob-lem turn out to be a linear system with multiple righthand sides with updates. Utilizing the characteristics ofthe problem itself enables us to apply more efficient al-gorithm to obtain the solution. Basic convergence resultsfollow immediately from the original formulation. Numeri-cal validation is presented for some simple one dimensionalsignal restoration simulations.

Hongbin Guo

84 LA09 Abstracts

Arizona State Universityhb guo@asu.edu

Rosemary A. RenautArizona State UniversityDepartment of Mathematics and Statisticsrenaut@asu.edu

Youzuo LinArizona State Universityylin44@asu.edu or ylin@mathpost.asu.edu

MS43

Accelerating the EM Algorithm

The EM algorithm is widely used for numerically approx-imating maximum-likelihood estimates in the context ofmissing information. This talk will focus on the EM al-gorithm applied to estimating unknown parameters in a(finite) mixture density, i.e., a probability density function(PDF) associated with a statistical population that is amixture of subpopulations, using “unlabeled” observationson the mixture. In the particular case when the subpopu-lation PDFs are from common parametric families, the EMalgorithm becomes a fixed-point iteration that has a num-ber of appealing properties. However, the convergence ofthe iterates is only linear and may be unacceptably slow ifthe subpopulations in the mixture are not “well-separated”in a certain sense. In this talk, we will review the EM al-gorithm for mixture densities, discuss a certain method foraccelerating convergence of the iterates, and report on nu-merical experiments.

Homer F. WalkerWorcester Polytechnic Institutewalker@wpi.edu

MS44

Development and History of Sparse Direct Meth-ods

Direct methods for the solution of large sparse systemswere used by the linear programming community from the1950s but it was not until the 1960s that they were appliedto a wider range of applications including the solution ofstiff ODEs and power systems. We sketch these early ori-gins and highlight the key points in the development ofsparse direct techniques and software. We also examinethe history of the direct v iterative debate and its currentresolution.

Iain DuffRutherford Appleton Laboratory, Oxfordshire, UK OX110QXand CERFACS, Toulouse, Franceiain.duff@stfc.ac.uk

MS44

Some History of Conjugate Gradients and OtherKrylov Subspace Methods

In the late 1940’s and early 1950’s, newly available comput-ing machines generated intense interest in solving “large’systems of linear equations. Among the algorithms devel-oped were several related methods, all of which generatedbases for Krylov subspaces and used the bases to minimizeor orthogonally project a measure of error. The best knownof these algorithms is conjugate gradients. We discuss the

origins of these algorithms, emphasizing research themesthat continue to have central importance.

Dianne P. O’LearyUniversity of Maryland, College ParkDepartment of Computer Scienceoleary@cs.umd.edu

MS44

Antecedents of the LR and QR Algorithms

The seminal idea in LR and QR is the reversal of factorsto form a new matrix. In 1882 Darboux introduced atransform that takes a 2nd order linear operator (ODE)and writes it as the composition of two 1st order linearoperators PQ and forms a new 2nd order operator QPas the desired transform. If you discretize properly youget an instance of the basic LR transform applied to atridiagonal matrix. In the 20th century a few physicists(Weyl,Schroedinger, Infeld) applied the transform to ob-tain eigenfunctions for operators that occur in applications.In that work the transform was applied a small numberof times, the goal was not to find eigenvalues, and shifts,though present, were incidental and not a tool to accelerateconvergence.

Beresford N. ParlettUniversity of CaliforniaDepartment of Mathematicsparlett@math.berkeley.edu

MS44

The Dual Flow Between Linear Algebra and Opti-mization

Optimization and linear algebra have been closely con-nected for more than 60 years, ever since Courant’s 1943 in-troduction of the quadratic penalty function and Dantzig’s1947 creation of the simplex method for linear program-ming. A multitude of linear algebraic subproblems appearin optimization methods, which often impose special struc-ture on the associated matrices. This talk will highlight aselection of symmetrically productive ties between linearalgebra and optimization, ranging from classical to ultra-modern.

