Toeplitz equations by conjugate gradients with circulant preconditioner
SIAM Journal on Scientific and Statistical Computing
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
Numerical solution of the space fractional Fokker-Planck equation
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on boundary and interior layers - computational and asymptotic methods (BAIL 2002)
Finite difference approximations for fractional advection-dispersion flow equations
Journal of Computational and Applied Mathematics
The accuracy and stability of an implicit solution method for the fractional diffusion equation
Journal of Computational Physics
Iterative Methods for Toeplitz Systems (Numerical Mathematics and Scientific Computation)
Iterative Methods for Toeplitz Systems (Numerical Mathematics and Scientific Computation)
Finite difference methods for two-dimensional fractional dispersion equation
Journal of Computational Physics
A second-order accurate numerical approximation for the fractional diffusion equation
Journal of Computational Physics
Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)
An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)
Numerical solutions for fractional reaction-diffusion equations
Computers & Mathematics with Applications
Implicit finite difference approximation for time fractional diffusion equations
Computers & Mathematics with Applications
Finite difference approximations for a fractional advection diffusion problem
Journal of Computational Physics
Finite difference approximations for two-sided space-fractional partial differential equations
Applied Numerical Mathematics
Finite Element Method for the Space and Time Fractional Fokker-Planck Equation
SIAM Journal on Numerical Analysis
A direct O(Nlog2N) finite difference method for fractional diffusion equations
Journal of Computational Physics
Multigrid method for fractional diffusion equations
Journal of Computational Physics
Journal of Computational Physics
Fractional-Order Anisotropic Diffusion for Image Denoising
IEEE Transactions on Image Processing
Journal of Computational Physics
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The implicit finite difference scheme with the shifted Grunwald formula, which is unconditionally stable, is employed to discretize fractional diffusion equations. The resulting systems are Toeplitz-like and then the fast Fourier transform can be used to reduce the computational cost of the matrix-vector multiplication. The preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems. The spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant; hence the convergence rate of the proposed iterative algorithm is superlinear. Numerical experiments are carried out to demonstrate that our circulant preconditioner works very well, even though for cases of variable diffusion coefficients.