PhD Thesis - ICI No Ordre: 1879 PhD Thesis Sp¢´ecialit¢´e : Informatique Sparse preconditioners for

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  • No Ordre: 1879

    PhD Thesis

    Spécialité : Informatique

    Sparse preconditioners for dense linear

    systems from electromagnetic applications

    présentée le 23 Avril 2002 à

    l’Institut National Polytechnique de Toulouse

    par

    Bruno CARPENTIERI

    CERFACS

    devant le Jury composé de :

    G. Alléon EADS M. Daydé Professeur à l’ENSEEIHT I. S. Duff Project Leader at CERFACS

    Group Leader at Rutherford Appleton Laboratory Président L. Giraud CERFACS G. Meurant CEA Rapporteur Y. Saad Professor at the University of Minnesota Rapporteur S. Piperno INRIA-CERMICS

    CERFACS report: TH/PA/02/48

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    Acknowledgments

    I wish to express my sincere gratitude to Iain S. Duff and Luc Giraud because they introduced me to the subject of this thesis and guided my research with vivid interest. They taught me the enjoyment both for rigour and for simplicity, and let me experience the freedom and the excitement of personal discovery. Without their professional advice and their trust in me, this thesis would not be possible.

    My sincere thanks go to Michel Daydé for his continued support in the development of my research at CERFACS.

    I am grateful to Gerard Meurant and Yousef Saad who accepted to act as referees for my thesis. It was an honour for me to benefit from their feedback on my research work.

    I wish to thank Guillaume Alléon and Serge Piperno who opened me the door of an enriching collaboration with EADS and INRIA-CERMICS, respectively, and accepted to take part in my jury. Guillaume Sylvand at INRIA-CERMICS deserves thanks for providing me with codes and valuable support.

    Grateful acknowledgments are made for the EMC Team at CERFACS for their interest in my work, in particular to Mbarek Fares who provided me with the CESC code, and Francis Collino and Florence Millot for many fertile discussions.

    I would like to thank sincerely all the members of the Parallel Algorithms Team and CSG at CERFACS for their professional and friendly support, and Brigitte Yzel for helping me many times gently. The Parallel Algorithms Team represented a stimulating environment to develop my thesis. I am grateful to many visitors or colleagues who, at different stages, shared my enjoyment for this research.

    Above all, I wish to express my deep gratitude to my family and friends for their presence and continued support.

    This work was supported by INDAM under a grant ”Borsa di Studio per l’Estero A.A. 1998-’99” (Provvedimento del Presidente del 30 Aprile 1998), and by CERFACS.

    - B. C.

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    To my family

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    Don’t just say ”it is impossible” without putting a sincere effort.

    Observe the word ”Impossible” carefully .. You can see ”I’m possible”.

    What really matters is your attitude and your perception.

    Anonymous

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    Abstract

    In this work, we investigate the use of sparse approximate inverse preconditioners for the solution of large dense complex linear systems arising from integral equations in electromagnetism applications.

    The goal of this study is the development of robust and parallelizable preconditioners that can easily be integrated in simulation codes able to treat large configurations. We first adapt to the dense situation the preconditioners initialy developed for sparse linear systems. We compare their respective numerical behaviours and propose a robust pattern selection strategy for Frobenius-norm minimization preconditioners.

    Our approach has been implemented by another PhD student in a large parallel code that exploits a fast multipole calculation for the matrix vector product in the Krylov iterations. This enables us to study the numerical scalability of our preconditioner on large academic and industrial test problems in order to identify its limitations. To remove these limitations we propose an embedded scheme. This inner-outer technique enables to significantly reduce the computational cost of the simulation and improve the robustness of the preconditioner. In particular, we were able to solve a linear system with more than a million unknowns arising from a simulation on a real aircraft. That solution was out of reach with our initial technique.

    Finally we perform a preliminary study on a spectral two-level preconditioner to enhance the robustness of our preconditioner. This numerical technique exploits spectral information of the preconditioned systems to build a low-rank update of the preconditioner.

    Keywords : Krylov subspace methods, preconditioning techniques, sparse approximate inverse, Frobenius-norm minimization method, nonzero pattern selection strategies, electromagnetic scattering applications, boundary element method, fast multipole method.

