Contour Integral Method for the Simulation of Accelerator Cavities · 2017. 11. 17. · Contour...

Contour Integral Method for the Simulation ofAccelerator CavitiesV. Pham-Xuan, W. Ackermann and H. De Gersem

Institut für Theorie Elektromagnetischer Felder

DESY meeting (14.11.2017)

Outline of the Talk

MotivationIterative methodsContour integral methods

FormulationMathematical modelEigenvalue algorithmsContour integral methodsMultigrid method as a preconditioner

Implementation

Preliminary Results

Possible Improvements

Presentation Outline

Implementation

Preliminary Results

MotivationProblem statement

Problem statement: we have to solve a nonlinear eigenvalue problem (NEP)where

• the problem is large and sparse;

• the number of eigenvalues is large;• prior information about eigenvalues is available;• in several applications, one is only interested in a few eigenvalues within a

certain range.

Figure: Chain of cavities (from [1])

Available methods:• Iterative methods: Jacobi-Davidson [2], Arnoldi, Lanczos, etc.• Contour integral methods: Beyn methods [3], resolvent sampling based Rayleigh-

Ritz method (RSRR) [4], etc.

Problem statement: we have to solve a nonlinear eigenvalue problem (NEP)where• the problem is large and sparse;• the number of eigenvalues is large;

• prior information about eigenvalues is available;• in several applications, one is only interested in a few eigenvalues within a

certain range.

Problem statement: we have to solve a nonlinear eigenvalue problem (NEP)where• the problem is large and sparse;• the number of eigenvalues is large;• prior information about eigenvalues is available;

• in several applications, one is only interested in a few eigenvalues within acertain range.

Problem statement: we have to solve a nonlinear eigenvalue problem (NEP)where• the problem is large and sparse;• the number of eigenvalues is large;• prior information about eigenvalues is available;• in several applications, one is only interested in a few eigenvalues within a

certain range.Figure: Chain of cavities (from [1])

Available methods:• Iterative methods: Jacobi-Davidson [2], Arnoldi, Lanczos, etc.

• Contour integral methods: Beyn methods [3], resolvent sampling based Rayleigh-Ritz method (RSRR) [4], etc.

Implementation

Preliminary Results

MotivationIterative methods (Jacobi-Davidson)

Lossless accelerator cavity: eigenvalues are on real axis

• choose an initial guess

• expand the search space ...

• until an approximate solution isfound

• the solution becomes the new initialguess

• continue expanding the searchspace ...

initial guess

Lossy accelerator cavity: eigenvalues are in the complex plane

• choose another initial guess

• find another approximate solution

• if we choose unsuitable initial guess

• the algorithm will converge to ...

• a previously determinedeigenvalue!!!!!

initial guess

initial guess • choose an initial guess

initial guess

Implementation

Preliminary Results

MotivationContour integral methods

An accurate computation of eigenpairs inside a region enclosed by a non-self-intersecting curve.

• choose a region to look foreigenvalues

• the region can be of any shape, e.grectangle ...

• circle/ellipse

• most computation is spent to solvelinear equation systems at differentinterpolation points which can beparallelized.

• circle/ellipse

Implementation

Preliminary Results

Implementation

Preliminary Results

FormulationMathematical model

The combination of Maxwell-Ampère equation and the Maxwell-Faraday equationresults in the double-curl equation

∇× 1µ∇× ~E − jωσ~E = εω2~E (1)

Applying the Galerkin’s approach to discretize (1) results in an eigenvalue problem

A3D~x + jωµ0C3D~x − ω2µ0ε0B3D~x = 0 (2)

which includes only losses from volumetric lossy material. Special treatment is car-ried out to incorporate 2D losses at port interfaces into (2), resulting in a nonlineareigenvalue problem (NEP)

P(ω)~x = 0 (3)

∇× 1µ∇× ~E − jωσ~E = εω2~E (1)

P(ω)~x = 0 (3)

∇× 1µ∇× ~E − jωσ~E = εω2~E (1)

P(ω)~x = 0 (3)

∇× 1µ∇× ~E − jωσ~E = εω2~E (1)

P(ω)~x = 0 (3)

Implementation

Preliminary Results

FormulationEigenvalue algorithms

To solve P(z)x = 0 most of the standard eigenvalue algorithms exploit a projectionprocedure in order to extract approximate eigenvectors from a given subspace.

