Contour Integral Method for the Simulation of Accelerator Cavities
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1 Contour Integral Method for the Simulation of Accelerator Cavities V. Pham-Xuan, W. Ackermann and H. De Gersem Institut für Theorie Elektromagnetischer Felder DESY meeting ( ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 1 / 28
2 Outline of the Talk Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 2 / 28
3 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 2 / 28
4 Motivation Problem statement Problem statement: we have to solve a nonlinear eigenvalue problem (NEP) where November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 3 / 28
5 Motivation Problem 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; Figure: Chain of cavities (from [1]) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 3 / 28
6 Motivation Problem 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; Figure: Chain of cavities (from [1]) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 3 / 28
7 Motivation Problem 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]) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 3 / 28
8 Motivation Problem 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. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 3 / 28
9 Motivation Problem 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. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 3 / 28
10 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 3 / 28
11 Motivation Iterative methods (Jacobi-Davidson) Lossless accelerator cavity: eigenvalues are on real axis November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 4 / 28
12 Motivation Iterative methods (Jacobi-Davidson) Lossless accelerator cavity: eigenvalues are on real axis choose an initial guess initial guess November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 4 / 28
13 Motivation Iterative methods (Jacobi-Davidson) Lossless accelerator cavity: eigenvalues are on real axis choose an initial guess expand the search space... initial guess November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 4 / 28
14 Motivation Iterative methods (Jacobi-Davidson) Lossless accelerator cavity: eigenvalues are on real axis choose an initial guess expand the search space... initial guess November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 4 / 28
15 Motivation Iterative methods (Jacobi-Davidson) Lossless accelerator cavity: eigenvalues are on real axis initial guess choose an initial guess expand the search space... until an approximate solution is found November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 4 / 28
16 Motivation Iterative methods (Jacobi-Davidson) Lossless accelerator cavity: eigenvalues are on real axis initial guess choose an initial guess expand the search space... until an approximate solution is found the solution becomes the new initial guess November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 4 / 28
17 Motivation Iterative methods (Jacobi-Davidson) Lossless accelerator cavity: eigenvalues are on real axis initial guess choose an initial guess expand the search space... until an approximate solution is found the solution becomes the new initial guess continue expanding the search space... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 4 / 28
18 Motivation Iterative methods (Jacobi-Davidson) Lossless accelerator cavity: eigenvalues are on real axis initial guess choose an initial guess expand the search space... until an approximate solution is found the solution becomes the new initial guess continue expanding the search space... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 4 / 28
19 Motivation Iterative methods (Jacobi-Davidson) Lossless accelerator cavity: eigenvalues are on real axis initial guess choose an initial guess expand the search space... until an approximate solution is found the solution becomes the new initial guess continue expanding the search space... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 4 / 28
20 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
21 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane choose an initial guess initial guess November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
22 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
23 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
24 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... until an approximate solution is found November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
