A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time-space fractional diffusion equation

TL;DR

This paper introduces a novel preconditioner for efficiently solving block lower triangular Toeplitz systems from time-space fractional diffusion equations, significantly reducing storage and computation costs.

Contribution

It develops a new block bi-diagonal Toeplitz preconditioner with low storage requirements and a skew-circulant preconditioner for fast inverse calculations, enhancing solution efficiency.

Findings

Preconditioners significantly improve convergence speed.

Storage requirement is reduced to O(N).

Numerical experiments confirm efficiency gains.

Abstract

A block lower triangular Toeplitz system arising from time-space fractional diffusion equation is discussed. For efficient solutions of such the linear system, the preconditioned biconjugate gradient stabilized method and flexible general minimal residual method are exploited. The main contribution of this paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is developed for the block lower triangular Toeplitz system, whose storage is of $\mathcal{O}(N)$ with $N$ being the spatial grid number; (ii) A new skew-circulant preconditioner is designed to fast calculate the inverse of the block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical experiments are given to demonstrate the efficiency of our preconditioners.