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.

翻译：本文提出一种求解亥姆霍兹方程的低秩方法。该方法基于WaveHoltz方法，通过对相关波动方程的解施加时域滤波来计算亥姆霍兹方程的解。波动方程采用高阶多块求和-分部有限差分法进行离散化。在二维情形中，我们使用奇异值分解进行数值解的压缩；在三维情形中，则采用张量列车进行压缩。为控制秩增长，我们在时间步进过程中采用截断步进法，并对WaveHoltz不动点迭代使用低秩安德森加速。我们进行了大量数值实验，验证了该迭代格式在二维和三维常波速及分段常波速自由空间与半空间问题中的收敛性和有效性。