25 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... until an approximate solution is found choose another initial guess November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
26 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... until an approximate solution is found choose another initial guess continue expanding the search space... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
27 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... until an approximate solution is found choose another initial guess continue expanding the search space... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
28 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... until an approximate solution is found choose another initial guess continue expanding the search space... find another approximate solution November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
29 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... until an approximate solution is found if we choose unsuitable initial guess November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
30 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... until an approximate solution is found if we choose unsuitable initial guess the algorithm will converge to... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
31 Motivation Iterative methods (Jacobi-Davidson) Lossy accelerator cavity: eigenvalues are in the complex plane initial guess choose an initial guess expand the search space... until an approximate solution is found if we choose unsuitable initial guess the algorithm will converge to... a previously determined eigenvalue!!!!! November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
32 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 5 / 28
33 Motivation Contour integral methods An accurate computation of eigenpairs inside a region enclosed by a non-selfintersecting curve. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 6 / 28
34 Motivation Contour integral methods An accurate computation of eigenpairs inside a region enclosed by a non-selfintersecting curve. choose a region to look for eigenvalues the region can be of any shape, e.g rectangle... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 6 / 28
35 Motivation Contour integral methods An accurate computation of eigenpairs inside a region enclosed by a non-selfintersecting curve. choose a region to look for eigenvalues the region can be of any shape, e.g rectangle... circle/ellipse November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 6 / 28
36 Motivation Contour integral methods An accurate computation of eigenpairs inside a region enclosed by a non-selfintersecting curve. choose a region to look for eigenvalues the region can be of any shape, e.g rectangle... circle/ellipse most computation is spent to solve linear equation systems at different interpolation points which can be parallelized. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 6 / 28
37 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 6 / 28
38 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 6 / 28
39 Formulation Mathematical model The combination of Maxwell-Ampère equation and the Maxwell-Faraday equation results in the double-curl equation 1 µ E jωσ E εω 2 E (1) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 7 / 28
40 Formulation Mathematical model The combination of Maxwell-Ampère equation and the Maxwell-Faraday equation results in the double-curl equation 1 µ E jωσ E εω 2 E (1) Applying the Galerkin s approach to discretize (1) results in an eigenvalue problem A 3D x + jωµ 0 C 3D x ω 2 µ 0 ε 0 B 3D x 0 (2) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 7 / 28
41 Formulation Mathematical model The combination of Maxwell-Ampère equation and the Maxwell-Faraday equation results in the double-curl equation 1 µ E jωσ E εω 2 E (1) Applying the Galerkin s approach to discretize (1) results in an eigenvalue problem A 3D x + jωµ 0 C 3D x ω 2 µ 0 ε 0 B 3D x 0 (2) which includes only losses from volumetric lossy material. Special treatment is carried out to incorporate 2D losses at port interfaces into (2), resulting in a nonlinear eigenvalue problem (NEP) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 7 / 28
