A novel α-absolute value preconditioner for all-at-once systems from heat equations

Abstract

In this paper, we generalize a fast Fourier transforms (FFTs) based preconditioner and propose a novel α-absolute value preconditioner for all-at-once systems from heat equations. Our preconditioner is symmetric positive definite and can also be computed efficiently using fast Fourier transforms or discrete sine transforms (DSTs). Furthermore, we prove that the corresponding preconditioned matrix has a unitary-plus-low-rank decomposition, and the eigenvalues of the preconditioned matrix are clustered at ±1, which leads to fast convergence rate of the preconditioned MINRES method. Finally, numerical examples on all-at-once systems from heat equations with different discretization scheme demonstrate the effectiveness of our new preconditioner.