Margaret H. WrightNew York UniversityCourant Institute of Mathematical Sciencesmhw@cs.nyu.edu

MS45

Regularizing Iterations for ODF Reconstruction

We use regularizing iterations to reconstruct local crys-tallographic orientations from X-ray diffraction measure-ments, and we demonstrate that right preconditioning isnecessary to provide satisfactory reconstructions. Ourright preconditioner is not a traditional one that acceler-ates convergence; its purpose is to modify the smoothnessproperties of the reconstruction. We also show that a newstopping criterion, based on the information available inthe residual vector, provides a robust choice of the numberof iterations.

Per Christian HansenDTU InformaticsTechnical University of Denmarkpch@imm.dtu.dk

LA09 Abstracts 85

Henning Osholm SørensenMaterials Research DivisionRis{o} DTUosho@risoe.dtu.dk

MS45

A Generalized Hybrid Regularization Approach forPartial Fourier MRI Reconstruction

Accelerated MR imaging techniques require regularizationto ensure robust solutions which must be computed effi-ciently in a manner consistent with the speed of data acqui-sition. One difficulty is the efficient determination of one ormore regularization parameters. We develop a generaliza-tion of the LSQR-Hybrid approach for the partial-Fourierreconstruction problem that allows us to efficiently selecttwo parameters via information generated during variousruns of the LSQR-Hybrid algorithm. Phantom and in-vivoresults will be presented.

Misha E. KilmerDepartment of MathematicsTufts Universitymisha.kilmer@tufts.edu

W. Scott HogeBrigham and Womens HospitalHarvard Medical School, Boston, MAshoge@ieee.org

MS45

Golub-Kahan Hybrid Regularization in Image Pro-cessing

Ill-posed problems arise in many image processing applica-tions, including microscopy, medicine and astronomy. It-erative methods are typically recommended for these largescale problems, but they can be difficult to use in practice.Lanczos based hybrid methods have been proposed to slowthe introduction of noise in the iterates. In this talk wediscuss the behavior of Lanczos based hybrid methods forlarge scale problems in image processing.

James G. NagyMathematics and Computer ScienceEmory Universitynagy@mathcs.emory.edu

Julianne ChungUniversity of Marylandjmchung@cs.umd.edu

MS45

Golub-Kahan Iterative Bidiagonalization and Re-vealing Noise in the Data

Consider an ill-posed problem with a noisy right-hand side(observation vector), where the size of the (white) noise isunknown. We show how the information from the Golub-Kahan iterative bidiagonalization can be used for revealingthe unknown level of the noise. Such information can beuseful in construction of stopping criteria in solving largeill-posed problems.

Zdenek StrakosInstitute of Computer ScienceAcademy of Sciences of the Czech Republic

z.strakos@gmail.com

Iveta HnetynkovaCharles University, Praguehnetynkova@cs.cas.cz

Martin PlesingerTechnical University Liberecmartin.plesinger@gmail.com

MS46

Randomized Algorithms in Linear Algebra: FromApproximating the Singular Value Decompositionto Solving Regression Problems

The introduction of randomization in the design and anal-ysis of algorithms for matrix computations (such as matrixmultiplication, least-squares regression, the Singular ValueDecomposition (SVD), etc.) over the last decade provideda new paradigm and a complementary perspective to tradi-tional numerical linear algebra approaches. These novel ap-proaches were motivated by technological developments inmany areas of scientific research that permit the automaticgeneration of large data sets, which are often modeled asmatrices. In this talk we will outline how such approachescan be used to approximate problems ranging from ma-trix multiplication and the Singular Value Decomposition(SVD) of matrices to approximately solving least-squaresproblems and systems of linear equations.

Petros DrineasComputer Science DepartmentRennselaer Polytechnic Institutedrinep@cs.rpi.edu

MS46

Fast Factorization Of Low-Rank Matrices via Ran-domized Sampling

The talk will describe a set of techniques based on ran-domized sampling that dramatically accelerate several keymatrix computations, including many needed for solvingpartial differential equations and for extracting informationfrom large data-sets (such as those arising from genomics,the link structure of the World Wide Web, etc). The algo-rithms are applicable to any matrix that can in principlebe approximated by a low-rank matrix, are as accurateas deterministic methods, and are for practical purposes100% reliable (the risk of ”failure” can easily be renderedless than e.g. 1e− 12). Several numerical examples will bepresented in which the randomized methods are applied tosolve problems arising in potential theory, acoustic scatter-ing, image processing, and other application areas.

