Close Search Advanced
Journals
Resources
About Us
Open Access

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations
Preview
Add to basket

Year:    2016

East Asian Journal on Applied Mathematics, Vol. 6 (2016), Iss. 2 : pp. 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.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/eajam.060815.180116a

East Asian Journal on Applied Mathematics, Vol. 6 (2016), Iss. 2 : pp. 109–130

Published online:    2016-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    22

Keywords:    High-dimensional two-sided fractional diffusion equation implicit finite difference method unconditionally stable multilevel circulant preconditioner GMRES method.

  1. Tensorized low-rank circulant preconditioners for multilevel Toeplitz linear systems from high-dimensional fractional Riesz equations

    Zhang, Lei | Zhang, Guo-Feng | Liang, Zhao-Zheng

    Computers & Mathematics with Applications, Vol. 110 (2022), Iss. P.64

    https://doi.org/10.1016/j.camwa.2022.01.003 [Citations: 0]

  2. Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

    Pougkakiotis, Spyridon | Pearson, John W. | Leveque, Santolo | Gondzio, Jacek

    SIAM Journal on Matrix Analysis and Applications, Vol. 41 (2020), Iss. 3 P.1443

    https://doi.org/10.1137/19M128288X [Citations: 2]

  3. Tensor-Train Format Solution with Preconditioned Iterative Method for High Dimensional Time-Dependent Space-Fractional Diffusion Equations with Error Analysis

    Chou, Lot-Kei | Lei, Siu-Long

    Journal of Scientific Computing, Vol. 80 (2019), Iss. 3 P.1731

    https://doi.org/10.1007/s10915-019-00994-3 [Citations: 4]

  4. A novel banded preconditioner for coupled tempered fractional diffusion equation generated from the regime-switching CGMY model

    Chen, Xu | Gong, Xin-Xin | She, Zi-Run | She, Zi-Hang

    Numerical Algorithms, Vol. (2024), Iss.

    https://doi.org/10.1007/s11075-024-01769-0 [Citations: 0]

  5. On τ-preconditioner for a novel fourth-order difference scheme of two-dimensional Riesz space-fractional diffusion equations

    Huang, Yuan-Yuan | Qu, Wei | Lei, Siu-Long

    Computers & Mathematics with Applications, Vol. 145 (2023), Iss. P.124

    https://doi.org/10.1016/j.camwa.2023.06.015 [Citations: 3]

  6. Block preconditioning strategies for time–space fractional diffusion equations

    Chen, Hao | Zhang, Tongtong | Lv, Wen

    Applied Mathematics and Computation, Vol. 337 (2018), Iss. P.41

    https://doi.org/10.1016/j.amc.2018.05.001 [Citations: 3]

  7. An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations

    Li, Fangli | Fu, Hongfei | Liu, Jun

    Mathematical Methods in the Applied Sciences, Vol. 44 (2021), Iss. 4 P.2934

    https://doi.org/10.1002/mma.6306 [Citations: 3]

  8. A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

    Tang, Shi-Ping | Huang, Yu-Mei

    Computers & Mathematics with Applications, Vol. 144 (2023), Iss. P.210

    https://doi.org/10.1016/j.camwa.2023.05.028 [Citations: 4]

  9. Fast ADI method for high dimensional fractional diffusion equations in conservative form with preconditioned strategy

    Chou, Lot-Kei | Lei, Siu-Long

    Computers & Mathematics with Applications, Vol. 73 (2017), Iss. 3 P.385

    https://doi.org/10.1016/j.camwa.2016.11.034 [Citations: 16]

  10. A preconditioned implicit difference scheme for semilinear two‐dimensional time–space fractional Fokker–Planck equations

    Zhang, Chengjian | Zhou, Yongtao

    Numerical Linear Algebra with Applications, Vol. 28 (2021), Iss. 4

    https://doi.org/10.1002/nla.2357 [Citations: 1]

  11. An Efficient Second-Order Convergent Scheme for One-Side Space Fractional Diffusion Equations with Variable Coefficients

    Lin, Xue-lei | Lyu, Pin | Ng, Michael K. | Sun, Hai-Wei | Vong, Seakweng

    Communications on Applied Mathematics and Computation, Vol. 2 (2020), Iss. 2 P.215

    https://doi.org/10.1007/s42967-019-00050-9 [Citations: 5]

  12. Kronecker product based preconditioners for boundary value method discretizations of space fractional diffusion equations

    Chen, Hao | Huang, Qiuyue

    Mathematics and Computers in Simulation, Vol. 170 (2020), Iss. P.316

    https://doi.org/10.1016/j.matcom.2019.11.007 [Citations: 1]

