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

Xue Lei Lin, Michael K. Ng*, Hai Wei Sun

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

48 Citations (Scopus)
23 Downloads (Pure)


In this paper, we study Toeplitz-like linear systems arising from time-dependent one-dimensional and two-dimensional Riesz space-fractional diffusion equations with variable diffusion coefficients. The coefficient matrix is a sum of a scalar identity matrix and a diagonal-times-Toeplitz matrix which allows fast matrix-vector multiplication in iterative solvers. We propose and develop a splitting preconditioner for this kind of matrix and analyze the spectra of the preconditioned matrix. Under mild conditions on variable diffusion coefficients, we show that the singular values of the preconditioned matrix are bounded above and below by positive constants which are independent of temporal and spatial discretization step-sizes. When the preconditioned conjugate gradient squared method is employed to solve such preconditioned linear systems, the method converges linearly within an iteration number independent of the discretization step-sizes. Numerical examples are given to illustrate the theoretical results and demonstrate that the performance of the proposed preconditioner is better than other tested solvers.

Original languageEnglish
Pages (from-to)1580-1614
Number of pages35
JournalSIAM Journal on Matrix Analysis and Applications
Issue number4
Publication statusPublished - 21 Dec 2017

Scopus Subject Areas

  • Analysis

User-Defined Keywords

  • Diagonal-times-Toeplitz matrices
  • Preconditioners
  • Space-fractional diffusion equations Krylov subspace methods
  • Variable coecients


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