An improved two‐grid preconditioner for the solution of three‐dimensional Helmholtz problems in heterogeneous media

Abstract

SUMMARYIn this paper, we address the solution of three‐dimensional heterogeneous Helmholtz problems discretized with second‐order finite difference methods with application to acoustic waveform inversion in geophysics. In this setting, the numerical simulation of wave propagation phenomena requires the approximate solution of possibly very large indefinite linear systems of equations. For that purpose, we propose and analyse an iterative two‐grid method acting on the Helmholtz operator where the coarse grid problem is solved inaccurately. A cycle of a multigrid method applied to a complex shifted Laplacian operator is used as a preconditioner for the approximate solution of this coarse problem. A single cycle of the new method is then used as a variable preconditioner of a flexible Krylov subspace method. We analyse the properties of the resulting preconditioned operator by Fourier analysis. Numerical results demonstrate the effectiveness of the algorithm on three‐dimensional applications. The proposed numerical method allows us to solve three‐dimensional wave propagation problems even at high frequencies on a reasonable number of cores of a distributed memory computer. Copyright © 2012 John Wiley & Sons, Ltd.

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

“…while neglecting terms that are of magnitude lower than O( 2 h 2 ) relatively to the magnitude of the entries of H. The first and second lines of (19) are identical to those in our discretization of (4) using (13). The second line suggests that the mass matrix that multiplies Δ should be discretized with an averaging operator 1 2 (a +1 + a −1 ) if we wish to make the two discretizations more similar.…”

“…The larger is, the more efficient the solution of the shifted system using multigrid is, but the quality of (24) as a preconditioner deteriorates. The compromise suggested in the work of Erlangga et al 12 is to use = 0.5, together with rather standard multigrid cycles, whereas the works of Calandra et al 13 and Treister and Haber 8 suggest applying more elaborate cycles.…”

“…One important observation should be made regarding (14): At the immediate neighborhood of the source point (distance of one node), we cannot use the second-order upwind scheme because ∇ changes signs. Therefore, for those nodes, we simply use (13). This situation is similar to the treatment of the same points when calculating using fast marching.…”

“…That is because the discretization of the problem requires a very fine mesh and a large number of unknowns, in addition to the indefiniteness of the associated matrix. [13][14][15] In recent years, there has been a great effort to develop efficient solvers for systems arising from (1), using several different approaches to tackle the problem. One of the most common approaches is the shifted Laplacian multigrid preconditioner, 12,13,[16][17][18][19][20][21][22] which modifies the equation by adding complex values to the diagonal of the matrix.…”

A short note on a pipelined polarized-trace algorithm for 3D Helmholtz

“…• On the other had, recently introduced preconditioners can be subdivided into multigrid-based preconditioners, such as the ones proposed by Erlangga et al (2006); Sheikh et al (2013); Calandra et al (2013), which are simple to implement but have super-linear complexity and need special tuning; and sweeping-like preconditioners such as the ones proposed by Gander and Nataf (2005); Engquist and Ying (2011a,b); Liu and Ying (2015); Chen and Xiang (2013); Stolk (2013); ZepedaNúñez and Demanet (2016); Vion and Geuzaine (2014).…”