Per-Gunnar MartinssonUniversity of Colorado at BoulderApplied Mathematics Departmentmartinss@colorado.edu

MS46

The Design and Analysis of Randomized Algo-rithms for High-dimensional Data: New Insightsfrom Multilinear Algebra and Statistics

In recent years, the spectral analysis of appropriately de-fined kernels has emerged as a principled way to ex-tract the low-dimensional structure often prevalent in high-dimensional data. From selecting so-called landmark train-

86 LA09 Abstracts

ing examples in supervised machine learning, to solvinglarge least-squares problems from classical statistics, ran-domized algorithms provide an appealing means of dimen-sionality reduction, helping to overcome the computationallimitations currently faced by practitioners with massivedatasets. In this talk we describe how new insights frommultilinear algebra and statistics can be brought to bearon the design and analysis of such algorithms, in orderto provide intuitive insight as well as concise formalisms.We discuss in particular the role of determinants and com-pound matrices as it arises in a variety of recent settingsof interest to the theoretical computer science community,and illustrate the practical implications of our results byway of example applications drawn from signal and imageprocessing. (Joint work with Mohamed-Ali Belabbas andother SISL members.)

Patrick J. WolfeHarvard UniversityDepartment of Statisticswolfe@stat.harvard.edu

MS47

Tournaments, Landau’s Theorem, Rado’s Theo-rem, and Partial Tournaments

Landau’s classical theorem gives necessary and sufficientconditions for the existence of a tournament with a pre-scribed score sequence. The theorem has been strength-ened to give the existence of a tournament with specialproperties. Rado’s theorem gives necessary and sufficientconditions for the existence of an independent transversalof a family of subsets of a set on which a matroid is de-fined. Rado’s theorem can be used to give a nice proof ofLandau’s theorem, indeed to give conditions for a partialtournament to be completed to a tournament with a pre-scribed score sequence. In this talk, we shall discuss thesetopics.

Richard A. BrualdiUniversity of WisconsinDepartment of Mathematicsbrualdi@math.wisc.edu

MS47

Hadamard Diagonalizable Graphs

The general question of interest is to characterize the undi-rected graphs G such that the Laplacian matrix associatedwith G can be diagonalized by some Hadamard matrix.During this presentation, I will survey many interestingand fundamental properties about these graphs along witha partial characterization of the cographs that can be di-agonalized by a Hadamard matrix.

Shaun M. FallatUniversity of ReginaCanadasfallat@math.uregina.ca

Steve KirklandUniversity of ReginaDept of Math and Statskirkland@math.uregina.ca

Sasmita BarikUniversity of Reginabarik@math.uregina.ca

MS47

Near Threshold Graphs

A conjecture of Grone and Merris states that for any graphG, its Laplacian spectrum, Λ(G), is majorized by its conju-gate degree sequence, D∗(G). That conjecture prompts aninvestigation of the relationship between Λ(G) and D∗(G),and Merris has characterized the graphs G for which themultisets Λ(G) and D∗(G) are equal. In this talk, we pro-vide a constructive characterization of the graphs G forwhich Λ(G) and D∗(G) share all but two elements.

Steve KirklandHamilton InstituteNational University of Irelandstephen.kirkland@nuim.ie

MS47

Scrambling Index of Primitive Matrices

The scrambling index of a primitive matrix A is the small-est positive integer k such that Ak(At)k = J , where At

denotes the transpose of A and J denotes all-ones matrix.We prove two upper bounds on the scrambling index ofprimitive matrices. The first bound is in terms of Booleanrank, and the second bound is in terms of the diameter andgirth of the adjacency matrix of A.

Jian ShenTexas State UniversitySan Marcos, TXjs48@txstate.edu

Mahmud AkelbekTexas State Universityam44@txstate.edu

Sandra FitalWeber State Universitysfitalakelbek@weber.edu

MS48

Fixed Points Theorems for Nonnegative Tensorsand Newton Methods

We discuss two theorems for nonnegative nonsymmetrictensors. First, the existence of a unique positive singularvalue and the corresponding singular vectors. Second, therescaling of nonnegative tensors to balanced tensors, i.e.the tensor version of Sinkhorn’s diagonal scaling to dou-bly stochastic matrices. In both cases the values of thesesolutions can be found efficiently by using the fix point the-orem. We then discuss the Newton method to speed up thecomputations for these fixed points.

