Compressed Absorbing Boundary Conditions for the Helmholtz Equation

TL;DR

This paper introduces a fast, memory-efficient method to approximate absorbing boundary conditions for the Helmholtz equation by directly fitting a surface operator using a few exterior solves, reducing computational complexity.

Contribution

It proposes a novel approach to compress absorbing boundary conditions by bypassing costly elimination procedures and directly fitting the operator with low-rank approximations from exterior solves.

Findings

Complexity grows logarithmically with frequency

Algorithm is nearly linear in matrix dimension

Provides a fast, memory-efficient compression scheme

Abstract

Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an absorbing layer to an operator at the boundary by layer-stripping elimination of the exterior unknowns, but the linear algebra involved is costly. We propose to bypass the elimination procedure, and directly fit the surface-to-surface operator in compressed form from a few exterior Helmholtz solves with random Dirichlet data. We obtain a concise description of the absorbing boundary condition, with a complexity that grows slowly (often, logarithmically) in the frequency parameter. We then obtain a fast (nearly linear in the dimension of the matrix) algorithm for the application of the absorbing boundary condition using partitioned low rank matrices. The result, modulo a precomputation, is a fast and memory-efficient compression scheme of an absorbing boundary condition for the Helmholtz equation.