LR-WaveHoltz: A Low-Rank Helmholtz Solver

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 seek to compress the solution in matrix form, and in three dimensions using tensor trains.

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.