Solving the three-dimensional high-frequency Helmholtz equation using contour integration and polynomial preconditioning

Xiao Liu, Yuanzhe Xi, Yousef Saad, Maarten V. de Hoop

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We propose an iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solution in certain invariant subspaces is approximated by solving complex-shifted linear systems, resulting in faster GMRES iterations due to the restricted spectrum. The shifted systems are solved by exploiting a polynomial fixed-point iteration, which is a robust scheme even if the magnitude of the shift is small. Numerical tests in three dimensions indicate that O(n1/3) matrix-vector products are needed to solve a high-frequency problem with a matrix size n with high accuracy. The method has a small storage requirement, can be applied to both dense and sparse linear systems, and is highly parallelizable.

Original languageEnglish (US)
Pages (from-to)58-82
Number of pages25
JournalSIAM Journal on Matrix Analysis and Applications
Volume41
Issue number1
DOIs
StatePublished - Jan 1 2020

Keywords

  • Cauchy integral
  • Helmholtz preconditioner
  • Polynomial iteration
  • Shifted Laplacian

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