LR-WaveHoltz: A Low-Rank Helmholtz Solver

Abstract: We propose a low-rank method for solving the Helmholtz equation. Our approach is based on the WaveHoltz method, which computes Helmholtz solutions by applying a time-domain filter to the solution of a related wave equation. The wave equation is discretized by high-order multiblock summation-by-parts finite differences. In two dimensions we use the singular value decomposition and in three dimensions we use tensor trains to compress the numerical solution. To control rank growth we use step-truncation during time stepping and a low-rank Anderson acceleration for the WaveHoltz fixed point iteration. We have carried out extensive numerical experiments demonstrating the convergence and efficacy of the iterative scheme for free- and half-space problems in two and three dimensions with constant and piecewise constant wave speeds.