Approximate aneigenspace Q obtain matrix Q

Project the originalNEP to a reduced NEP

using Rayleigh-Ritz procedure:PQ(z) = QHP(z)Q

Solve the reduced NEPsolve PQ(z)g = 0for eigenvalues z and eigenvectors g

Compute eigenpairsof the original NEP

same eigenvalues as for the reduced problem;eigenvectors x = Qg

Implementation

Preliminary Results

FormulationContour integral methods

Some basic spectral theory

The resolvent P(z)−1 reveals the existence of eigenvalues, indicates whereeigenvalues are located, and show how sensitive these eigenvalues are to per-tubation.

As explained in [3], from Keldysh’s theorem, we know that the resolvent functionP(z)−1 can be written (for simple eigenvalues λi ) as

P(z)−1 =∑

1z − λi

+ R(z) (4)

where• vi and wi are suitably scaled right and left eigenvectors, respectively, cor-

responding to the (simple) eigenvalue λi

• R(z) and P(z) are analytic functions

The resolvent P(z)−1 reveals the existence of eigenvalues, indicates whereeigenvalues are located, and show how sensitive these eigenvalues are to per-tubation.As explained in [3], from Keldysh’s theorem, we know that the resolvent functionP(z)−1 can be written (for simple eigenvalues λi ) as

P(z)−1 =∑

1z − λi

+ R(z) (4)

where• vi and wi are suitably scaled right and left eigenvectors, respectively, cor-

responding to the (simple) eigenvalue λi

• R(z) and P(z) are analytic functions

Q = (q1, q2, ..., qk )span{q1, q2, ..., qk} ⊇ span{v1, v2, ..., vn(Γ)}

P(z)−1 =∑

vi wHi

1z − λi

+ R(z)

Applying Cauchy’s integral formula

f (z)P(z)−1dz =n(Γ)∑i=1

f (λi )vi wHi

In practice, we evaluate the integral

f (z)P(z)−1V̂dz

Using interpolation, we obtain

f (z)P(z)−1V̂dz =nint∑i=1

ξi f (zi )P(zi )−1V̂

Q = (q1, q2, ..., qk )

span{q1, q2, ..., qk} ⊇ span{v1, v2, ..., vn(Γ)}

P(z)−1 =∑

vi wHi

1z − λi

+ R(z)

f (z)P(z)−1dz =n(Γ)∑i=1

f (λi )vi wHi

f (z)P(z)−1V̂dz

P(z)−1 =∑

vi wHi

1z − λi

+ R(z)

f (z)P(z)−1dz =n(Γ)∑i=1

f (λi )vi wHi

f (z)P(z)−1V̂dz

P(z)−1 =∑

vi wHi

1z − λi

+ R(z)

f (z)P(z)−1dz =n(Γ)∑i=1

f (λi )vi wHi

f (z)P(z)−1V̂dz

P(z)−1 =∑

vi wHi

1z − λi

+ R(z)

f (z)P(z)−1dz =n(Γ)∑i=1

f (λi )vi wHi

f (z)P(z)−1V̂dz

P(z)−1 =∑

vi wHi

1z − λi

+ R(z)

f (z)P(z)−1dz =n(Γ)∑i=1

f (λi )vi wHi

f (z)P(z)−1V̂dz

P(z)−1 =∑

vi wHi

1z − λi

+ R(z)

f (z)P(z)−1dz =n(Γ)∑i=1

f (λi )vi wHi

f (z)P(z)−1V̂dz

Implementation

Preliminary Results

FormulationMultigrid method as a preconditioner

The most expensive operation is to compute

X = P−1(zi )V (5)

equivalent to solving the linear system

P(zi )X = V (6)

• Direct inverse becomes prohibitively expensive for large problems.

