A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time-space fractional diffusion equation
For researchers solving time-space fractional diffusion equations, this work provides a more efficient preconditioning method for the resulting linear systems.
This paper develops a limited-memory block bi-diagonal Toeplitz preconditioner for solving block lower triangular Toeplitz systems arising from time-space fractional diffusion equations, achieving O(N) storage and efficient inversion via a skew-circulant preconditioner. Numerical experiments demonstrate efficiency.
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.