Shmuel FriedlandUniversity of Illinois at Chicagofriedlan@uic.edu

MS48

Trust-region Method for the Best Multilinear RankApproximation of Tensors

We work on reliable and efficient algorithms for the bestlow multilinear rank approximation of higher-order ten-sors. This approximation is used for dimensionality re-duction and signal subspace estimation. In this talk, weexpress it as the solution of a minimization problem on a

LA09 Abstracts 87

quotient manifold. Applying the Riemannian trust-regionscheme with the truncated conjugate gradient method forthe trust-region subproblems, superlinear convergence isachieved. We comment on the advantages of the algorithm,discuss the issue of local optima and show some applica-tions.

Mariya Ishteva

K.U.Leuven, ESAT/SCDLeuven, BelgiumMariya.Ishteva@esat.kuleuven.be

Lieven De LathauwerKatholieke Universiteit LeuvenLieven.DeLathauwer@kuleuven-kortrijk.be

P.-A. AbsilDepartment of Mathematical EngineeringUniversit’{e} catholique de Louvainabsil@inma.ucl.ac.be

Sabine Van HuffelK.U.Leuven, Belgiumsabine.vanhuffel@esat.kuleuven.be

MS48

Multi-multilinear Methods: Computing with Sumsof Products of Sums of Products

Representing a multivariate function as a sum of prod-ucts of one-variable functions bypasses the curse of dimen-sionality. However, for the wavefunction of a quantum-mechanical system consisting of K non-interacting subsys-tems, one can show that the number of terms needed growsexponentially in K. A representation with sums of prod-ucts of sums of products can handle this case, and poten-tially provide a size-consistent representation for generalsystems.

Martin J. MohlenkampOhio Universitymjm@math.ohiou.edu

MS48

Effective Ranks of Tensors and New Decomposi-tions in Higher Dimensions

Decompositions and approximations of d-dimensional ten-sors are crucial either in structure recovery problems andmerely for a compact representation of tensors. How-ever, the well-known decompositions have serious draw-backs: the Tucker decompositions suffer from exponen-tial dependence on the dimensionality d while fixed-rankcanonical approximations are not stable. In this talkwe present new decompositions that are stable and havethe same number of representation parameters as canon-ical decompositions for the same tensor. Moreover, thenew format possesses nice stability properties of the SVD(as opposed to the canonical format) and is convenientfor basic operations with tensors (see: I.V.Oseledets andE.E.Tyrtyshnikov, Breaking the curse od dimensionality,or how to use SVD in many dimensions, Research Re-port 09-03, Kowloon Tong, Hong Kong: ICM HKBU, 2009(www.math.hkbu.edu.hk/ICM/pdf/09-03.pdf). Last notthe least, the new decomposition provides a useful method(comparable and even superior to wavelets) for compres-

sion of low-dimensional data.

Eugene TyrtyshnikovInstitute of Numerical Mathematics, Russian Academy ofSci.tee@inm.ras.ru

Ivan OseledetsInstitute of Numerical MathematicsRussian Academy of Sciences, Moscowivan.oseledets@gmail.com

MS49

Numerical Computation with Semiseparable Ma-trices: Distributed Algorithms for Quantum Trans-port with Atomistic Basis Sets (with Ragu Balakr-ishnan)

Two of the most demanding computational problems inquantum transport analysis involve mathematical oper-ations with semiseparable matrices. First, we considerdetermining the diagonal of a semiseparable matrix thatis the inverse of a block-tridiagonal matrix. The secondoperation corresponds to computing matrix products in-volving semiseparable matrices. We present a parallel in-version algorithm for block-tridiagonal matrices as wellas distributed approaches for computing matrix productsbased upon an underlying compact representation for thesemiseparable matrices.

Stephen CauleyPurdue UniversityElectrical and Computer Engineeringstcauley@purdue.edu

MS49

Semiseparable Matrices and Fast Transforms forOrthogonal Polynomials and Associated Functions

A new connection between classical orthogonal polynomi-als or the corresponding associated functions and semisep-arable matrices will be developed. It will be shown howthis can be exploited to obtain fast FFT-like algorithmsfor these functions. The results will be demonstrated withtwo recent methods to compute discrete Fourier transformson the sphere S2 and on the rotation group SO(3).

