Preconditioning Helmholtz linear systems

Daniel Osei-Kuffuor, Yousef Saad

Research output: Contribution to journalArticlepeer-review

44 Scopus citations


Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILU-based preconditioners ineffective. This paper presents a study of ways to enhance standard preconditioners by altering the diagonal by imaginary shifts. Prior work indicates that modifying the diagonal entries during the incomplete factorization process, by adding to it purely imaginary values can improve the quality of the preconditioner in a substantial way. Here we propose simple algebraic heuristics to perform the shifting and test these techniques with the ARMS and ILUT preconditioners. Comparisons are made with applications stemming from the diffraction of an acoustic wave incident on a bounded obstacle (governed by the Helmholtz Wave Equation).

Original languageEnglish (US)
Pages (from-to)420-431
Number of pages12
JournalApplied Numerical Mathematics
Issue number4
StatePublished - Apr 2010

Bibliographical note

Funding Information:
✩ This work was supported by DOE under grants DE-FG02-03ER25585 and DE-FG-08ER25841 and by the Minnesota Supercomputer Institute.


  • Complex diagonal shifts
  • Diagonal perturbation
  • Helmholtz equation
  • Incomplete LU factorization
  • Indefinite systems
  • Preconditioning


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