Analysis of Two-Level Complex Shifted Laplace Preconditioner and Deflation-Based Preconditioner for Helmholtz Equation
DOI:
https://doi.org/10.21015/vtm.v10i2.1304Abstract
A Long time deflation preconditioner is used to speed up the convergence of the Krylov subspace method. The discretization of Helmholtz equation with Dirichlet boundary condition by finite difference method obtained any linear system. Resolving a large wavenumber requires a larger number of Grid points, i.e. large linear systems. Thus due to the large linear system, many (sparse) direct methods have taken more memory, So we have used the (iterative technique) Krylov subspace method. One of the problems of the Krylov subspace method is the required preconditioner for better convergence. We use (CSLP) as a preconditioner and drive eigenvalues of (CSLP). However, with increasing wavenumber CSLP shows slow convergence behavior. Then we use another projection-type preconditioner as a deflation preconditioner. A rigorous Fourier analysis (RFA) is a separate research idea to examine the con- vergence of the iterative method included in this article. We analyze the deflation preconditioner with a complex shifted Laplace preconditioner (CSLP) which exhibition spectral behavior of the preconditioner, which is favorable to the Krylov method.References
Bayliss, A., Goldstein, C. I. and Turkel, E. [1983], ‘An iterative method for the Helmholtz equation’,
Journal of Computational Physics 49, 443 – 457.
Davis, T. A. [2006], Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2), SIAM,
Philadelphia, PA, USA.
Erlangga, Y. A. [2005], A robust and effecient iterative method for numerical solution of Helmholtz
equation, PhD Thesis, DIAM, TU Delft.
Erlangga, Y. A. and Nabben, R. [2008], ‘Deflation and Balancing Preconditioners for Krylov Subspace
Methods Applied to Nonsymmetric Matrices’, SIAM J. Matrix Anal. Appl. 30, 684–699.
Erlangga, Y. A., Oosterlee, C. W. and Vuik, C. [2006], ‘A novel multigrid based preconditioner for heterogeneous
Helmholtz problems’, SIAM J. Sci. Comput 27, 1471–1492.
Erlangga, Y., Vuik, C. and Oosterlee, C. [2004], ‘On a class of preconditioners for solving the Helmholtz
equation’, Appl. Numer. Math. 50(3-4), 409–425.
Gijzen, M. B., Erlangga, Y. A. and Vuik, C. [2007], ‘Spectral Analysis of the Discrete Helmholtz Operator
Preconditioned with a Shifted Laplacian’, SIAM Journal on Scientific Computing 29, 1942–1958.
Humayoun, M., Burney, S. A., Sheikh, A. and Ghafoor, A. [2021], ‘Ritz Vectors-Based Deflation Preconditioner
for Linear System with Multiple Right-Hand Sides’, STATISTICS, COMPUTING AND INTERDISCIPLINARY
RESEARCH 3(2), 155–168.
Ihlenburg, F. and Babuska, I. [1995], ‘Finite element solution to the Helmholtz equation with high wave
numbers’, Computers and Mathematics with Applications 30, 9–37.
Mardoche, M. M. M. [2001], ‘Incomplete factorization-based preconditionings for solving the
Helmholtz equation’, International Journal for Numerical Methods in Engineering 50, 1077–1101.
Pillwein, V. and Takacs, S. [2014], ‘A local Fourier convergence analysis of a multigrid method using
symbolic computation’, Journal of Symbolic Computation 63, 1–20.
URL: https://linkinghub.elsevier.com/retrieve/pii/S0747717113001752
Saad, Y. [1996], Iterative Methods for Linear system, PWS Publishing Company.
Shaikh, A., Sheikh, A. H., Ali, A. and Zeb, S. [2019], ‘Critical Review of Preconditioners for Helmholtz
Equation and their Spectral Analysis’, Indian Journal of Science and Technology 12(20), 1–8.
Sheikh, A. H. [2014], Development Of The Helmholtz Solver Based On A Shifted Laplace Preconditioner
And A Multigrid Deflation Technique, PhD thesis, Delft University of Technology, The Netherlands.
URL: https://doi.org/10.4233/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1
Sheikh, A. H., Lahaye, D., Ramos, L. G., Nabben, R. and Vuik, C. [2016], ‘Accelerating the shifted Laplace
preconditioner for the Helmholtz equation by multilevel deflation’, Journal of Computational Physics
, 473–490. Publisher: Elsevier.
Sheikh, A. H., Lahaye, D. and Vuik, C. [2013], ‘On the convergence of shifted Laplace preconditioner
combined with multilevel deflation’, Numerical Linear Algebra with Applications 20, 645–662.
Sheikh, A. H., Vuik, C. and Lahaye, D. [2009], Fast iterative solution methods for the Helmholtz equation,
Technical Report 09-11, DIAM, TU Delft.
Sheikh, A. H., Vuik, C. and Lahaye, D. [2011], A scalable Helmholtz solver combining the shifted Laplace
preconditioner with Multigrid deflation, Technical Report 11-01, DIAM, TU Delft Netherlands.
Siyal, W. A., Sheikh, A. H., Mallah, M., Sandilo, S. H. and Shaikh, A. G. [2019], ‘Convergence Analysis of
Multigrid Method for Shifted Laplace at Various Levels Using Fourier Modes’, International Journal of
Computer Science and Network Security 19(9), 57–64.
URL: http://paper.ijcsns.org/07
book/201909/20190907.pdf
Siyal, W., Sheikh, A. H., Amur, K. B., Shaikh, G, A. and Malookani, R. A. [2020], ‘On the Efficiency of
Multigrid Solver for Shifted Laplace Equation in a Heterogeneous Medium’, International Journal of
Applied Mathematics and Statistics, 59(3), 102–114.
Stüben, K. [2001], An introduction to algebraic multigrid, in U. Trottenberg, C. Oosterlee and
A. Schüller, eds, ‘Multigrid’, Academic Press, San Diego, CA, pp. 413–528.
Vorst, H. A. v. d. [2003], Iterative Krylov Methods for Large Linear Systems, Cambridge University Press,
Cambridge.
Wienands, R. [2001], Extended local Fourier analysis for multigrid: Optimal smoothing, coarse grid
correction, and preconditioning, PhD Thesis, University of Cologne.
Wienands, R. and Joppich, W. [2005], Practical Fourier Analysis for Multigrid Methods, Numerical Insights,
Chapman & Hall/CRC, Boca Raton, Florida, USA.
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