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

Abstract

In this paper, a first order implicit finite difference scheme with Krylov subspace linear system solver is employed to solving time-dependent space-fractional diffusion equations in high dimensions where the initial condition and source term are in tensor-train (TT) format with low TT-ranks. In the time-marching process, TT-format of the solution is maintained and the increment of TT-ranks due to addition is moderated by rounding. The error introduced by rounding is shown to be consistent with the first order finite difference scheme. On the other hand, the linear systems involved in the solution process are shown to possess Toeplitz-like structure so that the complexity and required memory for Krylov subspace solver can be optimized. Further reduction in complexity is made by utilizing a circulant preconditioner which accelerates the convergence rate of Krylov subspace method dramatically. Numerical examples for problems up to 20 dimensions are presented.