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Article: Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations
Title | Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations |
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Authors | |
Keywords | Fast Fourier transform Circulant matrix Approximate inverse Toeplitz matrix Krylov subspace methods Fractional diffusion equation |
Issue Date | 2014 |
Citation | SIAM Journal on Scientific Computing, 2014, v. 36, n. 6, p. A2698-A2719 How to Cite? |
Abstract | © 2014 Society for Industrial and Applied Mathematics. The fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal-times-Toeplitz matrices. Standard circulant preconditioners may not work for such Toeplitz-like linear systems. The main aim of this paper is to propose and develop approximate inverse preconditioners for such Toeplitz-like matrices. An approximate inverse preconditioner is constructed to approximate the inverses of weighted Toeplitz matrices by circulant matrices, and then combine them together rowby-row. Because of Toeplitz structure, both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast Fourier transforms. Theoretically, we show that the spectra of the resulting preconditioned matrices are clustered around one. Thus Krylov subspace methods with the proposed preconditioner converge very fast. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show that its performance is better than the other testing preconditioners. |
Persistent Identifier | http://hdl.handle.net/10722/277013 |
ISSN | 2022 Impact Factor: 3.1 2020 SCImago Journal Rankings: 1.674 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Pan, Jianyu | - |
dc.contributor.author | Ke, Rihuan | - |
dc.contributor.author | Ng, Michael K. | - |
dc.contributor.author | Sun, Hai Wei | - |
dc.date.accessioned | 2019-09-18T08:35:20Z | - |
dc.date.available | 2019-09-18T08:35:20Z | - |
dc.date.issued | 2014 | - |
dc.identifier.citation | SIAM Journal on Scientific Computing, 2014, v. 36, n. 6, p. A2698-A2719 | - |
dc.identifier.issn | 1064-8275 | - |
dc.identifier.uri | http://hdl.handle.net/10722/277013 | - |
dc.description.abstract | © 2014 Society for Industrial and Applied Mathematics. The fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal-times-Toeplitz matrices. Standard circulant preconditioners may not work for such Toeplitz-like linear systems. The main aim of this paper is to propose and develop approximate inverse preconditioners for such Toeplitz-like matrices. An approximate inverse preconditioner is constructed to approximate the inverses of weighted Toeplitz matrices by circulant matrices, and then combine them together rowby-row. Because of Toeplitz structure, both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast Fourier transforms. Theoretically, we show that the spectra of the resulting preconditioned matrices are clustered around one. Thus Krylov subspace methods with the proposed preconditioner converge very fast. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show that its performance is better than the other testing preconditioners. | - |
dc.language | eng | - |
dc.relation.ispartof | SIAM Journal on Scientific Computing | - |
dc.subject | Fast Fourier transform | - |
dc.subject | Circulant matrix | - |
dc.subject | Approximate inverse | - |
dc.subject | Toeplitz matrix | - |
dc.subject | Krylov subspace methods | - |
dc.subject | Fractional diffusion equation | - |
dc.title | Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations | - |
dc.type | Article | - |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.1137/130931795 | - |
dc.identifier.scopus | eid_2-s2.0-84919691062 | - |
dc.identifier.volume | 36 | - |
dc.identifier.issue | 6 | - |
dc.identifier.spage | A2698 | - |
dc.identifier.epage | A2719 | - |
dc.identifier.eissn | 1095-7200 | - |
dc.identifier.isi | WOS:000346838800009 | - |