Jens KeinerUniversitaet zu LuebeckInstitut fur Mathematikkeiner@math.uni-luebeck.de

MS49

Hierarchical Matrix Preconditioners

Hierarchical (H-) matrices provide a powerful technique tocompute and store approximations to dense matrices in adata-sparse format. The basic idea is the appoximationof matrix data in hierarchically structured subblocks bylow rank representations. The usual matrix operations canbe computed approximately with almost linear complexity.We use such an h-arithmetic to set up preconditioners forthe iterative solution of sparse linear systems as they arisein the finite element discretization of PDEs.

Sabine Le BorneTennessee Technological UniversityDepartment of MathematicsSLeborne@tntech.edu

88 LA09 Abstracts

MS49

Root-finding, Structured Matrices and AdditivePreprocessing

The classical problem of polynomial root-finding was re-cently attacked highly successfully based on effective algo-rithms for approximation of eigenvalues of rank structuredmatrices. We advance this approach with some techniquesthat employ the displacement structure of the associatedmatrices and their additive preprocessing.

Victor PanCUNYLehmanv y pan@yahoo.com

MS50

Hybrid Techniques in the Solution of Large ScaleProblems

The most challenging problems for numerical linear algebraarguably arise from the discretization of partial differentialequations from three-dimensional modelling. Often theseproblems are intractable to either direct methods (becauseof fill-in) and direct methods (because of non-convergence).In this talk we will review some recent work by researchersat CERFACS and ENSEEIHT on the combined use of di-rect and iterative methods for solving very large linear sys-tems of equations arising in three-dimensional modelling.

Iain DuffRutherford Appleton Laboratory, Oxfordshire, UK OX110QXand CERFACS, Toulouse, Franceiain.duff@stfc.ac.uk

MS50

The Impact of Adaptive Solver Libraries on FutureMany-cores

Solving linear systems takes many forms. Matrix proper-ties (numerical and physical) and computational properties(program and architecture) affect performance delivered.The number of combinations is huge and growing over time.Adaptive libraries are essential to provide rational pathsthrough this thicket no individual can grasp the optimiza-tion issues. A parameter-space exhaustively-trained selec-tion system can make fast run-time selections for users. Wedescribe an adaptive Spike-Pardiso (PSpike) combinationpoly-algorithm providing excellent performance comparedto other direct and iterative solvers.

David KuckIntelParallel and Distributed Solutions Divisiondavid.kuck@intel.com

MS50

On the PSPIKE Parallel Sparse Linear SystemSolver

The availability of large-scale computing platforms com-prised of thousands of multicore processors motivates theneed for highly scalable sparse linear system solvers forsymmetric indefinite matrices. The solvers must opti-mize parallel performance, processor performance, as wellas memory requirements, while being robust across broadclasses of applications. We will present a new parallel solverthat combines the desirable characteristics of direct meth-

ods (robustness) and iterative solvers (computational cost),while alleviating their drawbacks (memory requirements,lack of robustness).

Olaf SchenkDepartment of Computer Science, University of BaselSwitzerlandolaf.schenk@unibas.ch

MS51

A Parallel Version of GPBiCR Method Suitable forDistributed Parallel Computing

Abstract not available at time of publication.

Tong-Xiang GuLaboratory of Computational PhysicsInstitute of Appl. Physics and Comp. Math., Beijing,Chinatxgu@iapcm.ac.cn

MS51

Structure Exploited Algorithms for Nonsymmet-ric Algebraic Riccati Equation Arising in TransportTheory

The nonsymmetric algebraic Riccati equation arising intransport theory is a special algebraic Riccati equationwhose coefficient matrices having some special structures.By reformulating the Riccati equation, they were foundthat its arbitrary solution matrix is of a Cauchy-like formand the solution matrix can be computed from a vectorform Riccati equation instead of the matrix form that. Inthis talk, we review some classical-type and Newton-typeiterative methods for solving the vector form Riccati equa-tion and present some work under investigation .

