Transfer operator approach to ray-tracing in circular domains

TL;DR

This paper introduces a transfer operator approach to ray-tracing in circular domains, analyzing convergence of numerical schemes and demonstrating relevance to more complex geometries through examples.

Contribution

It develops a rigorous analysis of finite-rank approximation convergence for ray-tracing operators in circular geometries, with implications for more general shapes.

Findings

Finite-rank Fourier basis approximations converge with a power law.

Convergence rate depends on the smoothness of the source distribution.

Numerical examples illustrate applicability to complex geometries.

Abstract

The computation of wave-energy distributions in the mid-to-high frequency regime can be reduced to ray-tracing calculations. Solving the ray-tracing problem in terms of an operator equation for the energy density leads to an inhomogeneous equation which involves a Perron-Frobenius operator defined on a suitable Sobolev space. Even for fairly simple geometries, let alone realistic scenarios such as typical boundary value problems in room acoustics or for mechanical vibrations, numerical approximations are necessary. Here we study the convergence of approximation schemes by rigorous methods. For circular billiards we prove that convergence of finite-rank approximations using a Fourier basis follows a power law where the power depends on the smoothness of the source distribution driving the system. The relevance of our studies for more general geometries is illustrated by numerical examples.