Localized Sparsifying Preconditioner for Periodic Indefinite Systems

TL;DR

This paper presents a localized sparsifying preconditioner for indefinite systems on periodic structures, improving accuracy and efficiency by utilizing local potential information and FFT-based local stencil computation.

Contribution

The work introduces a novel localized sparsifying preconditioner with FFT-based local stencil computation, enhancing accuracy and reducing setup time for indefinite periodic systems.

Findings

Iteration number grows mildly with problem size

Solving pseudospectral systems becomes as efficient as sparse systems

Preconditioner improves convergence for indefinite systems

Abstract

This paper introduces the localized sparsifying preconditioner for the pseudospectral approximations of indefinite systems on periodic structures. The work is built on top of the recently proposed sparsifying preconditioner with two major modifications. First, the local potential information is utilized to improve the accuracy of the preconditioner. Second, an FFT based method to compute the local stencil is proposed to reduce the setup time of the algorithm. Numerical results show that the iteration number of this improved method grows only mildly as the problem size grows, which implies that solving pseudospectral approximation systems is computationally as efficient as solving sparse systems, up to a mildly growing factor.