Linzhang LuSchool of Mathematical ScienceXiamen University, Xiamen, P.R. Chinalzlu@xmu.edu.cn llz@gznu.edu.cn

MS51

On Hybrid Preconditioning Methods for LargeSparse Saddle-Point Problems

Based on the block triangular product approximation to anenergy matrix, a class of hybrid preconditioning methodsis designed for accelerating the MINRES method for solv-ing the saddle point problems. The quasi-optimal valuesfor the parameters involved in the new preconditioners areestimated, so that the numerical conditioning and the spec-tral property of the saddle-point matrix of the linear systemcan be substantially improved. Several practical hybridpreconditioners and the corresponding preconditioning it-erative methods are constructed and studied, too.

Zeng-Qi WangShanghai Jiaotong Universitywangzengqi@sjtu.edu.cn

MS51

A Symmetric Homotopy and Hybrid Method forSolving Mixed Trigonometric Polynomial Systems

Abstract not available at time of publication.

Bo Yu

LA09 Abstracts 89

School of Mathematical ScienceDalian University of Technology, Dalian, P.R. Chinayubo@dlut.edu.cn

MS52

Subset Selection: Deterministic vs. Randomized

Subset selection methods try identify those columns of amatrix that are ”most” linearly independent. Many deter-ministic subset selection methods are based on a QR de-composition with column pivoting. In contrast, random-ized methods consist of two stages, where the first stagesamples a smaller set of columns according to a proba-bility distribution, and the second stage performs subsetselection on these columns. We analyze and compare theperformance of deterministic and randomized methods forsubset selection.

Ilse IpsenNorth Carolina State UniversityDepartment of Mathematicsipsen@ncsu.edu

MS52

Randomized Algorithms for Matrices and Large-Scale Data Applications

Randomization can be a powerful resource in the design ofalgorithms for linear algebra problems. Although much ofthis work has roots in convex analysis and theoretical com-puter science, and thus has relied on techniques very differ-ent than those traditionally used in numerical linear alge-bra, recent research has begun to “bridge the gap” betweenthe numerical linear algebra and theoretical computer sci-ence perspectives on these matrix problems. This has ledto practical numerical implementations of algorithms forvery traditional matrix problems arising in scientific com-puting and numerical linear algebra, and it has led to novelalgorithms for new matrix problems arising in large-scalescientific and Internet data applications. Several examplesof this paradigm will be described.

Michael MahoneyStanford UniversityApplied and Computational Mathematicsmmahoney@cs.stanford.edu

MS52

A Near-optimal Performance Analysis for Ran-domized Matrix Approximation

Simple randomized algorithms are exceptionally efficientat identifying the numerical range of a matrix, which isa basic building block for constructing approximations oflarge matrices. This talk describes a unified analysis ofthe most common algorithms. The argument decouplesthe linear algebra from the probability, which permits theapplication of powerful results from random matrix theory.The new methods lead to sharper and more transparentconclusions than previous analyses.

Joel TroppApplied and Computational MathematicsCaltechjtropp@acm.caltech.edu

MS53

Stochastic Binormalization of Symmetric Matrices

A symmetric matrix A is binormalized if the norm of eachrow (and column) is the same. Many matrices can be bi-normalized by symmetric diagonal scaling, and the result-ing matrix frequently has a smaller condition number. In2004, Livne and Golub introduced an algorithm to findsuch a diagonal scaling matrix D. The algorithm must ac-cess the elements of A individually. We first answer threeopen questions from their paper concerning the existenceand uniqueness of a binormalizing D. Then we introducea stochastic algorithm to find D while accessing A onlythrough matrix-vector products. Finally, we introduce alimited-memory quasi-Newton method that incorporatesstochastic binormalization.

Andrew BradleyStanford Universityambrad@stanford.edu

MS53

Linear Algebra Issues in SQP Methods for Nonlin-ear Optimization

We consider some linear algebraic issues associated withthe formulation and analysis of sequential quadratic pro-gramming (SQP) methods for large-scale nonlinearly con-strained optimization. Recent developments in methodsfor mixed-integer nonlinear programming and optimiza-tion subject to differential equation constraints has led toa heightened interest in methods that may be “hot started’from a good approximate solution. In this context we fo-cus on SQP methods that are best able to use “black-box’linear algebra software. Such methods provide an effectiveway of exploiting recent advances in linear algebra softwarefor multicore and GPU-based computer architectures.

