On the optimality of shifted Laplacian in the class of expansion preconditioners for the Helmholtz equation
For researchers solving Helmholtz equations, this paper provides a theoretical justification for the optimality of the widely-used CSL preconditioner within a broader class, though the result is incremental.
The paper introduces expansion preconditioners (EX(m)) for Helmholtz problems, generalizing the complex shifted Laplace preconditioner. It shows that while more terms improve convergence, the classic CSL (EX(1)) is the most efficient in practice due to cost-benefit trade-offs.
This paper introduces and explores the class of expansion preconditioners EX(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 the original Helmholtz operator inverse. The expansion preconditioner is shown to significantly improve Krylov solver convergence rates for the Helmholtz problem for growing values of the number of series terms m. However, the addition of multiple terms in the expansion also increases the computational cost of applying the preconditioner. A thorough cost-benefit analysis of the addition of extra terms in the EX(m) preconditioner proves that the CSL or EX(1) preconditioner is the practically most efficient member of the expansion preconditioner class. Additionally, possible extensions to the expansion preconditioner class that further increase preconditioner efficiency are suggested.