42 Formulation Mathematical model The combination of Maxwell-Ampère equation and the Maxwell-Faraday equation results in the double-curl equation 1 µ E jωσ E εω 2 E (1) Applying the Galerkin s approach to discretize (1) results in an eigenvalue problem A 3D x + jωµ 0 C 3D x ω 2 µ 0 ε 0 B 3D x 0 (2) which includes only losses from volumetric lossy material. Special treatment is carried out to incorporate 2D losses at port interfaces into (2), resulting in a nonlinear eigenvalue problem (NEP) P(ω) x 0 (3) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 7 / 28
43 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 7 / 28
44 Formulation Eigenvalue algorithms To solve P(z)x 0 most of the standard eigenvalue algorithms exploit a projection procedure in order to extract approximate eigenvectors from a given subspace. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 8 / 28
45 Formulation Eigenvalue algorithms To solve P(z)x 0 most of the standard eigenvalue algorithms exploit a projection procedure in order to extract approximate eigenvectors from a given subspace. Approximate an obtain matrix Q eigenspace Q November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 8 / 28
46 Formulation Eigenvalue algorithms To solve P(z)x 0 most of the standard eigenvalue algorithms exploit a projection procedure in order to extract approximate eigenvectors from a given subspace. Approximate an obtain matrix Q eigenspace Q Project the original NEP to a reduced NEP using Rayleigh-Ritz procedure: P Q (z) Q H P(z)Q November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 8 / 28
47 Formulation Eigenvalue algorithms To solve P(z)x 0 most of the standard eigenvalue algorithms exploit a projection procedure in order to extract approximate eigenvectors from a given subspace. Approximate an obtain matrix Q eigenspace Q Project the original NEP to a reduced NEP Solve the reduced NEP using Rayleigh-Ritz procedure: P Q (z) Q H P(z)Q solve P Q (z)g 0 for eigenvalues z and eigenvectors g November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 8 / 28
48 Formulation Eigenvalue algorithms To solve P(z)x 0 most of the standard eigenvalue algorithms exploit a projection procedure in order to extract approximate eigenvectors from a given subspace. Approximate an obtain matrix Q eigenspace Q Project the original NEP to a reduced NEP Solve the reduced NEP Compute eigenpairs of the original NEP using Rayleigh-Ritz procedure: P Q (z) Q H P(z)Q solve P Q (z)g 0 for eigenvalues z and eigenvectors g same eigenvalues as for the reduced problem; eigenvectors x Qg November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 8 / 28
49 Formulation Eigenvalue algorithms To solve P(z)x 0 most of the standard eigenvalue algorithms exploit a projection procedure in order to extract approximate eigenvectors from a given subspace. Approximate an obtain matrix Q eigenspace Q Project the original NEP to a reduced NEP Solve the reduced NEP Compute eigenpairs of the original NEP using Rayleigh-Ritz procedure: P Q (z) Q H P(z)Q solve P Q (z)g 0 for eigenvalues z and eigenvectors g same eigenvalues as for the reduced problem; eigenvectors x Qg November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 8 / 28
50 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 8 / 28
51 Formulation Contour integral methods Some basic spectral theory The resolvent P(z) 1 reveals the existence of eigenvalues, indicates where eigenvalues are located, and show how sensitive these eigenvalues are to pertubation. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 9 / 28
52 Formulation Contour integral methods Some basic spectral theory The resolvent P(z) 1 reveals the existence of eigenvalues, indicates where eigenvalues are located, and show how sensitive these eigenvalues are to pertubation. As explained in [3], from Keldysh s theorem, we know that the resolvent function P(z) 1 can be written (for simple eigenvalues λ i ) as P(z) 1 v i w H i i 1 + R(z) (4) z λ i where v i and w i are suitably scaled right and left eigenvectors, respectively, corresponding to the (simple) eigenvalue λ i R(z) and P(z) are analytic functions November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 9 / 28
53 Formulation Contour integral methods Some basic spectral theory November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 10 / 28
54 Formulation Contour integral methods Some basic spectral theory Q (q 1, q 2,..., q k ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 10 / 28
55 Formulation Contour integral methods Some basic spectral theory Q (q 1, q 2,..., q k ) span{q 1, q 2,..., q k } span{v 1, v 2,..., v n(γ) } November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 10 / 28
