Preconditioning Techniques for Diagonal-times-Toeplitz Matrices in Fractional Diffusion Equations

Jianyu Pan, Rihuan Ke, Michael K. Ng*, Hai Wei Sun

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

125 Citations (Scopus)
68 Downloads (Pure)


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 row-by-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

Original languageEnglish
Pages (from-to)A2698-A2719
Number of pages22
JournalSIAM Journal on Scientific Computing
Issue number6
Publication statusPublished - 18 Nov 2014

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Approximate inverse
  • Circulant matrix
  • Fast Fourier transform
  • Fractional diffusion equation
  • Krylov subspace methods
  • Toeplitz matrix


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