A parallel fast algorithm for computing the Helmholtz integral operator in 3-D layered media

Abstract

In this paper, we propose a parallel fast algorithm for computing the product of the discretized Helmholtz integral operator in layered media and a vector in O(NqNz2NxNylogNxNy) operations. Here NxNyNz is the number of sources and Nq is the number of quadrature points used in the evaluation of the Sommerfeld integral in the definition of layered media Green’s function (for problems in thin-layer media, Nz=O(1)). Such a product forms the key step of many iterative solvers (such as the Krylov subspace based GMRES and BiCGSTAB) for linear systems arising from the integral equation methods for the Helmholtz equations. The fast solver is based on two important techniques which reduce the cost of quadrature summations in the Sommerfeld contour integral for Green’s functions in 3-D layered media. The first technique is the removal of surface pole effects along the real axis integration contour by identifying the pole locations with a discrete wavelet transform; In the second technique, we apply a window-based high frequency filter to shorten the contour length. As a result, the integral operator for the 3-D layered media can be efficiently written as a sum of 2-D Hankel cylindrical integral operators, and the latter can be calculated by either a tree-code or a 2-D wideband fast multipole method in a fast manner. Numerical results show the efficiency and parallelism of the proposed fast algorithm.