• For large-scale problems, iterative methods are preferable.• Linear systems generated by Maxwell’s equations are extremely ill-conditioned.• Krylov iterative solvers with simple preconditioners often stagnate or diverge

as when applied to these linear systems.• Suitable preconditioners/iterative solvers should be applied to improve the con-

vergence of the iterative solvers.

X = P−1(zi )V (5)

P(zi )X = V (6)

• Direct inverse becomes prohibitively expensive for large problems.• For large-scale problems, iterative methods are preferable.

• Linear systems generated by Maxwell’s equations are extremely ill-conditioned.• Krylov iterative solvers with simple preconditioners often stagnate or diverge

X = P−1(zi )V (5)

P(zi )X = V (6)

• Direct inverse becomes prohibitively expensive for large problems.• For large-scale problems, iterative methods are preferable.• Linear systems generated by Maxwell’s equations are extremely ill-conditioned.

• Krylov iterative solvers with simple preconditioners often stagnate or divergeas when applied to these linear systems.

• Suitable preconditioners/iterative solvers should be applied to improve the con-vergence of the iterative solvers.

X = P−1(zi )V (5)

P(zi )X = V (6)

• Direct inverse becomes prohibitively expensive for large problems.• For large-scale problems, iterative methods are preferable.• Linear systems generated by Maxwell’s equations are extremely ill-conditioned.• Krylov iterative solvers with simple preconditioners often stagnate or diverge

as when applied to these linear systems.

• Suitable preconditioners/iterative solvers should be applied to improve the con-vergence of the iterative solvers.

X = P−1(zi )V (5)

P(zi )X = V (6)

• Direct inverse becomes prohibitively expensive for large problems.• For large-scale problems, iterative methods are preferable.• Linear systems generated by Maxwell’s equations are extremely ill-conditioned.• Krylov iterative solvers with simple preconditioners often stagnate or diverge

reduced 1st orderfull 1st order

reduced 2nd order

0 6 12 20

Pär Ingelström, "A New Set of H(curl) ConformingHierarchical Basis Functions for TetrahedralMeshes, IEEE Transactions on Microwave Theoryand Techniques, vol. 54, no. 1, Jan. 2006.

W-cycle

Pär Ingelström et al., "Comparison ofHierarchical Basis Functions for EfficientMultilevel Solvers", IET Science,Measurement and Technology, vol. 1, no.1, Jan. 2007.

M−1b = e (7)

This equation is repeatedly computed ateach iteration where M is thepreconditioner, b is the input and e is theoutput. The output is computed bysolving systems of the type

Piiei = bi −∑i 6=j

Pijej (8)

where i and j refer to the order of the trialand test functions.

Implementation

Preliminary Results

ImplementationNonlinear Eigenvalue Solver for Accelerator Cavities(NES4AC)

CST cavity design, FEM discretization

CEM3D [5] generate matrices P(zi )

NES4AC Solve nonlinear eigenvalue problem

Parallell level 1:multiple contours

Parallel level 2: compute nodes in parallel (implemented in codes)

Parallel level 3: parallelize computation at each node (implemented in the codes)

P(λ)x=vor x=P(z)-1v

Parallel level 2: compute nodes in parallel (implemented)

