An efficient two-level preconditioner for multi-frequency wave propagation problems

Manuel Baumann*, Martin B. van Gijzen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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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 analytic formula for the 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
Pages (from-to)316-332
Number of pages17
JournalApplied Numerical Mathematics
Publication statusPublished - 2019

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.


  • Frequency-domain formulation
  • Multi-shift GMRES
  • Preconditioner design
  • Shift-and-invert preconditioner
  • Visco-elastic wave equation


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