Philip E. Gill, Elizabeth WongUniversity of California, San DiegoDepartment of Mathematicspgill@ucsd.edu, elwong@ucsd.edu

MS53

An Active-set Convex QP Solver Based on Regu-larized KKT Systems

Implementations of the simplex method depend on “basisrepair’ to steer around near-singular basis matrices, andKKT-based QP solvers must deal with near-singular KKTsystems. However, few sparse-matrix packages have therequired rank-revealing features. For convex QP, we ex-plore the idea of avoiding singular KKT systems by apply-ing primal and dual regularization to the QP problem. Asimplified single-phase active-set algorithm can then be de-veloped. Warm starts are straightforward from any givenactive set, and the range of applicable KKT solvers ex-pands. QPBLUR is a prototype QP solver that makes useof the block-LU KKT updates in QPBLU (Hanh Huynh’sPhD dissertation, 2008) but employs regularization and thesimplified active-set algorithm. The aim is to provide anew QP subproblem solver for SNOPT for problems withmany degrees of freedom. Numerical results confirm therobustness of the single-phase regularized QP approach.

Christopher MaesStanford Universitycmaes@stanford.edu

90 LA09 Abstracts

Michael A. SaundersSystems Optimization Laboratory (SOL)Dept of Management Sci and Eng, Stanfordsaunders@stanford.edu

MS53

Linear Algebra Computations in Sparse Optimiza-tion

”Sparse optimization” is a term often used to describe opti-mization problem in which we seek an approximate solutionthat is sparse, in the sense of having relatively few nonze-ros in the vector of unknowns. The iterative optimizationmethods proposed for these problems make use of manytools from numerical linear algebra, including methods forlinear equations, least squares, and singular value decom-positions. This talk considers several sparse optimizationproblems of current interest, including compressed sensingand matrix completion, and outlines several algorithms foreach problem. The role of linear algebra methods in thesealgorithms is highlighted, and the talk considers whetheralternative techniques may be more effective.

Stephen J. WrightUniversity of WisconsinDept. of Computer Sciencesswright@cs.wisc.edu

MS54

Randomized Preprocessing in Tensor Problems

We develop and analyze the techniques of randomized pre-processing and show both theoretically and experimentallythat it facilitates some most fundamental computations innumerical linear algebra such as the solution of general andstructured linear systems of equations and matrix eigen-solving. In the case of structured inputs the computationalimprovement is dramatic. The approach seems to be alsopromising for multi-linear algebraic computations.

Victor PanCUNYLehmanv y pan@yahoo.com

MS54

Quasi-Newton-Grassamann and Krylov-typeMethods for Tensors Approximation

In this talk we will describe two different methods for com-puting low multilinear rank approximations of a given ten-sor. The underlying structure of the tensor approximationproblem involves a product of Grassmann manifolds, onwhich the related objective function is defined on. The firstmethod, explicitly utilizing this fact, is a quasi-Newton al-gorithm that operates on product of Grassmann manifolds.The quasi-Newton methods are based on BFGS and limitedmemory BFGS updates. We will give a general descriptionof the algorithms and discuss optimality of the BFGS up-date on Grassmannians. In the second method, for tensorapproximation, we will generalize Krylov methods to ten-sors. For large and sparse matrix problems Krylov basedmethods are, in many cases, the only practically usefulmethods. As sparse tensors occur frequently in informa-tion sciences, tensor-Krylov based methods will enable thecomputation of low rank Tucker models of huge tensors.

Berkant SavasUniversity of Texas at Austin

berkant@cs.utexas.edu

MS54

Krylov Subspace Methods for Linear Systems withKronecker Product Structure

The numerical solution of linear systems with certain Kro-necker product structures is considered. Such structuresarise, for example, from the finite element discretization ofa linear PDE on a d-dimensional hypercube. A standardKrylov subspace method applied to such a linear systemsuffers from the curse of dimensionality and has a compu-tational cost that grows exponentially with d. The key tobreaking the curse is to note that the solution can oftenbe very well approximated by a vector of low tensor rank.We propose and analyse a new class of methods, so calledtensor Krylov subspace methods, which exploit this factand attain a computational cost that grows linearly withd.