z5z6z7

z11z12 z13

z14z15

P−1(z)V dz

z3z4z5z6

z10z11

z12 z13 z14z15

subcomm 1z1, A1, B1

z2, A2, B2

z3, A3, B3

z4, A4, B4

subcomm 2z5, A5, B5

z6, A6, B6

z7, A7, B7

z8, A8, B8

subcomm 3z9, A9, B9

z10, A10, B10

z11, A11, B11

z12, A12, B12

subcomm 4z13, A13, B13

z14, A14, B14

z15, A15, B15

z16, A16, B16

P−1(z)V dz

z3z4z5z6

z10z11

z12 z13 z14z15

subcomm 1z1, A1, B1

z2, A2, B2

z3, A3, B3

z4, A4, B4

subcomm 2z5, A5, B5

z6, A6, B6

z7, A7, B7

z8, A8, B8

subcomm 3z9, A9, B9

z10, A10, B10

z11, A11, B11

z12, A12, B12

z14, A14, B14

z15, A15, B15

z16, A16, B16

P−1(z4)V

P−1(z3)V

P−1(z2)V

P−1(z1)V

P−1(z8)V

P−1(z7)V

P−1(z6)V

P−1(z5)V

P−1(z12)V

P−1(z11)V

P−1(z10)V

P−1(z9)V

P−1(z16)V

P−1(z15)V

P−1(z14)V

P−1(z13)V

P−1(z)V dz

Parallel level 3: parallelize computation at each node (implemented)

z3z4z5z6

z10z11

z12 z13 z14z15

subcomm 1z1, A1, B1

z2, A2, B2

z3, A3, B3

z4, A4, B4

subcomm 2z5, A5, B5

z6, A6, B6

z7, A7, B7

z8, A8, B8

subcomm 3z9, A9, B9

z10, A10, B10

z11, A11, B11

z12, A12, B12

z14, A14, B14

z15, A15, B15

z16, A16, B16

P−1(z4)V

P−1(z3)V

P−1(z2)V

P−1(z1)V

P−1(z8)V

P−1(z7)V

P−1(z6)V

P−1(z5)V

P−1(z12)V

P−1(z11)V

P−1(z10)V

P−1(z9)V

P−1(z16)V

P−1(z15)V

P−1(z14)V

P−1(z13)V

P−1(z)V dz

Parallel level 3: parallelize computation at each node (implemented)

z3z4z5z6

z10z11

z12 z13 z14z15

subcomm 1z1, A1, B1

z2, A2, B2

z3, A3, B3

z4, A4, B4

subcomm 2z5, A5, B5

z6, A6, B6

z7, A7, B7

z8, A8, B8

subcomm 3z9, A9, B9

z10, A10, B10

z11, A11, B11

z12, A12, B12

z14, A14, B14

z15, A15, B15

z16, A16, B16

P−1(z4)V

P−1(z3)V

P−1(z2)V

P−1(z1)V

P−1(z8)V

P−1(z7)V

P−1(z6)V

P−1(z5)V

P−1(z12)V

P−1(z11)V

P−1(z10)V

P−1(z9)V

P−1(z16)V

P−1(z15)V

P−1(z14)V

P−1(z13)V

P−1(zi)V

P−1(z)V dz

NES4AC highlights:

• extends the functionality of CEM3D [5].

• parallelized and developed in C++.

• based on PETSc (Portable, Extensible Toolkit for Scientific Computation) v3.3.0and LAPACK.

• adopts the parallel scheme of the contour integral method from SLEPc (Scal-able Library for Eigenvalue Problem Computations).

• uses the superLU_DIST for the computation of LU decompositions.

• including three contour integral algorithms for eigenvalue solution: Beyn1 (fora few eigenvalues), Beyn2 (for many eigenvalues) and RSRR.

• with two types of closed contour: ellipse and rectangle.

Implementation

Preliminary Results

Preliminary ResultsNonlinear Eigenvalues Problems in [6]Butterfly Problem• Name: butterfly (Quartic matrix polynomial with T-even structure)• P(λ) = λ4A4 + λ3A3 + λ2A2 + λA1 + A0

• Size: 64• Region: circle(-1.0,-0.5,0.7)• Number of eigenvalues: 55• Algorithm parameters:

• N = 60 (number of integration points)• L = 100:2:200 (number of columns of the random matrix)• K = 2 (for BEYN2)• Lorg = 20 (for RSRR)• Lred = 100 (for RSRR)• Nred = 30 (for RSRR)• Kred = 2 (for RSRR)• Rank tolerance = 1.0× 10−12

