領域分割型並列有限要素解析のための固有モード縮約に基づく反復法前処理
Deflated 共役勾配法は，任意の既知である線形独立な複数のベクトルを係数行列から縮約する前処理を施した共役勾配法である．本研究では，deflated 共役勾配法に入力する基底として，並列計算の分割領域における固有値解析で取得した低次固有モードを利用する subdomain 固有モード deflation を提案する．提案手法の計算性能は，薄板モデルの構造解析に適用することで，反復回数と計算時間を用いて評価する．数値例より，提案手法が良好な前処理性能および並列計算性能を示すことを確認し，標準的な共役勾配法と比較して，総計算時間を最大85%削減することが示された．...
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| Published in | 日本計算工学会論文集 Vol. 2024; no. 1; p. 20241011 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | Japanese |
| Published |
一般社団法人 日本計算工学会
07.08.2024
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1347-8826 |
| DOI | 10.11421/jsces.2024.20241011 |
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| Abstract | Deflated 共役勾配法は，任意の既知である線形独立な複数のベクトルを係数行列から縮約する前処理を施した共役勾配法である．本研究では，deflated 共役勾配法に入力する基底として，並列計算の分割領域における固有値解析で取得した低次固有モードを利用する subdomain 固有モード deflation を提案する．提案手法の計算性能は，薄板モデルの構造解析に適用することで，反復回数と計算時間を用いて評価する．数値例より，提案手法が良好な前処理性能および並列計算性能を示すことを確認し，標準的な共役勾配法と比較して，総計算時間を最大85%削減することが示された． |
|---|---|
| AbstractList | Deflated 共役勾配法は，任意の既知である線形独立な複数のベクトルを係数行列から縮約する前処理を施した共役勾配法である．本研究では，deflated 共役勾配法に入力する基底として，並列計算の分割領域における固有値解析で取得した低次固有モードを利用する subdomain 固有モード deflation を提案する．提案手法の計算性能は，薄板モデルの構造解析に適用することで，反復回数と計算時間を用いて評価する．数値例より，提案手法が良好な前処理性能および並列計算性能を示すことを確認し，標準的な共役勾配法と比較して，総計算時間を最大85%削減することが示された． |
| Author | 森田, 直樹 村井, 拓海 三目, 直登 |
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| References | (8) Hestenes M. R. and Stiefel E., Methods of conjugate gradients for solving linear systems, Journal of research of the National Bureau of Standards, 49-6, 1952, pp. 409-436. (23) Cybermedia center blog archive squid, osaka university, http://www.hpc.cmc.osaka-u.ac.jp/squid/, 2023 年 12 月 6 日閲覧. (17) Cortes, G. B. D., Vuik, C., and Jansen, J. D., On pod-based deflation vectors for dpcg applied to porous media problems, Journal of Computational and Applied Mathematics, 330, 2018, pp. 193-213. (14) Vanek, P., Mandel, J., and Brezina, M., Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing, 56-3, 1996, pp. 179-196. (1) Nicolaides, R. A., Deflation of conjugate gradients with applications to boundary value problems, SIAM Journal on Numerical Analysis, 24-2, 1987, pp. 355-365. (5) Knyazev, A. V., Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM Journal on Scientific Computing, 23-2, 2001, pp. 517-541. (4) Golub, G. H. and Underwood, R., The block lanczos method for computing eigenvalues, Mathematical software, 1977, pp. 361-377. (15) Concus, P., Golub, G. H., and Meurant, G., Block preconditioning for the conjugate gradient method, SIAM Journal on Scientific and Statistical Computing, 6-1, 1985, pp. 220-252. (22) 森田直樹, 集路幸正, 田中克治, 馬込望, 新舘京平, 柴沼一樹, 三目直登, 領域分割型並列シミュレーションのためのグラフ構造に基づく統一的ライブラリと多手法への展開, 日本計算工学会論文集, 2024, p. 20241008, 2024. (3) Igarashi, H. and Watanabe, K., Deflation techniques for computational electromagnetism: Theoretical considerations, IEEE transactions on magnetics, 47-5, 2011, pp. 1438-1441. (12) Tang, J. M., Nabben, R., Vuik, C., and Erlangga, Y. A., Theoretical and numerical comparison of various projection methods derived from deflation, domain decomposition and multigrid methods. Reports of the Department of Applied Mathematical Analysis, 07-04, 2007. (16) Baggio, R., Franceschini, A., Spiezia, N., and Janna, C., Rigid body modes deflation of the preconditioned conjugate gradient in the solution of discretized structural problems, Computers & Structures, 185, 2017, pp. 15-26. (13) Kaasschieter, E. F., Preconditioned conjugate gradients for solving singular systems. Journal of Computational and Applied mathematics, 24-1, 1988, pp. 265-275. (24) Karypis, G. and