title: Fast permutation preconditioning for fractional diffusion equations.
abstract: In this paper, an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable, is used to discretize the fractional diffusion equations with constant diffusion coefficients. The coefficient matrix possesses the Toeplitz structure and the fast Toeplitz matrix-vector product can be utilized to reduce the computational complexity from [Formula: see text] to [Formula: see text], where N is the number of grid points. Two preconditioned iterative methods, named bi-conjugate gradient method for Toeplitz matrix and bi-conjugate residual method for Toeplitz matrix, are proposed to solve the relevant discretized systems. Finally, numerical experiments are reported to show the effectiveness of our preconditioners.
authors.affiliation: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731 Sichuan People's Republic of China.
authors.forename: Sheng-Feng
authors.initials: SF
authors.lastname: Wang
authors.affiliation: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731 Sichuan People's Republic of China.
authors.forename: Ting-Zhu
authors.initials: TZ
authors.lastname: Huang
authors.affiliation: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731 Sichuan People's Republic of China.
authors.forename: Xian-Ming
authors.initials: XM
authors.lastname: Gu
authors.affiliation: Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang, 641100 Sichuan People's Republic of China.
authors.forename: Wei-Hua
authors.initials: WH
authors.lastname: Luo
keywords: BiCGT method; BiCRT method; Fast Fourier transforms; Fractional diffusion equations; Shifted Grünwald formula; Toeplitz matrix