Approximation of the high-frequency Helmholtz kernel by nested directional interpolation

TL;DR

This paper introduces an efficient approximation scheme for high-frequency Helmholtz kernels using polynomial interpolation and plane waves, enabling exponential convergence and supporting multilevel methods for fast Helmholtz integral computations.

Contribution

The paper develops a novel approximation scheme combining polynomial interpolation with plane waves for Helmholtz kernels, with proven exponential convergence and applicability to multilevel fast methods.

Findings

Exponential convergence of the approximation scheme.

Supports multilevel approximation techniques.

Applicable to 2D and 3D Helmholtz kernels.

Abstract

We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and postmultiplication by plane waves. It is shown to converge exponentially in the polynomial degree and supports multilevel approximation techniques. Our convergence analysis may be employed to establish exponential convergence of certain classes of fast methods for discretizations of the Helmholtz integral operator that feature polylogarithmic-linear complexity.