How to Choose the Shift in the Shifted Laplace Preconditioner for the Helmholtz Equation Combined with Deflation

Research output: Chapter in Book/Conference proceedings/Edited volumeChapterScientific

Abstract

In recent work we showed that the performance of the complex shifted Laplace preconditioner for the discretized Helmholtz equation can be significantly improved by combining it multiplicatively with a deflation procedure that employs multigrid vectors. In this chapter we argue that in this combination the preconditioner improves the convergence of the outer Krylov acceleration through a new mechanism. This mechanism allows for a much larger damping and facilitates the approximate solve with the preconditioner. The convergence of the outer Krylov acceleration is not significantly delayed and occasionally even accelerated. To provide a basis for these claims, we analyze for a one-dimensional problem a two-level variant of the method in which the preconditioner is applied after deflation and in which both the preconditioner and the coarse grid problem are inverted exactly. We show that in case that the mesh is sufficiently fine to resolve the wave length, the spectrum after deflation consists of a cluster surrounded by two tails that extend in both directions along the real axis. The action of the inverse of the preconditioner is to shrink the length of the tails while at the same time rotating them and shifting the center of the cluster towards the origin. A much larger damping parameter than in algorithms without deflation can be used.
Original languageEnglish
Title of host publicationModern Solvers for Helmholtz Problems
Subtitle of host publicationBook Part II
EditorsDomenico Lahaye, Jok Tang, Kees Vuik
Place of PublicationCham
PublisherSpringer
Pages85-112
Number of pages28
ISBN (Electronic)978-3-319-28832-1
ISBN (Print)978-3-319-28831-4
DOIs
Publication statusPublished - 2017

Publication series

NameGeosystems Mathematics
ISSN (Print)2510-1544

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