Daniel KressnerETH Zurichdaniel.kressner@sam.math.ethz.ch

Christine ToblerSeminar for Applied MathematicsETH Zurichchristine.tobler@sam.math.ethz.ch

MS54

P-factor Analysis for Nonlinear Tensor Mappings

This work presents the main concept and results of thep-regulariry theory (also known as p-factor tensor analy-sis of nonlinear mappings). This approach is based on theconstruction of a tensor type operator (p-factor operator).The main result of this theory gives a detailed descriptionof the structure of the zero set of an irregular nonlinearmapping F(x). Applications include a new numerical p-factor method for solving essentially nonlinear problemsand p-order optimality conditions for nonlinear optimiza-tion problems.

Alexey TretyakovUniversity of Podlasie, PolandComputer Center of RAS, Moscowtret@ap.siedlce.pl

MS55

Preconditioners for 3D Modeling Problems Basedon Generalized Schur Interpolation

We consider 3D modeling problems that lead to hierarchi-cally multibanded matrices. As prototype example we usethe matrix obtained for the Poisson equation on a regular3D grid with a 27 point stencil. Using lexicographic or-dering the matrix obtained is block tridiagonal, wherebyeach sub-block is again block tridiagonal and each sub-sub-block is a tridiagonal matrix. In the past no good pre-conditioner for such (often occurring) matrices has beendetermined, to the best of our knowledge. We proposea new method based on hierarchical Schur complementa-tion, whereby each Schur complement is approximated bya low complexity representation, this time using an originalSchur matrix interpolation method.

Patrick DewildeInstitute for Advanced Study, Technische Universitat

LA09 Abstracts 91

Munchenand CAS Section, EEMCS, TU Delftp.m.dewilde@ewi.tudelft.nl

MS55

Communication Complexity for Parallel and Se-quential Eigenvalue/Singular Value Algorithms

The flops complexity of an algorithm does not always deter-mine how fast the algorithm will run in practice; communi-cation costs associated to algorithms can make a world ofdifference. Roughly speaking, one must take into accountall movement of data between fast memory, where actualcomputations occur, and slow memory, where the data isstored for later access. Ideally, desirable algorithms mini-mize all three following key components: flops, bandwidth(proportional to the total number of data words moved),and latency (proportional to the total number of messagespassed to and from layers of memory). Recently, therehas been a lot of work on finding O(n3) algorithms whichachieve optimal communication costs (see the work of Dem-mel, Grigori, Hoemmen, and Langou, and also that of Bal-lard, Demmel, Holtz, and Schwartz). This talk will give acommunication-optimizing algorithm (in the big-Oh sense)for computing eigenvalue, generalized eigenvalue, and sin-gular value decompositions. The algorithm is based on anearlier algorithm by Bai, Demmel, and Gu, and it achievescommunication optimality by using a randomized rank-revealing decomposition introduced by Demmel, Dumitriu,and Holtz. In addition to being O(n3) and big-Oh optimal,this algorithm has the benefit that, with high probability,it is stable. This is joint work with Grey Ballard and JamesDemmel.

Ioana DumitriuUniversity of Washington, Seattledumitriu@math.washington.edu

MS55

A Fast Implicit Qr Eigenvalue Algorithm for aClass of Structured Matrices

The talk presents a fast adaptation of the implicit QReigenvalue algorithm for certain classes of rank structuredmatrices including companion matrices. This is a jointwork with D. Bini, P. Boito, Y. Eidelman and I. Gohberg.

Luca Gemignani

Univeristy of Pisa (Italy)Department of Mathematicsgemignan@dm.unipi.it

MS55

Signal Flow Graph Approach to Inversion of Qua-siseparable Vandermonde Matrices

We use the language of signal flow graph representationof digital filter structures to solve three purely mathemat-ical problems, including fast inversion of certain polyno-mialVandermonde matrices, deriving an analogue of theHorner and Clenshaw rules for polynomial evaluation in a(H,m)quasiseparable basis, and computation of eigenvec-tors of (H,m) quasiseparable classes of matrices. Whilealgebraic derivations are possible, using elementary op-erations (specifically, flow reversal) on signal flow graphsprovides a unified derivation, and reveals connections with

systems theory.

Vadim OlshevskyDept. of MathematicsU. of Connecticut, Storrs.olshevsky@uconn.edu