A locally optimal preconditioned Newton-Schur method for symmetric elliptic eigenvalue problems

Chen, Wenbin; Shao, Nian; Xu, Xuejun

doi:10.1090/mcom/3860

A locally optimal preconditioned Newton-Schur method for symmetric elliptic eigenvalue problems
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by Wenbin Chen, Nian Shao and Xuejun Xu

Math. Comp. 92 (2023), 2655-2684

DOI: https://doi.org/10.1090/mcom/3860

Published electronically: July 5, 2023

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Abstract:

A locally optimal preconditioned Newton-Schur method is proposed for solving symmetric elliptic eigenvalue problems. Firstly, the Steklov-Poincaré operator is used to project the eigenvalue problem on the domain $\Omega$ onto the nonlinear eigenvalue subproblem on $\Gamma$, which is the union of subdomain boundaries. Then, the direction of correction is obtained via applying a non-overlapping domain decomposition method on $\Gamma$. Four different strategies are proposed to build the hierarchical subspace $U_{k+1}$ over the boundaries, which are based on the combination of the coarse-subspace with the directions of correction. Finally, the approximation of eigenpair is updated by solving a local optimization problem on the subspace $U_{k+1}$. The convergence rate of the locally optimal preconditioned Newton-Schur method is proved to be $\gamma =1-c_{0}T_{h,H}^{-1}$, where $c_{0}$ is a constant independent of the fine mesh size $h$, the coarse mesh size $H$ and jumps of the coefficients; whereas $T_{h,H}$ is the constant depending on stability of the decomposition. Numerical results confirm our theoretical analysis.

