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A sine transform based preconditioned MINRES method for all-at-once systems from constant and variable-coefficient evolutionary PDEs

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Abstract

In this work, we propose a simple yet generic preconditioned Krylov subspace method for a large class of nonsymmetric block Toeplitz all-at-once systems arising from discretizing evolutionary partial differential equations. Namely, our main result is to propose two novel symmetric positive definite preconditioners, which can be efficiently diagonalized by the discrete sine transform matrix. More specifically, our approach is to first permute the original linear system to obtain a symmetric one and subsequently develop desired preconditioners based on the spectral symbol of the modified matrix. Then, we show that the eigenvalues of the preconditioned matrix sequences are clustered around \(\pm 1\), which entails rapid convergence when the minimal residual method is devised. Alternatively, when the conjugate gradient method on the normal equations is used, we show that our preconditioner is effective in the sense that the eigenvalues of the preconditioned matrix sequence are clustered around unity. An extension of our proposed preconditioned method is given for high-order backward difference time discretization schemes, which can be applied on a wide range of time-dependent equations. Numerical examples are given, also in the variable-coefficient setting, to demonstrate the effectiveness of our proposed preconditioners, which consistently outperforms an existing block circulant preconditioner discussed in the relevant literature.

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References

  1. Barakitis, N., Ekstrom, S.E., Vassalos, P.: Preconditioners for fractional diffusion equations based on the spectral symbol. Numerical Linear Algebra with Applications, e2441 (2022). https://doi.org/10.1002/nla.2441

  2. Barbarino, G.: A systematic approach to reduced GLT. BIT Numer. Math. 1–63 (2021). https://doi.org/10.1007/s10543-021-00896-7

  3. Benzi, M., Golub, G.H.: Bounds for the entries of matrix functions with applications to preconditioning. BIT Numer. Math. 39(3), 417–438 (1999). (https://link.springer.com/article/10.1023/A:1022362401426)

    Article  MathSciNet  Google Scholar 

  4. Bertaccini, D.: Reliable preconditioned iterative linear solvers for some integrators. Numer. Linear Algebra with Appl. 8(2), 111–125 (2001). https://doi.org/10.1002/1099-1506(200103)8:2>111::AID-NLA234>3.0.CO;2-Q

  5. Bertaccini, D., Ng, M.K.: Block \(\omega \)-circulant preconditioners for the systems of differential equations. Calcolo. 40(2), 71–90 (2003). (https://link.springer.com/article/10.1007/s100920300004)

    Article  MathSciNet  Google Scholar 

  6. Bini, D., Benedetto, F.: A new preconditioner for the parallel solution of positive definite Toeplitz systems. In Proc. Second ACM Symp. on Parallel Algorithms and Architectures, pp. 220–223 (1990). https://doi.org/10.1145/97444.97688

  7. Bini, D., Capovani, M.: Spectral and computational properties of band symmetric Toeplitz matrices. Linear Algebra Appl. 52, 99–126 (1983). https://doi.org/10.1016/0024-3795(83)90009-5

    Article  MathSciNet  Google Scholar 

  8. Brandts, J.H., da Silva, R.R.: Computable eigenvalue bounds for rank-\(k\) perturbations. Linear Algebra Appl. 432(12), 3100–3116 (2010). https://doi.org/10.1016/j.laa.2010.02.010

    Article  MathSciNet  Google Scholar 

  9. Chan, R.H., Ng, M.K.: Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38(3), 427–482 (1996). https://doi.org/10.1137/s0036144594276474

    Article  MathSciNet  Google Scholar 

  10. Chan, T.F.: An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Stat. Comput. 9(4), 766–771 (1988). https://doi.org/10.1137/0909051

    Article  MathSciNet  Google Scholar 

  11. Cocquet, P.H., Gander, M.J.: How large a shift is needed in the shifted Helmholtz preconditioner for its effective inversion by multigrid? SIAM J. Sci. Stat. Comput. 39(2), A438–A478 (2017). https://doi.org/10.1137/15M102085X

    Article  MathSciNet  Google Scholar 

  12. Danieli, F., Wathen, A.J.: All-at-once solution of linear wave equations. Numer. Linear Algebra with Appl. e2386 (2021). https://doi.org/10.1002/nla.2386

  13. Di Benedetto, F.: Analysis of preconditioning techniques for ill-conditioned Toeplitz matrices. SIAM J. Sci. Comput. 16(3), 682–697 (1995). https://doi.org/10.1137/0916041

    Article  MathSciNet  Google Scholar 

  14. Di Benedetto, F.: Solution of Toeplitz normal equations by sine transform based preconditioning. Linear Algebra and Its Appl. 285(1–3), 229–255 (1998). https://doi.org/10.1016/s0024-3795(98)10115-5

    Article  MathSciNet  Google Scholar 

  15. Di Benedetto, F., Serra-Capizzano, S.: A unifying approach to abstract matrix algebra preconditioning. Numer Math. 82, 57–90 (1999). https://doi.org/10.1007/s002110050411

