A multigrid‐based shifted Laplacian preconditioner for a fourth‐order Helmholtz discretization

Abstract

Abstract. In this paper, an iterative solution method for a fourth-order accurate discretization of the Helmholtz equation is presented. The method is a generalization of that presented in [10], where multigrid was employed as a preconditioner for a Krylov subspace iterative method. This multigrid preconditioner is based on the solution of a second Helmholtz operator with a complexvalued shift. In particular, we compare preconditioners based on a point-wise Jacobi smoother with those using an ILU(0) smoother, we compare using the prolongation operator developed by de Zeeuw in [37] with interpolation operators based on algebraic multigrid principles, and we compare the performance of the Krylov subspace method Bi-CGSTAB with the recently introduced induced dimension reduction method, IDR(s). These three improvements are combined to yield an efficient solver for heterogeneous high-wavenumber problems.Key words. Helmholtz equation, non-constant high wavenumber, complex-valued multigrid preconditioner, algebraic multigrid, ILU smoother AMS subject classifications. 65N55, 65F10, 78A45, 35J051. Introduction. Many authors, e.g. [5,7,13,19], have contributed to the development of appropriate multigrid methods for the Helmholtz equation, but an efficient multigrid treatment of heterogeneous problems with high wavenumers arising in engineering settings has not yet been proposed in the literature. The multigrid method [4,14] is known to be a highly efficient iterative method, for example, for discrete Poisson-type equations, even with fourth-order accurate discretizations [6,33]. The Helmholtz equation, however, does not belong to the class of PDEs for which off-the-shelf multigrid methods perform efficiently. Convergence degradation and, consequently, loss of O(N ) complexity are caused by difficulties encountered in the smoothing and coarse-grid correction components; see [7,33] for a discussion.We present an efficient numerical solution technique for the heterogeneous highwavenumber Helmholtz equation, discretized by fourth-order finite differences. Recently,in [10], a robust preconditioned Bi-CGSTAB method has been proposed for solving these problems, in which the preconditioner is based on a second Helmholtz equation with an imaginary shift. This preconditioner is a member of the family of shifted Laplacian operators, introduced in [20], and its inverse can be efficiently approximated by means of a multigrid iteration. Two-dimensional results, representative for geophysical applications, generated by second-order finite differences, have been presented in [26] and 3D results in [27].In this paper, we generalize this solver and include, in particular, a fourth-order discretization of the Helmholtz operator in our discussion. The multigrid preconditioner is enhanced, in the sense that we replace the point-wise Jacobi smoother in the multigrid preconditioner by a variant of the incomplete lower-upper factorization smoother, ILU(0). Furthermore, we evaluate the performance of a prolongation

A fast method for the solution of the Helmholtz equation

“…However, our tests show only slight variation (by an iteration or two) over different realizations. For the upwind discretization, these results are qualitatively similar to those seen for the shifted-Laplace preconditioners considered in [7,18,21], where results degrade as xh increases and, for fixed xh, scale like 1/h. However, the h-ellipticity bounds discussed above suggest that better performance is possible, with iteration counts that are independent of h for fixed xh, and only slight dependence on xh, at least in the upwind-discretization case.…”

“…In recent years, there has been a substantial interest in developing fast solvers for the Helmholtz equation [3,4,[6][7][8][9]18,21]. This interest is motivated primarily by geophysical applications [13], although it also serves as an important model problem for many fields where fully indefinite partial differential equations arise.…”

“…Recent work includes the combination of multigrid with Krylov space relaxation [6], multigrid applied to the least-squares formulation [9], and the wave-ray multigrid approach [3,4]. Another approach that has attracted significant interest both recently and in the past is the family of ''shifted-Laplacian'' preconditioners (see, for example [7,12,18,21], and the references cited therein). Despite many years of research, none of the above approaches is a ''silver bullet'' for this very difficult problem.…”

Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

“…One of the advantages of the shifted Laplacian preconditioner is that it can be applied when the wavenumber k is variable (i.e., the medium being modelled is inhomogeneous) as was done, for example, in [15,42], and [54] (with the last paper considering the higher-order case). In the present paper, however, all the theory is for constant k (although Example 5.5 contains an experiment where k is variable).…”

On the robustness of ILU smoothers on triangular grids

“…In particular, G. Wittum [36] was the first to rigorously prove the robustness of this smoother for an anisotropic model problem, and R. Stevenson [23] generalized the results in [36]. More recently, in [27], a fast preconditioner solver based on multigrid with an ILU smoother was proposed for solving heterogeneous high-wavenumber Helmholtz problems, and in [21], a new relaxation methodology based on a truncated ILU smoother is presented for multigrid preconditioning of discrete convection-diffusion problems.…”