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  • Contents

    1 Introduction 1 1.1 The physical problem and applications . . . . . . . . . . . . . 2 1.2 The mathematical problem . . . . . . . . . . . . . . . . . . . 2 1.3 Numerical solution of Maxwell’s equations . . . . . . . . . . . 5

    1.3.1 Differential equation methods . . . . . . . . . . . . . . 5 1.3.2 Integral equation methods . . . . . . . . . . . . . . . . 6

    1.4 Direct versus iterative solution methods . . . . . . . . . . . . 8 1.4.1 A sparse approach for solving scattering problems . . 9

    2 Iterative solution via preconditioned Krylov solvers of dense systems in electromagnetism 13 2.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . 13 2.2 Preconditioning based on sparsification strategies . . . . . . . 18

    2.2.1 SSOR . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 Incomplete Cholesky factorization . . . . . . . . . . . 25 2.2.3 AINV . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.4 SPAI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2.5 SLU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2.6 Other preconditioners . . . . . . . . . . . . . . . . . . 50

    2.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 50

    3 Sparse pattern selection strategies for robust Frobenius- norm minimization preconditioner 53 3.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . 54 3.2 Pattern selection strategies for Frobenius-norm minimization

    methods in electromagnetism . . . . . . . . . . . . . . . . . . 56 3.2.1 Algebraic strategy . . . . . . . . . . . . . . . . . . . . 56 3.2.2 Topological strategy . . . . . . . . . . . . . . . . . . . 58 3.2.3 Geometric strategy . . . . . . . . . . . . . . . . . . . . 60 3.2.4 Numerical experiments . . . . . . . . . . . . . . . . . . 61

    3.3 Strategies for the coefficient matrix . . . . . . . . . . . . . . . 65 3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 73

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    4 Symmetric Frobenius-norm minimization preconditioners in electromagnetism 77 4.1 Comparison with standard preconditioners . . . . . . . . . . . 77 4.2 Symmetrization strategies for Frobenius-norm minimization

    method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 88

    5 Combining fast multipole techniques and approximate inverse preconditioners for large parallel electromagnetics calculations. 91 5.1 The fast multipole method . . . . . . . . . . . . . . . . . . . . 92 5.2 Implementation of the Frobenius-norm minimization

    preconditioner in the fast multipole framework . . . . . . . . 94 5.3 Numerical scalability of the preconditioner . . . . . . . . . . . 96 5.4 Improving the preconditioner robustness using embedded

    iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 108

    6 Spectral two-level preconditioner 111 6.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . 111 6.2 Two-level preconditioner via low-rank spectral updates . . . . 114

    6.2.1 Additive formulation . . . . . . . . . . . . . . . . . . . 115 6.2.2 Numerical experiments . . . . . . . . . . . . . . . . . . 118 6.2.3 Symmetric formulation . . . . . . . . . . . . . . . . . . 136

    6.3 Multiplicative formulation of low-rank spectral updates . . . 139 6.3.1 Numerical experiments . . . . . . . . . . . . . . . . . . 140

    6.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 143

    7 Conclusions and perspectives 145

    A Numerical results with the two-level spectral preconditioner 153 A.1 Effect of the low-rank updates on the GMRES convergence . 154 A.2 Experiments with the operator WH = V H² M1 . . . . . . . . . 164 A.3 Cost of the eigencomputation . . . . . . . . . . . . . . . . . . 174 A.4 Sensitivity of the preconditioner to the accuracy of the

    eigencomputation . . . . . . . . . . . . . . . . . . . . . . . . . 179 A.5 Experiments with a poor preconditioner M1 . . . . . . . . . . 189 A.6 Numerical results for the symmetric formulation . . . . . . . 204 A.7 Numerical results for the multiplicative formulation . . . . . . 216

  • List of Tables

    2.1.1 Number of matrix-vector products needed by some unpreconditioned Krylov solvers to reduce the residual by a factor of 10−5. . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    2.2.2 Number of iterations using both symmetric and unsymmetric preconditioned Krylov methods to reduce the normwise backward error by 10−5 on Example 1.