[Paper Review] Generalization of the complex shifted Laplacian: on the class of expansion preconditioners for Helmholtz problems

Siegfried Cools, Wim Vanroose|arXiv (Cornell University)|Jan 19, 2015

Matrix Theory and Algorithms25 references3 citations

TL;DR

This paper introduces the expansion preconditioner class EX(m), a generalization of the complex shifted Laplacian (CSL) preconditioner for Helmholtz problems, based on a truncated Taylor series expansion of the inverse Helmholtz operator. While higher-order terms in EX(m) improve Krylov solver convergence, the increased application cost makes EX(1), equivalent to CSL, the most efficient practical choice.

ABSTRACT

In this note we propose the class of expansion preconditioners which forms a direct generalization of the CSL preconditioner. The construction of the EX(m) preconditioner is based upon a truncated Taylor series expansion of the original Helmholtz operator inverse. The EX(m) preconditioner is shown to significantly improve Krylov solver convergence rates for the Helmholtz problem for growing values of 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 the CSL or EX(1) preconditioner to be the practically most efficient member of the EX(m) class. We suggest possible extensions to the expansion preconditioner class which are particularly interesting forma theoretical viewpoint.

Motivation & Objective

• To generalize the complex shifted Laplacian (CSL) preconditioner into a broader class of preconditioners for Helmholtz problems.
• To analyze the trade-off between convergence improvement and computational cost when increasing the number of terms in the preconditioner expansion.
• To identify the optimal member of the EX(m) class in terms of practical efficiency for solving Helmholtz problems.
• To explore theoretical extensions of the expansion preconditioner framework for future research.

Proposed method

• Construct the EX(m) preconditioner via a truncated Taylor series expansion of the inverse Helmholtz operator.
• Formally define the preconditioner as a finite series of terms derived from the inverse operator's expansion.
• Apply the EX(m) preconditioner within Krylov subspace solvers to accelerate convergence for Helmholtz problems.
• Perform a cost-benefit analysis comparing solver convergence rates and preconditioner application costs across different values of m.
• Use the CSL preconditioner as the baseline, corresponding to the m=1 case of the EX(m) family.
• Explore theoretical extensions of the expansion preconditioner class for potential future developments.

Experimental results

Research questions

• RQ1 How does increasing the number of terms in the truncated Taylor series expansion affect the convergence rate of Krylov solvers for Helmholtz problems?
• RQ2 What is the trade-off between improved convergence and increased computational cost when using higher-order EX(m) preconditioners?
• RQ3 Is there an optimal value of m that maximizes the efficiency of the EX(m) preconditioner in practice?
• RQ4 How does the EX(m) preconditioner generalize the existing CSL preconditioner?
• RQ5 What theoretical extensions can be made to the expansion preconditioner framework to enhance its applicability?

Key findings

• The EX(m) preconditioner family generalizes the CSL preconditioner by incorporating higher-order terms from a Taylor series expansion of the inverse Helmholtz operator.
• Increasing the value of m leads to faster convergence of Krylov solvers for Helmholtz problems due to improved preconditioning.
• However, the computational cost of applying the preconditioner increases with m, particularly due to the need to evaluate multiple terms.
• Despite improved convergence, the higher cost of higher-order EX(m) preconditioners reduces their practical efficiency.
• The EX(1) preconditioner, equivalent to the CSL method, is found to be the most efficient member of the EX(m) class in practice.
• The study suggests theoretical extensions to the expansion preconditioner class that may be valuable for future research.