56 Formulation Contour integral methods Some basic spectral theory P(z) 1 v iw H i i 1 z λ i + R(z) Q (q 1, q 2,..., q k ) span{q 1, q 2,..., q k } span{v 1, v 2,..., v n(γ) } November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 10 / 28
57 Formulation Contour integral methods Some basic spectral theory P(z) 1 v iw H i i 1 + R(z) z λ i Applying Cauchy s integral formula n(γ) 1 f (z)p(z) 1 dz f (λ i)v iw H i 2πi Γ i1 Q (q 1, q 2,..., q k ) span{q 1, q 2,..., q k } span{v 1, v 2,..., v n(γ) } November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 10 / 28
58 Formulation Contour integral methods Some basic spectral theory P(z) 1 v iw H i i 1 + R(z) z λ i Applying Cauchy s integral formula n(γ) 1 f (z)p(z) 1 dz f (λ i)v iw H i 2πi Γ i1 In practice, we evaluate the integral 1 2πi f (z)p(z) 1 ˆV dz Γ Q (q 1, q 2,..., q k ) span{q 1, q 2,..., q k } span{v 1, v 2,..., v n(γ) } November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 10 / 28
59 Formulation Contour integral methods Some basic spectral theory Q (q 1, q 2,..., q k ) span{q 1, q 2,..., q k } span{v 1, v 2,..., v n(γ) } P(z) 1 v iw H i i 1 + R(z) z λ i Applying Cauchy s integral formula n(γ) 1 f (z)p(z) 1 dz f (λ i)v iw H i 2πi Γ i1 In practice, we evaluate the integral 1 2πi f (z)p(z) 1 ˆV dz Γ Using interpolation, we obtain n 1 int f (z)p(z) 1 ˆV dz ξ if (z i)p(z i) 1 ˆV 2πi Γ i1 November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 10 / 28
60 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 10 / 28
61 Formulation Multigrid method as a preconditioner The most expensive operation is to compute X P 1 (z i )V (5) equivalent to solving the linear system P(z i )X V (6) Direct inverse becomes prohibitively expensive for large problems. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 11 / 28
62 Formulation Multigrid method as a preconditioner The most expensive operation is to compute equivalent to solving the linear system X P 1 (z i )V (5) P(z i )X V (6) Direct inverse becomes prohibitively expensive for large problems. For large-scale problems, iterative methods are preferable. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 11 / 28
63 Formulation Multigrid method as a preconditioner The most expensive operation is to compute equivalent to solving the linear system X P 1 (z i )V (5) P(z i )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. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 11 / 28
64 Formulation Multigrid method as a preconditioner The most expensive operation is to compute equivalent to solving the linear system X P 1 (z i )V (5) P(z i )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. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 11 / 28
65 Formulation Multigrid method as a preconditioner The most expensive operation is to compute equivalent to solving the linear system X P 1 (z i )V (5) P(z i )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 convergence of the iterative solvers. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 11 / 28
66 Formulation Multigrid method as a preconditioner reduced 1st order full 1st order reduced 2nd order Pär Ingelström, "A New Set of H(curl) Conforming Hierarchical Basis Functions for Tetrahedral Meshes, IEEE Transactions on Microwave Theory and Techniques, vol. 54, no. 1, Jan November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 12 / 28
67 Formulation Multigrid method as a preconditioner i4 i3 M 1 b e (7) i2 i1 W-cycle Pär Ingelström et al., "Comparison of Hierarchical Basis Functions for Efficient Multilevel Solvers", IET Science, Measurement and Technology, vol. 1, no. 1, Jan This equation is repeatedly computed at each iteration where M is the preconditioner, b is the input and e is the output. The output is computed by solving systems of the type P ii e i b i i j P ij e j (8) where i and j refer to the order of the trial and test functions. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 13 / 28
68 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 13 / 28
69 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) CST cavity design, FEM discretization November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 14 / 28
70 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) CST cavity design, FEM discretization CEM3D [5] generate matrices P(z i ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 14 / 28
71 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) CST cavity design, FEM discretization CEM3D [5] generate matrices P(z i ) NES4AC Solve nonlinear eigenvalue problem November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 14 / 28
