Abstract.

Preconditioning for Krylov methods often relies on operator theory when mesh independent estimates are looked for. The goal of this paper is to contribute to the long development of the analysis of superlinear convergence of Krylov iterations when the preconditioned operator is a compact perturbation of the identity. Mesh independent superlinear convergence of GMRES and CGN iterations is derived for Galerkin solutions for complex non-Hermitian and noncoercive operators. The results are applied to noncoercive boundary value problems, including shifted Laplacian preconditioners for Helmholtz problems.

Keywords

  1. Krylov iteration
  2. preconditioning
  3. noncoercive operators
  4. mesh independence
  5. shifted Laplace

MSC codes

  1. 65F10
  2. 65J10
  3. 65N30

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References

1.
O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1994.
2.
O. Axelsson and J. Karátson, Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators, Numer. Math., 99 (2004), pp. 197–223.
3.
O. Axelsson and J. Karátson, Mesh independent superlinear PCG rates via compact-equivalent operators, SIAM J. Numer. Anal., 45 (2007), pp. 1495–1516.
4.
O. Axelsson and J. Karátson, Equivalent operator preconditioning for linear elliptic problems, Numer. Algorithms, 50 (2009), pp. 297–380.
5.
O. Axelsson and J. Karátson, Superlinear convergence of the GMRES for PDE-constrained optimization problems, Numer. Funct. Anal. Optim., 39 (2018), pp. 921–936.
6.
I. Babuška and A. K. Aziz, The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations, Academic Press, New York, 1972.
7.
B. Beckermann and A. B. J. Kuijlaars, Superlinear convergence of conjugate gradients, SIAM J. Numer. Anal., 39 (2001), pp. 300–329.
8.
M. Bernkopf, S. A. Sauter, C. Torres, and A. Veit, Solvability of Discrete Helmholtz Equations, https://doi.org/10.48550/arXiv.2105.02273, 2021.
9.
S. Cools and W. Vanroose, On the optimality of shifted Laplacian in a class of polynomial preconditioners for the Helmholtz equation, in Modern Solvers for Helmholtz Problems, Geosystems Mathematics, Birkhäuser/Springer, Cham, 2017, pp. 53–81.
10.
H. Egger, Preconditioning CGNE iteration for inverse problems, Numer. Linear Algebra Appl., 14, pp. 183–196.
11.
H. C. Elman and M. H. Schultz, Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations, SIAM J. Numer. Anal., 23 (1986), pp. 44–57.
12.
Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, A novel multigrid based preconditioner for heterogeneous Helmholtz problems, SIAM J. Sci. Comput., 27 (2006), pp. 1471–1492.
13.
Y. A. Erlangga, L. Garcia Ramos, and R. Nabben, The multilevel Krylov-multigrid method for the Helmholtz equation preconditioned by the shifted Laplacian, in Modern Solvers for Helmholtz Problems, Geosystems Mathematics, Birkhäuser/Springer, Cham, 2017, pp. 113–139.
14.
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Appl. Math. Sci. 159, Springer-Verlag, New York, 2004.
15.
O. G. Ernst and M. J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, in Numerical Analysis of Multiscale Problems, I. G. Graham, T. Y. Hou, O. Lakkis, and R. Scheichl, eds., Lecture Notes in Comput. Sci. 83, Springer-Verlag, New York, 2012.
16.
B. Engquist and L. Ying, Fast algorithms for high frequency wave propagation, in Numerical Analysis of Multiscale Problems, I. G. Graham, T. Y. Hou, O. Lakkis, and R. Scheichl, eds., Lecture Notes in Comput. Sci. 83, Springer-Verlag, New York, 2012.
17.
V. Faber, T. Manteuffel, and S. V. Parter, On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations, Adv. Appl. Math., 11 (1990), pp. 109–163.
18.
M. Gander, F. Magoulès, and F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput., 24 (2002), pp. 38–60.
19.
M. J. Gander, I. G. Graham, and E. A. Spence, Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed?, Numer. Math., 131 (2015), pp. 567–614.
20.
L. García Ramos and R. Nabben, On the spectrum of deflated matrices with applications to the deflated shifted Laplace preconditioner for the Helmholtz equation, SIAM J. Matrix Anal. Appl., 39 (2018), pp. 262–286.
21.
Cs. Gáspár, A regularized multi-level technique for solving potential problems by the method of fundamental solutions, Eng. Anal. Bound. Elem., 57 (2015), pp. 66–71.
22.
Cs. Gáspár, Regularization and multi-level tools in the method of fundamental solutions, in Meshfree Methods for Partial Differential Equations VII, Lect. Notes Comput. Sci. Eng. 100, Springer, Cham, 2015, pp. 145–162.
23.
T. Gergelits, K.-A. Mardal, B. F. Nielsen, and Z. Strakoš, Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator, SIAM J. Numer. Anal., 57 (2019), pp. 1369–1394.
24.
T. Gergelits, B. F. Nielsen, and Z. Strakoš, Generalized spectrum of second order differential operators, SIAM J. Numer. Anal., 58 (2020), pp. 2193–2211.
25.
M. B. van Gijzen, Y. A. Erlangga, and C. Vuik, Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian, SIAM J. Sci. Comput., 29 (2007), pp. 1942–1958.
26.
