Preconditioning the 2D Helmholtz equation with polarized traces

Abstract

We present a domain decomposition solver for the 2D Helmholtz equation, with a special choice of integral transmission condition that involves polarizing the waves into oneway components. This refinement of the transmission condition is the key to combining local direct solves into an efficient iterative scheme, which can then be deployed in a highperformance computing environment. The method involves an expensive, but embarrassingly parallel precomputation of local Green's functions, and a fast online computation of layer potentials in partitioned low-rank form. The online part has sequential complexity that scales sublinearly with respect to the number of volume unknowns, even in the high-frequency regime. The favorable complexity scaling continues to hold in the context of low-order finite difference schemes for standard community models such as BP and Marmousi2, where convergence occurs in 5 to 10 GMRES iterations.