72 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) 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)xv or xp(z) -1 v November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 15 / 28
73 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) Parallell level 1: multiple contours Parallel level 2: compute nodes in parallel (implemented) Parallel level 3: parallelize computation at each node (implemented in the codes) P(λ)xv or xp(z) -1 v November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 15 / 28
74 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) Parallell level 1: multiple contours z6 z 5 z 4 z 7 z 3 z 8 z 2 z 9 z 1 z 10 z 16 z 11z12z13 z 15 z 14 V P 1 (z)vdz Γ Parallel level 2: compute nodes in parallel (implemented) Parallel level 3: parallelize computation at each node (implemented in the codes) P(λ)xv or xp(z) -1 v November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 15 / 28
75 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) Parallell level 1: multiple contours z6 z 5 z 4 z 7 z 3 z 8 z 2 z 9 z 1 z 10 z 16 z 11z12 z 15 z13 z 14 V P 1 (z)vdz Γ Parallel level 2: compute nodes in parallel (implemented) subcomm 1 z1,a1,b1 z2,a2,b2 z3,a3,b3 z4,a4,b4 subcomm 2 z5,a5,b5 z6,a6,b6 z7,a7,b7 z8,a8,b8 subcomm 3 z9,a9,b9 z10, A10, B10 z11, A11, B11 z12, A12, B12 subcomm 4 z13, A13, B13 z14, A14, B14 z15, A15, B15 z16, A16, B16 Parallel level 3: parallelize computation at each node (implemented in the codes) P(λ)xv or xp(z) -1 v November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 15 / 28
76 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) Parallell level 1: multiple contours z6 z 5 z 4 z 7 z 3 z 8 z 2 z 9 z 1 z 10 z 16 z 11z12 z 15 z13 z 14 V P 1 (z)vdz Γ Parallel level 2: compute nodes in parallel (implemented) subcomm 1 z1,a1,b1 z2,a2,b2 z3,a3,b3 z4,a4,b4 subcomm 2 z5,a5,b5 z6,a6,b6 z7,a7,b7 z8,a8,b8 subcomm 3 z9,a9,b9 z10, A10, B10 z11, A11, B11 z12, A12, B12 subcomm 4 z13, A13, B13 z14, A14, B14 z15, A15, B15 z16, A16, B16 P 1 (z1)v P 1 (z5)v P 1 (z9)v P 1 (z13)v P 1 (z2)v P 1 (z6)v P 1 (z10)v P 1 (z14)v Parallel level 3: parallelize computation at each node (implemented in the codes) P 1 (z3)v P 1 (z4)v P 1 (z7)v P 1 (z8)v P 1 (z11)v P 1 (z12)v P 1 (z15)v P 1 (z16)v P(λ)xv or xp(z) -1 v November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 15 / 28
77 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) Parallell level 1: multiple contours z6 z 5 z 4 z 7 z 3 z 8 z 2 z 9 z 1 z 10 z 16 z 11z12 z 15 z13 z 14 V P 1 (z)vdz Γ Parallel level 2: compute nodes in parallel (implemented) subcomm 1 z1,a1,b1 z2,a2,b2 z3,a3,b3 z4,a4,b4 subcomm 2 z5,a5,b5 z6,a6,b6 z7,a7,b7 z8,a8,b8 subcomm 3 z9,a9,b9 z10, A10, B10 z11, A11, B11 z12, A12, B12 subcomm 4 z13, A13, B13 z14, A14, B14 z15, A15, B15 z16, A16, B16 P 1 (z1)v P 1 (z5)v P 1 (z9)v P 1 (z13)v P 1 (z2)v P 1 (z6)v P 1 (z10)v P 1 (z14)v Parallel level 3: parallelize computation at each node (implemented) P 1 (z3)v P 1 (z4)v P 1 (z7)v P 1 (z8)v P 1 (z11)v P 1 (z12)v P 1 (z15)v P 1 (z16)v P(λ)xv or xp(z) -1 v November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 15 / 28
78 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) Parallell level 1: multiple contours z6 z 5 z 4 z 7 z 3 z 8 z 2 z 9 z 1 z 10 z 16 z 11z12 z 15 z13 z 14 V P 1 (z)vdz Γ Parallel level 2: compute nodes in parallel (implemented) subcomm 1 z1,a1,b1 z2,a2,b2 z3,a3,b3 z4,a4,b4 subcomm 2 z5,a5,b5 z6,a6,b6 z7,a7,b7 z8,a8,b8 subcomm 3 z9,a9,b9 z10, A10, B10 z11, A11, B11 z12, A12, B12 subcomm 4 z13, A13, B13 z14, A14, B14 z15, A15, B15 z16, A16, B16 P 1 (z1)v P 1 (z5)v P 1 (z9)v P 1 (z13)v P 1 (z2)v P 1 (z6)v P 1 (z10)v P 1 (z14)v P(λ)xv or xp(z) -1 v Parallel level 3: parallelize computation at each node (implemented) P 1 (z3)v P 1 (z4)v P 1 (z7)v P 1 (z11)v P 1 (z8)v P 1 (z12)v P 1 (z i)v i P 1 (z15)v P 1 (z16)v November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 15 / 28
79 Implementation Nonlinear Eigenvalue Solver for Accelerator Cavities (NES4AC) NES4AC highlights: extends the functionality of CEM3D [5]. parallelized and developed in C++. based on PETSc (Portable, Extensible Toolkit for Scientific Computation) v3.3.0 and LAPACK. adopts the parallel scheme of the contour integral method from SLEPc (Scalable Library for Eigenvalue Problem Computations). uses the superlu_dist for the computation of LU decompositions. including three contour integral algorithms for eigenvalue solution: Beyn1 (for a few eigenvalues), Beyn2 (for many eigenvalues) and RSRR. with two types of closed contour: ellipse and rectangle. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 16 / 28
