Compressed absorbing boundary conditions via matrix probing

TL;DR

This paper introduces a method to efficiently approximate absorbing boundary conditions for the Helmholtz equation using matrix probing, reducing computational cost while maintaining accuracy in heterogeneous media.

Contribution

It proposes a novel approach to directly fit the boundary operator in compressed form from a few solves, bypassing costly elimination procedures.

Findings

Achieves accurate boundary condition approximation with fewer Helmholtz solves.

Complexity grows logarithmically with frequency, improving efficiency.

Provides a practical method for heterogeneous media applications.

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. The result is a concise description of the absorbing boundary condition, with a complexity that grows slowly (often, logarithmically) in the frequency parameter.