    Article  MathSciNet  Google Scholar 

  16. Dobrev, V.A., Kolev, Tz., Petersson, N.A., Schroder, J.B.: Two-level convergence theory for multigrid reduction in time (MGRIT). SIAM J. Sci. Comput. 39(5), S501–S527 . https://doi.org/10.1137/16m1074096

  17. Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262–279 (2016). https://doi.org/10.1016/j.jcp.2015.11.061

    Article  ADS  MathSciNet  Google Scholar 

  18. Falgout, R.D., Friedhoff, S., Kolev, T.V., MacLachlan, S.P., Schroder, J.B.: Parallel time integration with multigrid. SIAM J. Sci. Comput. 36(6), C635–C661 (2014). https://doi.org/10.1002/pamm.201410456

    Article  MathSciNet  Google Scholar 

  19. Ferrari, P., Furci, I., Hon, S., Mursaleen, M.A., Serra-Capizzano, S.: The eigenvalue distribution of special 2-by-2 block matrix-sequences with applications to the case of symmetrized Toeplitz structures. SIAM J. Matrix Anal. Appl. 40(3), 1066–1086 (2019). https://doi.org/10.1137/18m1207399

    Article  MathSciNet  Google Scholar 

  20. Gander, M.J.: 50 years of time parallel time integration. Multiple Shooting and Time Domain Decomposition Methods. In: Carraro, T., Geiger, M., S.Körkel, R.R. (eds.) 69–113. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-23321-5_3

  21. Gander, M.J., Halpern, L., Ryan, J., Tran, T.T.B.: A direct solver for time parallelization. Domain Decomposition Methods in Science and Engineering XXII. In: Dickopf, T., Gander, M.J., Halpern, L., Krause, R., Pavarino, L.F. (eds.) 491–499. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-18827-0_50

  22. Gander, M.J., Neumüller, M.: Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. 38(4), A2173–A2208 (2016). https://doi.org/10.1137/15m1046605

    Article  MathSciNet  Google Scholar 

  23. Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007). https://doi.org/10.1137/05064607x

    Article  MathSciNet  Google Scholar 

  24. Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. 1, Springer, Cham (2017). https://doi.org/10.1007/978-3-319-53679-8_8

  25. Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. 2, Springer, Cham (2018). https://doi.org/10.1007/978-3-030-02233-4

  26. Goddard, A., Wathen, A.: A note on parallel preconditioning for all-at-once evolutionary PDEs. Electron. Trans. Numer. Anal. 51, 135–150 (2019). https://doi.org/10.1553/etna_vol51s135

    Article  MathSciNet  Google Scholar 

  27. Hon, S.: Optimal block circulant preconditioners for block Toeplitz systems with application to evolutionary PDEs. J. Comput. Appl. Math. 407, 113965 (2021). https://doi.org/10.1016/j.cam.2021.113965

  28. Hon, S., Serra-Capizzano, S., Wathen, A.: Band-Toeplitz preconditioners for ill-conditioned Toeplitz systems. BIT Numer. Math. (2021). https://doi.org/10.1007/s10543-021-00889-6

  29. Horton, G., Vandewalle, S.: A space-time multigrid method for parabolic partial differential equations. SIAM J. Sci. Comput. 16(4), 848–864 (1995). https://doi.org/10.1137/0916050

    Article  MathSciNet  Google Scholar 

  30. Lin, X.L., Ng, M.K.: An all-at-once preconditioner for evolutionary partial differential equations. SIAM J. Sci. Comput. 43(4), A2766–A2784 (2021). https://doi.org/10.1137/20m1316354

    Article  MathSciNet  Google Scholar 

  31. Lin, X.L., Ng, M.K., Zhi, Y.: A parallel-in-time two-sided preconditioning for all-at-once system from a non-local evolutionary equation with weakly singular kernel. J. Comput. Phys. 434, 110221 (2021). https://doi.org/10.1016/j.jcp.2021.110221

    Article  MathSciNet  Google Scholar 

  32. Lions, J.L., Maday, Y., Turinici, G.: A ”parareal” in time discretization of PDE’s. Comptes rendus de l’Académie des sciences. Série 1, Mathématique 332(7), 661–668 (2001)

  33. Liu, J., Wu, S.L.: A fast block \(\alpha \)-circulant preconditioner for all-at-once systems from wave equations. SIAM J. Matrix Analysis and Appl. 41(4), 1912–1943 (2021). https://doi.org/10.1137/19m1309869

    Article  CAS  Google Scholar 

  34. Mazza, M., Pestana, J.: Spectral properties of flipped Toeplitz matrices and related preconditioning. BIT Numer. Math. 59, 463–482 (2018). https://doi.org/10.1007/s10543-018-0740-y