80 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 16 / 28
81 Preliminary Results Nonlinear Eigenvalues Problems in [6] Butterfly Problem Name: butterfly (Quartic matrix polynomial with T-even structure) P(λ) λ 4 A 4 + λ 3 A 3 + λ 2 A 2 + λa 1 + A 0 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 November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 17 / 28
82 Preliminary Results Nonlinear Eigenvalues Problems in [6] Butterfly Problem Eigenvalues for butterfly problem - case 1 Eigenvalues for butterfly problem - case matlab - polyeig beyn matlab - polyeig beyn2 imaginary imaginary imaginary real Eigenvalues for butterfly problem - case matlab - polyeig rsrr real beyn1 beyn2 rsrr Min. residual Max. residual ɛ P(λ)x P(λ) x real November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 18 / 28
83 Preliminary Results Nonlinear Eigenvalues Problems in [6] Schrödinger Name: Schrödinger (QEP from Schrödinger operator) P(λ) K 2λC + λ 2 B 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 November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 19 / 28
84 Preliminary Results Nonlinear Eigenvalues Problems in [6] Schrödinger Eigenvalues for Schrödinger problem - case matlab - polyeig 0.4 beyn1 Eigenvalues for Schrödinger problem - case matlab - polyeig 0.4 beyn2 imaginary imaginary real Eigenvalues for Schrödinger problem - case matlab - polyeig 0.4 rsrr imaginary real beyn1 beyn2 rsrr Min. residual n/a Max. residual n/a real November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 20 / 28
85 Preliminary Results Spherical Cavity Name: spherical cavity Electrical conductivity: Ohm/sq Radius: 1m Size: 7614/17830/61106/143354/ Target frequency: 125MHz Number of eigenvalues: 3 Algorithm parameters: Region: rectangle(1.0, 1.5, , 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 November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 21 / 28
86 Preliminary Results Spherical Cavity ɛ f f f analytical f analytical ɛ Q Q Q analytical Q analytical November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 22 / 28
87 Preliminary Results Multigrid preconditioner - Spherical Cavity # elements # dofs # dofs (1st order) Dis , Dis. 2 1,256 6,640 1,078 Dis. 3 2,918 16,300 2,755 Dis. 4 18, ,828 18,487 Table: Mesh refinement of a spherical cavity Dis. 1 Dis. 2 Dis. 3 Dis. 4 mg-gmres 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 November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 23 / 28
88 Preliminary Results Multigrid preconditioner - Tesla Cavity # elements # dofs # dofs (1st order) Dis. 1 48, ,026 50,067 Dis. 2 57, ,664 59,864 Dis. 3 76, ,812 80,552 Table: Mesh refinement of a Tesla cavity Dis. 1 Dis. 2 Dis. 3 mg-gmres GMRES 500 (0.2) 500 (0.15) 500 (0.2) Table: Number of iterations required to achieve a residual of 1.0e-8 November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 24 / 28
89 Presentation Outline Motivation Iterative methods Contour integral methods Formulation Mathematical model Eigenvalue algorithms Contour integral methods Multigrid method as a preconditioner Implementation Preliminary Results Possible Improvements November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 24 / 28
90 Possible Improvements Recycling Krylov subspace methods X i P 1 (z i )V P(z i )X i V P(z 1) X 1 V P(z 2) X 2 V P(z 3) X 3 V... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 25 / 28
91 Possible Improvements Recycling Krylov subspace methods X i P 1 (z i )V P(z i )X i V P(z 1) X 1 V Iterative method is repeatedly applied for different RHS. P(z 2) X 2 V P(z 3) X 3 V... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 25 / 28
92 Possible Improvements Recycling Krylov subspace methods X i P 1 (z i )V P(z i )X i V P(z 1) X 1 V Iterative method is repeatedly applied for different RHS. The matrix is unchanged. P(z 2) X 2 V P(z 3) X 3 V... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 25 / 28
93 Possible Improvements Recycling Krylov subspace methods X i P 1 (z i )V P(z i )X i V P(z 1) X 1 V P(z 2) X 2 V Iterative method is repeatedly applied for different RHS. The matrix is unchanged. Opportunity for applying recycling Krylov subspace methods or block Krylov subspace methods. P(z 3) X 3 V... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 25 / 28