Preliminary ResultsNonlinear Eigenvalues Problems in [6]Butterfly Problem

-2 -1 0 1real

Eigenvalues for butterfly problem - case 1

matlab - polyeigbeyn1

-2 -1 0 1real

matlab - polyeigrsrr

-2 -1 0 1real

beyn1 beyn2 rsrr

Min. residual 4.9 × 10−5 2.05 × 10−14 4.05 × 10−12

Max. residual 0.0015 1.72 × 10−13 4.38 × 10−9

ε = ‖P(λ)x‖‖P(λ)‖‖x‖

Preliminary ResultsNonlinear Eigenvalues Problems in [6]Schrödinger• Name: Schrödinger (QEP from Schrödinger operator)• P(λ) = K − 2λC + λ2B• Size: 1998• Region: circle(0.75,0.0,0.45)• Number of eigenvalues: 30• Algorithm parameters:

• N = 20 (number of integration points)• L = 50:2:100 (number of columns of the random matrix)• K = 2 (for BEYN2)• Lorg = 20 (for RSRR)• Lred = 50 (for RSRR)• Nred = 30 (for RSRR)• Kred = 2 (for RSRR)• Rank tolerance = 1.0× 10−12

Preliminary ResultsNonlinear Eigenvalues Problems in [6]Schrödinger

0.5 1 1.5real

Eigenvalues for Schrödinger problem - case 1

0.5 1 1.5real

matlab - polyeigrsrr

0.5 1 1.5real

beyn1 beyn2 rsrr

Min. residual n/a 1.08 × 10−15 1.97 × 10−17

Max. residual n/a 1.73 × 10−14 2.52 × 10−17

Preliminary ResultsSpherical Cavity

• Name: spherical cavity

• Electrical conductivity: 58× 106 Ohm/sq

• Radius: 1m

• Size: 7614/17830/61106/143354/278138

• Target frequency: 125MHz• Number of eigenvalues: 3• Algorithm parameters:• Region: rectangle(1.0, 1.5, 1.0× 10−15, 0.05)• N = 20 (number of integration points)• L = 40:10:100 (number of columns of the random matrix)• K = 2 (for BEYN2)• Lorg = 20 / Lred = 20 / Nred = 20 / Kred = 2 (for RSRR)• Rank tolerance = 1.0× 10−8

Preliminary ResultsSpherical Cavity

εf =‖f−fanalytical‖‖fanalytical‖

εQ =‖Q−Qanalytical‖‖Qanalytical‖

Preliminary ResultsMultigrid preconditioner - Spherical Cavity

# elements # dofs # dofs (1st order)Dis. 1 342 1,652 250Dis. 2 1,256 6,640 1,078Dis. 3 2,918 16,300 2,755Dis. 4 18,062 105,828 18,487

Table: Mesh refinement of a spherical cavity

Dis. 1 Dis. 2 Dis. 3 Dis. 4mg-GMRES 17 21 23 23

GMRES 500 (1.0e-5) 500 (1.0e-2) 500 (1.0e-2) 500 (5.0e-2)

Table: Number of iterations required to achieve a residual of 1.0e-8

Preliminary ResultsMultigrid preconditioner - Tesla Cavity

# elements # dofs # dofs (1st order)Dis. 1 48,819 286,026 50,067Dis. 2 57,394 339,664 59,864Dis. 3 76,113 373,812 80,552

Table: Mesh refinement of a Tesla cavity

Dis. 1 Dis. 2 Dis. 3mg-GMRES 30 25 24

GMRES 500 (0.2) 500 (0.15) 500 (0.2)

Table: Number of iterations required to achieve a residual of 1.0e-8

Implementation

Preliminary Results

Possible ImprovementsRecycling Krylov subspace methods

Xi = P−1(zi)V

P (zi)Xi = V

P (z1) X1 V

P (z2) X2 V

P (z3) X3 V

• Iterative method is repeatedly applied for differentRHS.

• The matrix is unchanged.

• Opportunity for applying recycling Krylov subspacemethods or block Krylov subspace methods.

