An Efficient Two-Level Preconditioner for Multi-Frequency Wave Propagation Problems

M. Baumann, M.B. van Gijzen

Research output: Book/ReportReportProfessional


We consider wave propagation problems that are modeled in the frequency-domain, and that need to be solved simultaneously for multiple frequencies within a fixed range. For this, a single shift-and-invert preconditioner at a so-called seed frequency is applied. The choice of the seed is crucial for the performance of preconditioned multi-shift GMRES and is closely related to the parameter choice for the Complex Shifted Laplace preconditioner. Based on a classical GMRES convergence bound, we present an optimal seed parameter that purely depends on the original frequency range. The new insight is exploited in a two-level preconditioning strategy: A shifted Neumann preconditioner with minimized spectral radius is additionally applied to multi-shift GMRES. Moreover, we present a reformulation of the multi-shift problem to a matrix equation solved with, for instance, global GMRES. Here, our analysis allows for rotation of the spectrum of the linear operator. Numerical experiments for the time-harmonic visco-elastic wave equation demonstrate the performance of the new preconditioners.
Original languageEnglish
Place of PublicationDelft
PublisherDelft University of Technology
Number of pages17
Publication statusPublished - 2017

Publication series

NameReports of the Delft Institute of Applied Mathematics
ISSN (Print)1389-6520


  • multi-shift GMRES
  • shift-and-invert preconditioner
  • preconditioner design
  • visco-elastic wave equation
  • frequency-domain formulation


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