94 Possible Improvements Recycling Krylov subspace methods X i P 1 (z i )V P(z i )X i V P(z 1) X 1 V P(z 2) X 2 V Iterative method is repeatedly applied for different RHS. The matrix is unchanged. Opportunity for applying recycling Krylov subspace methods or block Krylov subspace methods. The system-matrices are slightly changed for different interpolation points P(z 3) X 3 V... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 25 / 28
95 Possible Improvements Recycling Krylov subspace methods X i P 1 (z i )V P(z i )X i V P(z 1) X 1 V P(z 2) X 2 V P(z 3) X 3 V Iterative method is repeatedly applied for different RHS. The matrix is unchanged. Opportunity for applying recycling Krylov subspace methods or block Krylov subspace methods. The system-matrices are slightly changed for different interpolation points Opportunity for applying recycling Krylov subspace methods or block Krylov subspace methods.... November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 25 / 28
96 Thank you for your attention November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 26 / 28
97 References I [1] T. Flisgen, J. Heller, T. Galek, L. Shi, N. Joshi, N. Baboi, R. M. Jones, and U. van Rienen, Eigenmode Compendium of The Third Harmonic Module of the European X-Ray Free Electron Laser, Physical Review Accelerators and Beams, vol. 20, p , Apr [2] H. Voss, A Jacobi-Davidson Method for Nonlinear and Nonsymmetric Eigenproblems, Computers and Structures, vol. 85, no , pp , [3] W.-J. Beyn, An Integral Method for Solving Nonlinear Eigenvalue Problems, Linear Algebra and its Applications, vol. 436, no. 10, pp , November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 27 / 28
98 References II [4] J. Xiao, C. Zhang, T. M. Huang, and T. Sakurai, Solving Large-Scale Nonlinear Eigenvalue Problems by Rational Interpolation and Resolvent Sampling Based Rayleigh-Ritz Method, International Journal for Numerical Methods in Engineering, vol. 110, no. 8, pp , [5] T. Banova, W. Ackermann, and T. Weiland, Accurate Determination of Thousands of Eigenvalues for Large-Scale Eigenvalue Problems, IEEE Transactions on Magnetics, vol. 50, pp , Feb [6] T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur, NLEVP: A Collection of Nonlinear Eigenvalue Problems, ACM Transactions on Mathematical Software, vol. 39, no. 2, pp. 7:1 7:28, November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 28 / 28
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100 Appendix Contour integral methods Beyn1 (for a few eigenvalues) Define the matrices A 0 and A 1 C n k Then A 0 VW H ˆV and A 1 V ΛW H ˆV where Λ diag(λ 1,..., λ n(γ) ) V [ v 1 v n(γ) ] W [ w 1 w n(γ) ] A 0 1 P(z) 1 ˆV dz (9) 2πi Γ A 1 1 zp(z) 1 ˆV dz (10) 2πi Γ ˆV is a random matrix ˆV C n L. L is smaller than n and equal or greater and k November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 29 / 28
101 Appendix Contour integral methods Beyn1 (for a few eigenvalues) Beyn s method is based on the singular value decomposition of A 0 A 0 V 0 Σ 0 W H 0 (11) Beyn has shown that the matrix B V H 0 A 1W H 0 Σ 1 0 (12) is diagonalizable. Its eigenvalues are the eigenvalues of P inside the contour and its eigenvectors lead to the corresponding eigenvectors of P. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 30 / 28
102 Appendix Contour integral methods Beyn2 (for many eigenvalues) Define the matrices A p C n k A p 1 z p P(z) 1 ˆV dz (13) 2πi Γ Then A p V Λ p W H ˆV. The matrices B 0 and B 1 are defined as follows B 0 A 0 A K 1.. A K 1 A 2K 2 ; B 1 A 1 A K.. A K A 2K 1 (14) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 31 / 28
103 Appendix Contour integral methods Beyn2 (for many eigenvalues) Performing the singular value decomposition of B 0 B 0 V 0 Σ 0 W H 0 (15) Beyn has shown that the matrix D V H 0 B 1W H 0 Σ 1 0 (16) is diagonalizable. Its eigenvalues are the eigenvalues of P inside the contour and its eigenvectors lead to the corresponding eigenvectors of P. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 32 / 28
104 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 5 z 4 z 3 z 6 z 2 z 7 z 1 V z 8 z 12 z 9 z 10 z 11 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) ( ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
105 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 5 z 4 z 3 z 6 z 2 z 7 z 1 V z 8 z 12 z 9 z 10 z 11 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) ( ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