I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators, Vol. I, Oper. Theory Adv. Appl. 49, Birkhäuser Verlag, Basel, 1990.
27.
C. I. Goldstein, T. A. Manteuffel, and S. V. Parter, Preconditioning and boundary conditions without \(H_2\) estimates: \(L_2\) condition numbers and the distribution of the singular values, SIAM J. Numer. Anal., 30 (1993), pp. 343–376.
28.
I. G. Graham, E. A. Spence, and E. Vainikko, Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption, Math. Comp., 86 (2017), pp. 2089–2127.
29.
I. Harari and F. Magoulès, Numerical investigations of stabilized finite element computations for acoustics, Wave Motion, 39 (2004), pp. 339–349.
30.
R. M. Hayes, Iterative methods of solving linear problems on Hilbert space, in Contributions to the Solution of Systems of Linear Equations and the Determination of Eigenvalues, O. Taussky, ed., Applied Mathematics Series 39, National Bureau of Standards, 1954, pp. 71–103.
31.
R. Herzog and E. Sachs, Superlinear convergence of Krylov subspace methods for self-adjoint problems in Hilbert space, SIAM J. Numer. Anal., 53 (2015), pp. 1304–1324.
32.
R. Hiptmair and C. Urzúa-Torres, Compact equivalent inverse of the electric field integral operator on screens, Integral Equations Operator Theory, 92 (2020), 14.
33.
P. A. Krutitskii, The impedance problem for the propagative Helmholtz equation in interior multiply connected domain, Comput. Math. Appl., 46 (2003), pp. 1601–1610.
34.
A. B. J. Kuijlaars, Convergence analysis of Krylov subspace iterations with methods from potential theory, SIAM Rev., 48 (2006), pp. 3–40.
35.
B. Lee, T. A. Manteuffel, S. F. McCormick, and J. Ruge, First-order system least-squares (FOSLS) for the Helmholtz equation, SIAM J. Sci. Comput., 20 (2000), pp. 1927–1949.
36.
M. Levitin, L. Parnovski, I. Polterovich, and D. A. Sher, Sloshing, Steklov and Corners: Asymptotics of Steklov Eigenvalues for Curvilinear Polygons, https://arxiv.org/pdf/1908.06455.pdf, 2019.
37.
I. Livshits, Use of shifted Laplacian operators for solving indefinite Helmholtz equations, Numer. Math. Theory Methods Appl., 8 (2015), pp. 136–148.
38.
F. Magoulès, K. Meerbergen, and J.-P. Coyette, Application of a domain decomposition method with Lagrange multipliers to acoustic problems arising from the automotive industry, J. Comput. Acoust., 8 (2000), pp. 503–521.
39.
F. Magoulès, F.-X. Roux, and S. Salmon, Optimal discrete transmission conditions for a non-overlapping domain decomposition method for the Helmholtz equation, SIAM J. Sci. Comput., 25 (2004), pp. 1497–1515.
40.
J. Málek and Z. Strakoš, Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs, SIAM Spotlights 1, SIAM, Philadelphia, 2015.
41.
T. A. Manteuffel and S. V. Parter, Preconditioning and boundary conditions, SIAM J. Numer. Anal., 27 (1990), pp. 656–694.
42.
I. Moret, A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal., 34 (1997), pp. 513–516.
43.
F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publications, New York, 1990.
44.
Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003.
45.
S. A. Sauter, A refined finite element convergence theory for highly indefinite Helmholtz problems, Computing, 78 (2006), pp. 101–115.
46.
A. Schiela and S. Ulbrich, Operator preconditioning for a class of inequality constrained optimal control problems, SIAM J. Optim., 24 (2014), pp. 435–466.
47.
V. Simoncini and D. B. Szyld, On the occurrence of superlinear convergence of exact and inexact Krylov subspace methods, SIAM Rev., 47 (2005), pp. 247–272.
48.
H. Weyl, Das asymptotische verteilungsgesetz der eigenwerte linearer partieller differentialgleichungen, Math. Ann., 71 (1911), pp. 441–479.
49.
R. Winther, Some superlinear convergence results for the conjugate gradient method, SIAM J. Numer. Anal., 17 (1980), pp. 14–17.
50.
O. Widlund, A Lanczos method for a class of non-symmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801–812.

Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 1057 - 1079
ISSN (online): 1095-7170

History

Submitted: 22 December 2021
Accepted: 22 November 2022
Published online: 27 April 2023

Keywords

  1. Krylov iteration
  2. preconditioning
  3. noncoercive operators
  4. mesh independence
  5. shifted Laplace

MSC codes

  1. 65F10
  2. 65J10
  3. 65N30

Notes

Owe Axelsson passed away in June 2022. It was a privilege that we could work with him on this and previous papers, and we will never forget his unique inspiration and presence.

Authors

Affiliations

Owe Axelsson
Institute of Geonics AS CR, IT4 Innovations, Ostrava, Czech Republic.
Department of Applied Analysis and ELKH-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Budapest, Hungary; and Department of Analysis, Budapest University of Technology and Economics, Hungary.
Frédéric Magoulès
CentraleSupelec, Université Paris-Saclay, France, and University of Pécs, Pécs, Hungary.

Funding Information

Funding: This research has been supported by the National Research, Development and Innovation Office (NKFIH), grants K137699 and SNN125119.

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