References

I. Babuška and J. E. Osborn, Estimates for the errors in eigenvalue and eigenvector approximation by Galerkin methods, with particular attention to the case of multiple eigenvalues, SIAM J. Numer. Anal. 24 (1987), no. 6, 1249–1276. MR 917451, DOI 10.1137/0724082
I. Babuška and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), no. 186, 275–297. MR 962210, DOI 10.1090/S0025-5718-1989-0962210-8
I. Babuška and J. E. Osborn, Handbook of Numerical Analysis, vol. 2, Elsevier Science B.V., North-Holland, 1991.
Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst (eds.), Templates for the solution of algebraic eigenvalue problems, Software, Environments, and Tools, vol. 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. A practical guide. MR 1792141, DOI 10.1137/1.9780898719581
Randolph E. Bank, Analysis of a multilevel inverse iteration procedure for eigenvalue problems, SIAM J. Numer. Anal. 19 (1982), no. 5, 886–898. MR 672565, DOI 10.1137/0719064
Amir Beck and Marc Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci. 2 (2009), no. 1, 183–202. MR 2486527, DOI 10.1137/080716542
Constantine Bekas and Yousef Saad, Computation of smallest eigenvalues using spectral Schur complements, SIAM J. Sci. Comput. 27 (2005), no. 2, 458–481. MR 2202229, DOI 10.1137/040603528
J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), no. 175, 103–134. MR 842125, DOI 10.1090/S0025-5718-1986-0842125-3
James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, The construction of preconditioners for elliptic problems by substructuring. IV, Math. Comp. 53 (1989), no. 187, 1–24. MR 970699, DOI 10.1090/S0025-5718-1989-0970699-3
Susanne C. Brenner, The condition number of the Schur complement in domain decomposition, Numer. Math. 83 (1999), no. 2, 187–203. MR 1712684, DOI 10.1007/s002110050446
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Susanne C. Brenner and Li-Yeng Sung, Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions, Math. Comp. 69 (2000), no. 232, 1319–1339. MR 1710656, DOI 10.1090/S0025-5718-00-01236-9
Maksymilian Dryja and Olof B. Widlund, Some domain decomposition algorithms for elliptic problems, Iterative methods for large linear systems (Austin, TX, 1988) Academic Press, Boston, MA, 1990, pp. 273–291. MR 1038100
Jed A. Duersch, Meiyue Shao, Chao Yang, and Ming Gu, A robust and efficient implementation of LOBPCG, SIAM J. Sci. Comput. 40 (2018), no. 5, C655–C676. MR 3860901, DOI 10.1137/17M1129830
E. G. D’yakonov and A. V. Knyazev, Group iterative method for finding low-order eigenvalues, Moscow University Computational Mathematics and Cybernetics 15 (1982), 32–40.
Gene H. Golub and Charles F. Van Loan, Matrix computations, 4th ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. MR 3024913, DOI 10.56021/9781421407944
W. Hackbusch, On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multi-grid method, SIAM J. Numer. Anal. 16 (1979), no. 2, 201–215. MR 526484, DOI 10.1137/0716015
W. Hackbusch, Elliptic Eigenvalue Problems, Springer, Berlin, Heidelberg, 2017, pp. 329–354.
Xiaozhe Hu and Xiaoliang Cheng, Acceleration of a two-grid method for eigenvalue problems, Math. Comp. 80 (2011), no. 275, 1287–1301. MR 2785459, DOI 10.1090/S0025-5718-2011-02458-0
V. Kalantzis, Domain decomposition algorithms for the solution of sparse symmetric generalized eigenvalue problems, Ph.D. thesis, University of Minnesota, Minnesota, 2018.
V. Kalantzis, A domain decomposition Rayleigh–Ritz algorithm for symmetric generalized eigenvalue problems, SIAM J. Sci. Comput. 42 (2020), no. 6, C410–C435.
Vassilis Kalantzis, A spectral Newton-Schur algorithm for the solution of symmetric generalized eigenvalue problems, Electron. Trans. Numer. Anal. 52 (2020), 132–153. MR 4066335, DOI 10.1553/etna_{v}ol52s132
Vassilis Kalantzis, Ruipeng Li, and Yousef Saad, Spectral Schur complement techniques for symmetric eigenvalue problems, Electron. Trans. Numer. Anal. 45 (2016), 305–329. MR 3546384
Andrew V. Knyazev, Preconditioned eigensolvers—an oxymoron?, Electron. Trans. Numer. Anal. 7 (1998), 104–123. Large scale eigenvalue problems (Argonne, IL, 1997). MR 1667642
Andrew V. Knyazev, Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput. 23 (2001), no. 2, 517–541. Copper Mountain Conference (2000). MR 1861263, DOI 10.1137/S1064827500366124
A. V. Knyazev, M. E. Argentati, I. Lashuk, and E. E. Ovtchinnikov, Block locally optimal preconditioned eigenvalue xolvers (BLOPEX) in hypre and PETSc, SIAM J. Sci. Comput. 29 (2007), no. 5, 2224–2239. MR 2350029, DOI 10.1137/060661624