• The system-matrices are slightly changed fordifferent interpolation points

Xi = P−1(zi)V

P (zi)Xi = V

P (z1) X1 V

P (z2) X2 V

P (z3) X3 V

Xi = P−1(zi)V

P (zi)Xi = V

P (z1) X1 V

P (z2) X2 V

P (z3) X3 V

Xi = P−1(zi)V

P (zi)Xi = V

P (z1) X1 V

P (z2) X2 V

P (z3) X3 V

Xi = P−1(zi)V

P (zi)Xi = V

P (z1) X1 V

P (z2) X2 V

P (z3) X3 V

Xi = P−1(zi)V

P (zi)Xi = V

P (z1) X1 V

P (z2) X2 V

P (z3) X3 V

Thank you for your attention

References I

[1] T. Flisgen, J. Heller, T. Galek, L. Shi, N. Joshi, N. Baboi, R. M. Jones, andU. van Rienen, “Eigenmode Compendium of The Third Harmonic Module of theEuropean X-Ray Free Electron Laser,” Physical Review Accelerators andBeams, vol. 20, p. 042002, Apr 2017.

[2] H. Voss, “A Jacobi-Davidson Method for Nonlinear and NonsymmetricEigenproblems,” Computers and Structures, vol. 85, no. 17-18, pp. 1284–1292,2007.

[3] W.-J. Beyn, “An Integral Method for Solving Nonlinear Eigenvalue Problems,”Linear Algebra and its Applications, vol. 436, no. 10, pp. 3839 – 3863, 2012.

References II

[4] J. Xiao, C. Zhang, T. M. Huang, and T. Sakurai, “Solving Large-Scale NonlinearEigenvalue Problems by Rational Interpolation and Resolvent Sampling BasedRayleigh-Ritz Method,” International Journal for Numerical Methods inEngineering, vol. 110, no. 8, pp. 776–800, 2017.

[5] T. Banova, W. Ackermann, and T. Weiland, “Accurate Determination ofThousands of Eigenvalues for Large-Scale Eigenvalue Problems,” IEEETransactions on Magnetics, vol. 50, pp. 481–484, Feb. 2014.

[6] T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur, “NLEVP: ACollection of Nonlinear Eigenvalue Problems,” ACM Transactions onMathematical Software, vol. 39, no. 2, pp. 7:1–7:28, 2013.

AppendixContour integral methods

Beyn1 (for a few eigenvalues)

Define the matrices A0 and A1 ∈ Cn×k

P(z)−1V̂dz (9)

zP(z)−1V̂dz (10)

Then A0 = VW H V̂ and A1 = VΛW H V̂ where

• Λ = diag(λ1, ...,λn(Γ))• V =

[v1 · · · vn(Γ)

]• W =

[w1 · · · wn(Γ)

]V̂ is a random matrix V̂ ∈ Cn×L. L is smaller than n and equal or greater and k

Beyn1 (for a few eigenvalues)

Beyn’s method is based on the singular value decomposition of A0

A0 = V0Σ0W H0 (11)

Beyn has shown that the matrix

B = V H0 A1W H

0 Σ−10 (12)

is diagonalizable. Its eigenvalues are the eigenvalues of P inside the contour andits eigenvectors lead to the corresponding eigenvectors of P.

Beyn2 (for many eigenvalues)

Define the matrices Ap ∈ Cn×k

zpP(z)−1V̂dz (13)

Then Ap = VΛpW H V̂ . The matrices B0 and B1 are defined as follows

A0 · · · AK−1...

...AK−1 · · · A2K−2

; B1 =

A1 · · · AK...

...AK · · · A2K−1

Beyn2 (for many eigenvalues)

Performing the singular value decomposition of B0

B0 = V0Σ0W H0 (15)

Beyn has shown that the matrix

D = V H0 B1W H

0 Σ−10 (16)

is diagonalizable. Its eigenvalues are the eigenvalues of P inside the contour andits eigenvectors lead to the corresponding eigenvectors of P.

Resolvent Sampling based Rayleigh-Ritz method

P−1(z1) P

−1(z2) P−1(z3) P

−1(z4) P−1(z5) P

−1(z6) P−1(z7) P

−1(z8) P−1(z9) P

−1(z10)

The Beyn2 algorithm is robust andaccurate if a large L but a small K areused.

