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This paper introduces and explores the class of expansion preconditionersEX(m) that forms a direct generalization to the classic complex shifted Laplace(CSL) preconditioner for Helmholtz problems. The construction of the EX(m)preconditioner is based upon a truncated Taylor series expansion of theoriginal Helmholtz operator inverse. The expansion preconditioner is shown tosignificantly improve Krylov solver convergence rates for the Helmholtz problemfor growing values of the number of series terms m. However, the addition ofmultiple terms in the expansion also increases the computational cost ofapplying the preconditioner. A thorough cost-benefit analysis of the additionof extra terms in the EX(m) preconditioner proves that the CSL or EX(1)preconditioner is the practically most efficient member of the expansionpreconditioner class. Additionally, possible extensions to the expansionpreconditioner class that further increase preconditioner efficiency aresuggested.

On the optimality of shifted Laplacian in the class of expansion preconditioners for the Helmholtz equation