电邮这篇文章 K-OPTIMAL PRECONDITIONERS BASED ON APPROXIMATIONS OF INVERSE MATRICES
- 作者: Oseledets I.V1,2,3, Muravleva E.A4,2
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隶属关系:
- Artificial Intelligence Research Institute (AIRI)
- Skolkovo Institute of Science and Technology
- Marchuk Institute of Computational Mathematics RAS
- Sberbank AI Center for Science
- 期: 卷 65, 编号 7 (2025)
- 页面: 1143-1155
- 栏目: General numerical methods
- URL: https://bakhtiniada.ru/0044-4669/article/view/304081
- DOI: https://doi.org/10.31857/S0044466925070063
- EDN: https://elibrary.ru/JXYARI
- ID: 304081
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详细
The problem of constructing preconditioners of a special kind for solving systems of linear algebraic equations is considered. A new approach to the construction of preconditioners based on minimizing the K-number of conditionality for the A−1P matrix is proposed, where A is the initial matrix of the system, P is the preconditioner. It is proved that for circulant matrices, this approach is equivalent to constructing an optimal Chen circulant for the inverse matrix. Numerical experiments have been carried out on a series of test problems with Toeplitz matrices, showing that the proposed approach makes it possible to significantly reduce the number of iterations of the conjugate gradient method compared with the classical approach. The results obtained open up new possibilities for constructing effective preconditioners in other classes of matrices.
作者简介
I. Oseledets
Artificial Intelligence Research Institute (AIRI); Skolkovo Institute of Science and Technology; Marchuk Institute of Computational Mathematics RAS
Email: oseledets@uiri.net
Moscow, Russia; Moscow, Russia; Moscow, Russia
E. Muravleva
Sberbank AI Center for Science; Skolkovo Institute of Science and Technology
Email: EdnaMuravleva@sberbank.ru
Moscow, Russia; Moscow, Russia
参考
- Häusner P., Juscafresa A.N., Sjölund J. Learning incomplete factorization preconditioners for GMRES // arXiv preprint arXiv:2409.08262, 2024.
- Häusner P., Öktem O., Sjölund J. Neural incomplete factorization: learning preconditioners for the conjugate gradient method // arXiv preprint arXiv:2305.16368, 2023.
- Trifonov V., Rudikov A., Iliev O., Oseledets I., Muravleva E. Learning from linear algebra: A graph neural network approach to preconditioner design for conjugate gradient solvers // arXiv preprint arXiv:2405.15557, 2024.
- Katrutsa A., Daulbaev T., Oseledets I. Black-box learning of multigrid parameters // J. Comput. Appl. Math. 2020. V. 368. P. 112524.
- Gelfand I. Normierte Ringe // Marew. c6. 1941. Т. 9. № 1. С. 3–24.
- Kozyakin V.S. On accuracy of approximation of the spectral radius by the Gelfand formula // Linear Algebra Appl. 2009. V. 431. № 11. Р. 2134–2141.
- Hutchinson M.F. Astochastic estimator of the trace of the influence matrix for Laplacian smoothing splines // Comm. Statist. Simulation Comput. 1989. V. 18. № 3. Р. 1059–1076.
- Chan R.H.F., Jin X.Q. An introduction to iterative Toeplitz solvers // SIAM, 2007.
- Chan T.F. An optimal circulant preconditioner for Toeplitz systems // SIAM J. Sci. Stat. Comput. 1988. V. 9. № 4. Р. 766–771.
- Tyrhyshnikov E.E. Optimal and superoptimal circulant preconditioners // SIAM J. Matrix Anal. Appl. 1992. № 13. № 2. Р. 459–473.
- Kaporin I.E. New convergence results and preconditioning strategies for the conjugate gradient method // Numer. Linear Algebra Appl. 1994. V. 1. № 2. P. 179–210.
- Kaporin I. Superlinear convergence in minimum residual iterations // Numer. Linear Algebra Appl. 2005. V. 12. № 5–6. P. 453–470.
- Kaporin I.E. Preconditioning of the method of conjugate gradients for solving discrete analogues of differential problems // Differ. Uravn. 1990. V. 26. № 7. P. 1225–1236.
- Serra Capizzano S., Tyryshnikov E.E. Any circulant-like preconditioner for multilevel matrices is not superlinear // SIAM J. Matrix Anal. Appl. 2000. V. 21. № 2. P. 431–439.
- Kanopuu H.E. Использование полиномов Чебышёва и приближенного обратного треугольного разложения для предобусловливания метода сопряженных градиентов // Ж. вычисл. матем. и матем. физ. 2012. Т. 52. № 2. С. 179–204.
- Kanopuu H.E., Munoocoa O.IO. Неполное обратное треугольное разложение в параллельных алгоритмах предобусловленного метода сопряженных градиентов // Препринты ИПМ им. М.В. Келдыша. 2017. С. 37–28.
- Oseledets I., Tyryshnikov E. A unifying approach to the construction of circulant preconditioners // Linear Algebra Appl. 2006. V. 418. № 2–3. P. 435–449.
- Tilli P. A note on the spectral distribution of Toeplitz matrices // Linear Multilinear Algebra. 1998. V. 45. № 2–3. P. 147–159.
- Tyryshnikov E.E. A unifying approach to some old and new theorems on distribution and clustering // Linear Algebra Appl. 1996. V. 232. P. 1–43.
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