On the optimality of shifted Laplacian in the class of expansion preconditioners for the Helmholtz equation

TL;DR

This paper introduces the expansion preconditioners EX(m) for Helmholtz problems, generalizing the complex shifted Laplace preconditioner, and analyzes their efficiency and convergence improvements.

Contribution

It proposes a new class of expansion preconditioners EX(m) based on Taylor series, and demonstrates that the classic CSL preconditioner is most efficient among them.

Findings

EX(m) improves Krylov solver convergence for larger m

Adding more terms increases computational cost

CSL (EX(1)) is the most efficient preconditioner in the class

Abstract

This paper introduces and explores the class of expansion preconditioners EX(m) that forms a direct generalization to the classic complex shifted Laplace (CSL) preconditioner for Helmholtz problems. The construction of the EX(m) preconditioner is based upon a truncated Taylor series expansion of the original Helmholtz operator inverse. The expansion preconditioner is shown to significantly improve Krylov solver convergence rates for the Helmholtz problem for growing values of the number of series terms m. However, the addition of multiple terms in the expansion also increases the computational cost of applying the preconditioner. A thorough cost-benefit analysis of the addition of extra terms in the EX(m) preconditioner proves that the CSL or EX(1) preconditioner is the practically most efficient member of the expansion preconditioner class. Additionally, possible extensions to the expansion preconditioner class that further increase preconditioner efficiency are suggested.