106 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 5 z 4 z 3 z 6 z 2 z 7 z 1 V z 8 z 12 z 9 z 10 z 11 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) ( ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
107 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 5 z 4 z 3 z 6 z 2 z 7 z 1 V z 8 z 12 z 9 z 10 z 11 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) ( ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
108 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 5 z 4 z 3 z 6 z 2 z 7 z 1 V z 8 z 12 z 9 z 10 z 11 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) ( ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
109 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 7 z 6 z 8 z 5 z 4 z 3 z 9 z 10 z 11 z 2 z 12 z 1 V The Beyn2 algorithm is robust and accurate if a large L but a small K are used. However, for large-scale problems, a small L is essential to reduce the computational burden. 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) ( ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
110 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 7 z 6 z 8 z 5 z 4 z 3 z 9 z 10 z 11 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) z 2 z 12 z 1 V The Beyn2 algorithm is robust and accurate if a large L but a small K are used. However, for large-scale problems, a small L is essential to reduce the computational burden. Decrease L and increase K make the algorithm unstable and inaccurate. ( ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
111 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 7 z 6 z 8 z 5 z 4 z 3 z 9 z 10 z 11 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) z 2 z 12 z 1 V The Beyn2 algorithm is robust and accurate if a large L but a small K are used. However, for large-scale problems, a small L is essential to reduce the computational burden. Decrease L and increase K make the algorithm unstable and inaccurate. (,,,,,,,,, ) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
112 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 7 z 6 z 8 z 5 z 4 z 3 z 9 z 10 z 11 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) (,,,,,,,,, ) z 2 z 12 z 1 V The Beyn2 algorithm is robust and accurate if a large L but a small K are used. However, for large-scale problems, a small L is essential to reduce the computational burden. Decrease L and increase K make the algorithm unstable and inaccurate. RSRR reduce the number of columns of V. November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
113 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method z 7 z 6 z 8 z 5 z 4 z 3 z 9 z 10 z 11 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) (,,,,,,,,, ) z 2 z 12 z 1 V The Beyn2 algorithm is robust and accurate if a large L but a small K are used. However, for large-scale problems, a small L is essential to reduce the computational burden. Decrease L and increase K make the algorithm unstable and inaccurate. RSRR reduce the number of columns of V. Let Q C n k be an orthogonal basis of search space, then the original NEP can be converted to the following reduced NEP P Q (z)g 0 November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 33 / 28
114 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method (1) Initialization: Fix the contour Γ, the number N and the sampling points z i. Fix the number L and generate a n L random matrix U November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 34 / 28
115 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method (1) Initialization: Fix the contour Γ, the number N and the sampling points z i. Fix the number L and generate a n L random matrix U (2) Compute P(z i ) 1 U for i 0, 1,..., N 1 November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 34 / 28
116 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method (1) Initialization: Fix the contour Γ, the number N and the sampling points z i. Fix the number L and generate a n L random matrix U (2) Compute P(z i ) 1 U for i 0, 1,..., N 1 (3) Form S as follows November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 34 / 28
117 Appendix Contour integral methods Resolvent Sampling based Rayleigh-Ritz method (1) Initialization: Fix the contour Γ, the number N and the sampling points z i. Fix the number L and generate a n L random matrix U (2) Compute P(z i ) 1 U for i 0, 1,..., N 1 (3) Form S as follows S [ P(z 0 ) 1 U, P(z 1 ) 1 U,, P(z N 1 ) 1 U ] C n N L (17) November 14, 2017 TU Darmstadt Fachbereich 18 Institut für Theorie Elektromagnetischer Felder Vinh Pham-Xuan 34 / 28
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