Andrew V. Knyazev and Klaus Neymeyr, A geometric theory for preconditioned inverse iteration. III. A short and sharp convergence estimate for generalized eigenvalue problems, Linear Algebra Appl. 358 (2003), 95–114. Special issue on accurate solution of eigenvalue problems (Hagen, 2000). MR 1942725, DOI 10.1016/S0024-3795(01)00461-X
A. V. Knyazev and A. L. Skorokhodov, Preconditioned iterative methods in subspace for solving linear systems with indefinite coefficient matrices and eigenvalue problems, Soviet J. Numer. Anal. Math. Modelling 4 (1989), no. 4, 283–310. MR 1020265, DOI 10.1515/rnam.1989.4.4.283
Andrew V. Knyazev and Alexander L. Skorokhodov, Preconditioned gradient-type iterative methods in a subspace for partial generalized symmetric eigenvalue problems, SIAM J. Numer. Anal. 31 (1994), no. 4, 1226–1239. MR 1286225, DOI 10.1137/0731064
Qigang Liang and Xuejun Xu, A two-level preconditioned Helmholtz-Jacobi-Davidson method for the Maxwell eigenvalue problem, Math. Comp. 91 (2022), no. 334, 623–657. MR 4379971, DOI 10.1090/mcom/3702
Qun Lin and Hehu Xie, A multi-level correction scheme for eigenvalue problems, Math. Comp. 84 (2015), no. 291, 71–88. MR 3266953, DOI 10.1090/S0025-5718-2014-02825-1
S. Yu. Maliassov, On the Schwarz alternating method for eigenvalue problems, Russian J. Numer. Anal. Math. Modelling 13 (1998), no. 1, 45–56. MR 1611642, DOI 10.1515/rnam.1998.13.1.45
Stephen F. McCormick, A mesh refinement method for $Ax=\lambda Bx$, Math. Comp. 36 (1981), no. 154, 485–498. MR 606508, DOI 10.1090/S0025-5718-1981-0606508-4
Yu. E. Nesterov, A method for solving the convex programming problem with convergence rate $O(1/k^{2})$, Dokl. Akad. Nauk SSSR 269 (1983), no. 3, 543–547 (Russian). MR 701288
Beresford N. Parlett, The symmetric eigenvalue problem, Classics in Applied Mathematics, vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Corrected reprint of the 1980 original. MR 1490034, DOI 10.1137/1.9781611971163
B. T. Poljak, Some methods of speeding up the convergence of iterative methods, Ž. Vyčisl. Mat i Mat. Fiz. 4 (1964), 791–803 (Russian). MR 169403
A. Quarteroni and A. Valli, Theory and application of Steklov-Poincaré operators for boundary-value problems, Applied and industrial mathematics (Venice, 1989) Math. Appl., vol. 56, Kluwer Acad. Publ., Dordrecht, 1991, pp. 179–203. MR 1147198
Yousef Saad, Numerical methods for large eigenvalue problems, Classics in Applied Mathematics, vol. 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Revised edition of the 1992 original [ 1177405]. MR 3396212, DOI 10.1137/1.9781611970739.ch1
Nian Shao and Wenbin Chen, Convergence analysis of Newton-Schur method for symmetric elliptic eigenvalue problem, SIAM J. Numer. Anal. 61 (2023), no. 1, 315–342. MR 4550688, DOI 10.1137/21M1448847
Barry F. Smith, A domain decomposition algorithm for elliptic problems in three dimensions, Numer. Math. 60 (1991), no. 2, 219–234. MR 1133580, DOI 10.1007/BF01385722
Andrea Toselli and Olof Widlund, Domain decomposition methods—algorithms and theory, Springer Series in Computational Mathematics, vol. 34, Springer-Verlag, Berlin, 2005. MR 2104179, DOI 10.1007/b137868
Wei Wang and Xuejun Xu, A two-level overlapping hybrid domain decomposition method for eigenvalue problems, SIAM J. Numer. Anal. 56 (2018), no. 1, 344–368. MR 3749389, DOI 10.1137/16M1088302
Wei Wang and Xuejun Xu, On the convergence of a two-level preconditioned Jacobi-Davidson method for eigenvalue problems, Math. Comp. 88 (2019), no. 319, 2295–2324. MR 3957894, DOI 10.1090/mcom/3403
Hehu Xie, Lei Zhang, and Houman Owhadi, Fast eigenpairs computation with operator adapted wavelets and hierarchical subspace correction, SIAM J. Numer. Anal. 57 (2019), no. 6, 2519–2550. MR 4027848, DOI 10.1137/18M1194079
Jinchao Xu, Theory of multilevel methods, ProQuest LLC, Ann Arbor, MI, 1989. Thesis (Ph.D.)–Cornell University. MR 2637710
Jinchao Xu and Aihui Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70 (2001), no. 233, 17–25. MR 1677419, DOI 10.1090/S0025-5718-99-01180-1
Jinchao Xu and Aihui Zhou, Local and parallel finite element algorithms for eigenvalue problems, Acta Math. Appl. Sin. Engl. Ser. 18 (2002), no. 2, 185–200. MR 2008551, DOI 10.1007/s102550200018
Jinchao Xu and Jun Zou, Some nonoverlapping domain decomposition methods, SIAM Rev. 40 (1998), no. 4, 857–914. MR 1659681, DOI 10.1137/S0036144596306800
Yidu Yang and Hai Bi, Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems, SIAM J. Numer. Anal. 49 (2011), no. 4, 1602–1624. MR 2831063, DOI 10.1137/100810241
J. Zhou, X. Hu, L. Zhong, S. Shu, and L. Chen, Two-grid methods for Maxwell eigenvalue problems, SIAM J. Numer. Anal. 52 (2014), no. 4, 2027–2047. MR 3248033, DOI 10.1137/130919921