However, for large-scale problems, asmall L is essential to reduce thecomputational burden.

Decrease L and increase K make thealgorithm unstable and inaccurate.

RSRR reduce the number of columns ofV .

Let Q ∈ Cn×k be an orthogonal basis of search space, then the original NEP can be con-verted to the following reduced NEP

PQ(z)g = 0

P−1(z1) P

−1(z2) P−1(z3) P

−1(z4) P−1(z5) P

−1(z6) P−1(z7) P

−1(z8) P−1(z9) P

−1(z10)

PQ(z)g = 0

P−1(z1) P

−1(z2) P−1(z3) P

−1(z4) P−1(z5) P

−1(z6) P−1(z7) P

−1(z8) P−1(z9) P

−1(z10)

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

PQ(z)g = 0

P−1(z1) P

−1(z2) P−1(z3) P

−1(z4) P−1(z5) P

−1(z6) P−1(z7) P

−1(z8) P−1(z9) P

−1(z10)

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

= = = = = = = = = =

PQ(z)g = 0

P−1(z1) P

−1(z2) P−1(z3) P

−1(z4) P−1(z5) P

−1(z6) P−1(z7) P

−1(z8) P−1(z9) P

−1(z10)

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

= = = = = = = = = =

+ + + + + + + + +( )

PQ(z)g = 0

P−1(z1) P

−1(z2) P−1(z3) P

−1(z4) P−1(z5) P

−1(z6) P−1(z7) P

−1(z8) P−1(z9) P

−1(z10)

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

= = = = = = = = = =

+ + + + + + + + +( )

PQ(z)g = 0

P−1(z1) P

−1(z2) P−1(z3) P

−1(z4) P−1(z5) P

−1(z6) P−1(z7) P

−1(z8) P−1(z9) P

−1(z10)

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

= = = = = = = = = =

+ + + + + + + + +( )

PQ(z)g = 0

P−1(z1) P−1(z2) P−1(z3) P−1(z4) P−1(z5) P−1(z6) P−1(z7) P−1(z8) P−1(z9) P−1(z10)

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

= = = = = = = = = =

( ), , , , , , , , ,

PQ(z)g = 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

= = = = = = = = = =

( ), , , , , , , , ,

PQ(z)g = 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

= = = = = = = = = =

( ), , , , , , , , ,

PQ(z)g = 0

(1) Initialization: Fix the contour Γ, the number N and the sampling points zi . Fixthe number L and generate a n × L random matrix U

(2) Compute P(zi )−1U for i = 0, 1, ... , N − 1(3) Form S as follows

S =[P(z0)−1U, P(z1)−1U, · · · , P(zN−1)−1U

]∈ Cn×N·L (17)

(4) Generate the matrix Q via the truncated singular value decomposition S ≈QΣV H .

(5) Compute PQ(z) = QHP(z)Q, and solve the projected NEP PQ(λ)g = 0 usingthe SS-FULL algorithm to obtain n(Γ) eigenapairs (gj ,λj ).

(6) Compute the eigenpairs of the original NEP via the eigenpairs of the reducedNEP.

(2) Compute P(zi )−1U for i = 0, 1, ... , N − 1

(3) Form S as followsS =

[P(z0)−1U, P(z1)−1U, · · · , P(zN−1)−1U

]∈ Cn×N·L (17)

S =[P(z0)−1U, P(z1)−1U, · · · , P(zN−1)−1U

]∈ Cn×N·L (17)

S =[P(z0)−1U, P(z1)−1U, · · · , P(zN−1)−1U

]∈ Cn×N·L (17)

S =[P(z0)−1U, P(z1)−1U, · · · , P(zN−1)−1U

]∈ Cn×N·L (17)

S =[P(z0)−1U, P(z1)−1U, · · · , P(zN−1)−1U

]∈ Cn×N·L (17)

S =[P(z0)−1U, P(z1)−1U, · · · , P(zN−1)−1U

]∈ Cn×N·L (17)

Contour Integral Method for the Simulation of Accelerator Cavities · 2017. 11. 17. · Contour...

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