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Volume 6, Issue 2
Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Siu-Long Lei, Xu Chen & Xinhe Zhang

East Asian J. Appl. Math., 6 (2016), pp. 109-130.

Published online: 2018-02

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  • Abstract

High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant preconditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

  • AMS Subject Headings

65F10, 65L12, 65T50, 26A33

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-6-109, author = {Siu-Long Lei, Xu Chen and Xinhe Zhang}, title = {Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {6}, number = {2}, pages = {109--130}, abstract = {

High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant preconditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.060815.180116a}, url = {http://global-sci.org/intro/article_detail/eajam/10777.html} }
TY - JOUR T1 - Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations AU - Siu-Long Lei, Xu Chen & Xinhe Zhang JO - East Asian Journal on Applied Mathematics VL - 2 SP - 109 EP - 130 PY - 2018 DA - 2018/02 SN - 6 DO - http://doi.org/10.4208/eajam.060815.180116a UR - https://global-sci.org/intro/article_detail/eajam/10777.html KW - High-dimensional two-sided fractional diffusion equation, implicit finite difference method, unconditionally stable, multilevel circulant preconditioner, GMRES method. AB -

High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant preconditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

Siu-Long Lei, Xu Chen and Xinhe Zhang. (2018). Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations. East Asian Journal on Applied Mathematics. 6 (2). 109-130. doi:10.4208/eajam.060815.180116a
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