Kumar, V., Metis: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices, Computer Science & Engineering (CS&E) Technical Reports, 97-061, 1997. (25) Center for collision safety and analysis - 2010 toyota yaris, https://www.ccsa.gmu.edu/models/2010-toyota-yaris/, 2023 年 12 月 7 日閲覧. (20) Liu, W. and Vinter B., CSR5: An efficient storage format for cross-platform sparse matrix-vector multiplication, Proceedings of the 29th ACM on International Conference on Supercomputing, 2015, pp. 339-350. (2) Tang, J. M., MacLachlan, S. P., Nabben, R., and Vuik, C., A comparison of two-level preconditioners based on multigrid and deflation, SIAM Journal on Matrix Analysis and Applications, 31-4, 2010, pp. 1715-1739. (18) Baiges, J., Codina, R., and Idelsohn, S., A domain decomposition strategy for reduced order models. application to the incompressible navier-stokes equations, Computer Methods in Applied Mechanics and Engineering, 267, 2013, pp. 23-42. (21) monolis: monolithic linear solver based on domain decomposition, https://www.kz.tsukuba.ac.jp/~nmorita/monolis.html, 2023 年 12 月 22 日閲覧. (19) 奥田洋司, 柄谷和輝, 永田雅喜, 並列化プラグイン機能を利用した並列有限要素法コードの開発とその性能評価, 日本機械学会論文集, A 編, 68-671, 2002, pp. 1002-1009. (10) Agathos, K., Dodwell, T., Chatzi, E., and Bordas, S. P. A., An adapted deflated conjugate gradient solver for robust extended/generalised finite element solutions of large scale, 3D crack propagation problems, Computer Methods in Applied Mechanics and Engineering, 395, 2022, p. 114937. (11) Saad, Y., Yeung, M., Erhel, J., and Guyomarc’h, F., A deflated version of the conjugate gradient algorithm, SIAM Journal on Scientific Computing, 21-5, 2000, pp. 1909-1926. (7) Frank, J. and Vuik, C., On the construction of deflation-based preconditioners, SIAM Journal on Scientific Computing, 23-2, 2001, pp. 442-462. (9) Jonsthovel, T. B., van Gijzen, M. B., Vuik, C., and Scarpas, A., On the use of rigid body modes in the deflated preconditioned conjugate gradient method, SIAM Journal on Scientific Computing, 35-1, 2013, pp. B207-B225. (6) Bergamaschi, L., A survey of low-rank updates of preconditioners for sequences of symmetric linear systems, Algorithms, 13-4, 2020, p. 100. |
| References_xml | – reference: (3) Igarashi, H. and Watanabe, K., Deflation techniques for computational electromagnetism: Theoretical considerations, IEEE transactions on magnetics, 47-5, 2011, pp. 1438-1441. – reference: (12) Tang, J. M., Nabben, R., Vuik, C., and Erlangga, Y. A., Theoretical and numerical comparison of various projection methods derived from deflation, domain decomposition and multigrid methods. Reports of the Department of Applied Mathematical Analysis, 07-04, 2007. – reference: (23) Cybermedia center blog archive squid, osaka university, http://www.hpc.cmc.osaka-u.ac.jp/squid/, 2023 年 12 月 6 日閲覧. – reference: (9) Jonsthovel, T. B., van Gijzen, M. B., Vuik, C., and Scarpas, A., On the use of rigid body modes in the deflated preconditioned conjugate gradient method, SIAM Journal on Scientific Computing, 35-1, 2013, pp. B207-B225. – reference: (8) Hestenes M. R. and Stiefel E., Methods of conjugate gradients for solving linear systems, Journal of research of the National Bureau of Standards, 49-6, 1952, pp. 409-436. – reference: (17) Cortes, G. B. D., Vuik, C., and Jansen, J. D., On pod-based deflation vectors for dpcg applied to porous media problems, Journal of Computational and Applied Mathematics, 330, 2018, pp. 193-213. – reference: (16) Baggio, R., Franceschini, A., Spiezia, N., and Janna, C., Rigid body modes deflation of the preconditioned conjugate gradient in the solution of discretized structural problems, Computers & Structures, 185, 2017, pp. 15-26. – reference: (15) Concus, P., Golub, G. H., and Meurant, G., Block preconditioning for the conjugate gradient method, SIAM Journal on Scientific and Statistical Computing, 6-1, 1985, pp. 220-252. – reference: (11) Saad, Y., Yeung, M., Erhel, J., and Guyomarc’h, F., A deflated version of the conjugate gradient algorithm, SIAM Journal on Scientific Computing, 21-5, 2000, pp. 1909-1926. – reference: (20) Liu, W. and Vinter B., CSR5: An efficient storage format for cross-platform sparse matrix-vector multiplication, Proceedings of the 29th ACM on International Conference on Supercomputing, 2015, pp. 339-350. – reference: (1) Nicolaides, R. A., Deflation of conjugate gradients with applications to boundary value problems, SIAM Journal on Numerical Analysis, 24-2, 1987, pp. 355-365. – reference: (25) Center for collision safety and analysis - 2010 toyota yaris, https://www.ccsa.gmu.edu/models/2010-toyota-yaris/, 2023 年 12 月 7 日閲覧. – reference: (6) Bergamaschi, L., A survey of low-rank updates of preconditioners for sequences of symmetric linear systems, Algorithms, 13-4, 2020, p. 100. – reference: (10) Agathos, K., Dodwell, T., Chatzi, E., and Bordas, S. P. A., An adapted deflated conjugate gradient solver for robust extended/generalised finite element solutions of large scale, 3D crack propagation problems, Computer Methods in Applied Mechanics and Engineering, 395, 2022, p. 114937. – reference: (7) Frank, J. and Vuik, C., On the construction of deflation-based preconditioners, SIAM Journal on Scientific Computing, 23-2, 2001, pp. 442-462. – reference: (4) Golub, G. H. and Underwood, R., The block lanczos method for computing eigenvalues, Mathematical software, 1977, pp. 361-377. – reference: (18) Baiges, J., Codina, R., and Idelsohn, S., A domain decomposition strategy for reduced order models. application to the incompressible navier-stokes equations, Computer Methods in Applied Mechanics and Engineering, 267, 2013, pp. 23-42. – reference: (19) 奥田洋司, 柄谷和輝, 永田雅喜, 並列化プラグイン機能を利用した並列有限要素法コードの開発とその性能評価, 日本機械学会論文集, A 編, 68-671, 2002, pp. 1002-1009. – reference: (22) 森田直樹, 集路幸正, 田中克治, 馬込望, 新舘京平, 柴沼一樹, 三目直登, 領域分割型並列シミュレーションのためのグラフ構造に基づく統一的ライブラリと多手法への展開, 日本計算工学会論文集, 2024, p. 20241008, 2024. – reference: (21) monolis: monolithic linear solver based on domain decomposition, https://www.kz.tsukuba.ac.jp/~nmorita/monolis.html, 2023 年 12 月 22 日閲覧. – reference: (2) Tang, J. M., MacLachlan, S. P., Nabben, R., and Vuik, C., A comparison of two-level preconditioners based on multigrid and deflation, SIAM Journal on Matrix Analysis and Applications, 31-4, 2010, pp. 1715-1739. – reference: (13) Kaasschieter, E. F., Preconditioned conjugate gradients for solving singular systems. Journal of Computational and Applied mathematics, 24-1, 1988, pp. 265-275. – reference: (24) Karypis, G. and Kumar, V., Metis: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices, Computer Science & Engineering (CS&E) Technical Reports, 97-061, 1997. – reference: (5) Knyazev, A. V., Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM Journal on Scientific Computing, 23-2, 2001, pp. 517-541. – reference: (14) Vanek, P., Mandel, J., and Brezina, M., Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing, 56-3, 1996, pp. 179-196. |
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| Snippet | Deflated 共役勾配法は，任意の既知である線形独立な複数のベクトルを係数行列から縮約する前処理を施した共役勾配法である．本研究では，deflated 共役勾配法に入力する... |
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| SubjectTerms | Deflated Conjugate Gradient Method Eigenmode Deflation Finite Element Method Parallelization Preconditioning Subdomain Deflation |
| Title | 領域分割型並列有限要素解析のための固有モード縮約に基づく反復法前処理 |
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