    Article  MathSciNet  Google Scholar 

  35. McDonald, E., Hon, S., Pestana, J., Wathen, A.: Preconditioning for nonsymmetry and time-dependence. In Lecture Notes in Comput. Sci. Eng. 116, 81–91 (2017). Springer International Publishing. https://doi.org/10.1007/978-3-319-52389-7_7

  36. McDonald, E., Pestana, J., Wathen, A.: Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations. SIAM J. Sci. Comput. 40(2), A1012–A1033 (2018). https://doi.org/10.1137/16m1062016

    Article  MathSciNet  Google Scholar 

  37. Ng, M.K.: Iterative Methods for Toeplitz Systems. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2004)

    Book  Google Scholar 

  38. Ng, M.K., Pan, J.: Approximate inverse circulant-plus-diagonal preconditioners for Toeplitz-plus-diagonal matrices. SIAM J. Sci. Comput. 32(3), 1442–1464 (2010). https://doi.org/10.1137/080720280

    Article  MathSciNet  Google Scholar 

  39. Serra-Capizzano, S.: The GLT class as a generalized Fourier analysis and applications. Linear Algebra and Its Appl. 419(1), 180–233 (2006). https://doi.org/10.1016/j.laa.2006.04.012

    Article  MathSciNet  Google Scholar 

  40. Serra-Capizzano, S.: Toeplitz preconditioners constructed from linear approximation processes. SIAM J. Matrix Analysis and Appl. 20(2), 446–465 (1999). https://doi.org/10.1137/s0895479897316904

    Article  MathSciNet  Google Scholar 

  41. Serra-Capizzano, S.: Matrix algebra preconditioners for multilevel Toeplitz matrices are not superlinear. Special issue on structured and infinite systems of linear equations. Linear Algebra Appl. 343(344), 303–319 (2002). https://doi.org/10.1016/S0024-3795(01)00361-5

    Article  MathSciNet  Google Scholar 

  42. Serra-Capizzano, S., Tilli, P.: On unitarily invariant norms of matrix-valued linear positive operators. J. Inequalities and App. 7(3), 309–330 (2002). https://doi.org/10.1155/s1025583402000152

    Article  MathSciNet  Google Scholar 

  43. Serra-Capizzano, S., Tyrtyshnikov, E.: How to prove that a preconditioner cannot be superlinear. Math. Comput. 72(243), 1305–1316 (2003). https://doi.org/10.1090/s0025-5718-03-01506-0

    Article  ADS  MathSciNet  Google Scholar 

  44. Strang, G.: A proposal for Toeplitz matrix calculations. Stud. Appl. Math. 74(2), 171–176 (1986). https://doi.org/10.1002/sapm1986742171

    Article  Google Scholar 

  45. Wathen, A.: Preconditioning. Acta Numerica. 24, 329–376 (2015). https://doi.org/10.1017/s0962492915000021

    Article  MathSciNet  Google Scholar 

  46. Wu, S.L., Zhou, T.: Parallel implementation for the two-stage SDIRK methods via diagonalization. J. Comput. Phys. 428, 110076 (2021). https://doi.org/10.1016/j.jcp.2020.110076

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Funding

The work of S. Hon was supported in part by the Hong Kong RGC under grant 22300921, a start-up allowance from the Croucher Foundation, and a Tier 2 Start-up Grant from Hong Kong Baptist University. The work of S. Serra-Capizzano was supported in part by INDAM-GNCS. Furthermore, the work of Stefano Serra-Capizzano was funded from the European High-Performance Computing Joint Undertaking (JU) under grant agreement no. 955701. The JU receives support from the European Union’s Horizon 2020 research and innovation programme and Belgium, France, Germany, Switzerland. Stefano Serra-Capizzano is also grateful for the support of the Laboratory of Theory, Economics and Systems - Department of Computer Science at Athens University of Economics and Business.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

    Sean Hon, Po Yin Fung & Jiamei Dong

  2. Department of Science and High Technology, University of Insubria, Como, 6152, Lombardy, Italy

    Stefano Serra-Capizzano

  3. Department of Information Technology, Uppsala University, Uppsala, 75008, Uppland, Sweden

    Stefano Serra-Capizzano

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  1. Sean Hon
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  2. Po Yin Fung
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  4. Stefano Serra-Capizzano
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Contributions

S. Hon developed the methodology, wrote the main manuscript text, and conducted the main numerical experiments. P. Y. Fung improved the numerical experiments and modified the manuscript text. J. Dong modified the manuscript text. S. Serra-Capizzano validated the methodology and modified the manuscript text. All authors reviewed the manuscript.

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Correspondence to Sean Hon.

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Hon, S., Fung, P.Y., Dong, J. et al. A sine transform based preconditioned MINRES method for all-at-once systems from constant and variable-coefficient evolutionary PDEs. Numer Algor 95, 1769–1799 (2024). https://doi.org/10.1007/s11075-023-01627-5

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