  13. A Kronecker product splitting preconditioner for two-dimensional space-fractional diffusion equations

    Chen, Hao | Lv, Wen | Zhang, Tongtong

    Journal of Computational Physics, Vol. 360 (2018), Iss. P.1

    https://doi.org/10.1016/j.jcp.2018.01.034 [Citations: 17]

  14. A numerical framework for the approximate solution of fractional tumor-obesity model

    Arshad, Sadia | Baleanu, Dumitru | Defterli, Ozlem |

    International Journal of Modeling, Simulation, and Scientific Computing, Vol. 10 (2019), Iss. 01 P.1941008

    https://doi.org/10.1142/S1793962319410083 [Citations: 5]

  15. Splitting preconditioning based on sine transform for time-dependent Riesz space fractional diffusion equations

    Lu, Xin | Fang, Zhi-Wei | Sun, Hai-Wei

    Journal of Applied Mathematics and Computing, Vol. 66 (2021), Iss. 1-2 P.673

    https://doi.org/10.1007/s12190-020-01454-0 [Citations: 19]

  16. A fast preconditioned iterative method for two-dimensional options pricing under fractional differential models

    Chen, Xu | Ding, Deng | Lei, Siu-Long | Wang, Wenfei

    Computers & Mathematics with Applications, Vol. 79 (2020), Iss. 2 P.440

    https://doi.org/10.1016/j.camwa.2019.07.010 [Citations: 4]

  17. Kronecker product-based structure preserving preconditioner for three-dimensional space-fractional diffusion equations

    Chen, Hao | Lv, Wen

    International Journal of Computer Mathematics, Vol. 97 (2020), Iss. 3 P.585

    https://doi.org/10.1080/00207160.2019.1581177 [Citations: 3]

  18. Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations

    Jian, Huan-Yan | Huang, Ting-Zhu | Gu, Xian-Ming | Zhao, Xi-Le | Zhao, Yong-Liang

    Computers & Mathematics with Applications, Vol. 94 (2021), Iss. P.136

    https://doi.org/10.1016/j.camwa.2021.05.003 [Citations: 9]

  19. PCG method with Strang’s circulant preconditioner for Hermitian positive definite linear system in Riesz space fractional advection–dispersion equations

    Qu, Wei | Shen, Hai-Wei | Liang, Yong

    Computational and Applied Mathematics, Vol. 37 (2018), Iss. 4 P.4554

    https://doi.org/10.1007/s40314-018-0586-6 [Citations: 3]

  20. An implicit difference scheme with the KPS preconditioner for two-dimensional time–space fractional convection–diffusion equations

    Zhou, Yongtao | Zhang, Chengjian | Brugnano, Luigi

    Computers & Mathematics with Applications, Vol. 80 (2020), Iss. 1 P.31

    https://doi.org/10.1016/j.camwa.2020.02.014 [Citations: 8]

  21. Efficient preconditioners for Radau-IIA time discretization of space fractional diffusion equations

    Chen, Hao | Xu, Dongping

    Numerical Algorithms, Vol. 83 (2020), Iss. 4 P.1349

    https://doi.org/10.1007/s11075-019-00728-4 [Citations: 1]

  22. A Robust Preconditioner for Two-dimensional Conservative Space-Fractional Diffusion Equations on Convex Domains

    Chen, Xu | Deng, Si-Wen | Lei, Siu-Long

    Journal of Scientific Computing, Vol. 80 (2019), Iss. 2 P.1033

    https://doi.org/10.1007/s10915-019-00966-7 [Citations: 3]

  23. A Splitting Preconditioner for Toeplitz-Like Linear Systems Arising from Fractional Diffusion Equations

    Lin, Xue-lei | Ng, Michael K. | Sun, Hai-Wei

    SIAM Journal on Matrix Analysis and Applications, Vol. 38 (2017), Iss. 4 P.1580

    https://doi.org/10.1137/17M1115447 [Citations: 50]

  24. FAST SECOND-ORDER ACCURATE DIFFERENCE SCHEMES FOR TIME DISTRIBUTED-ORDER AND RIESZ SPACE FRACTIONAL DIFFUSION EQUATIONS

    Jian, Huanyan | Huang, Tingzhu | Zhao, Xile | Zhao, Yongliang

    Journal of Applied Analysis & Computation, Vol. 9 (2019), Iss. 4 P.1359

    https://doi.org/10.11948/2156-907X.20180247 [Citations: 1]

  25. Efficient preconditioner of one-sided space fractional diffusion equation

    Lin, Xue-Lei | Ng, Michael K. | Sun, Hai-Wei

    BIT Numerical Mathematics, Vol. 58 (2018), Iss. 3 P.729

    https://doi.org/10.1007/s10543-018-0699-8 [Citations: 22]