Transfer operator approach to ray-tracing in circular domains

Slipantschuk, Julia; Richter, Martin; Chappell, David J.; Tanner, Gregor; Just, Wolfram; Bandtlow, Oscar F.

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.

Citation

Slipantschuk, J., Richter, M., Chappell, D. J., Tanner, G., Just, W., & Bandtlow, O. F. (2020). Transfer operator approach to ray-tracing in circular domains. Nonlinearity, 33(11), 5773-5790. https://doi.org/10.1088/1361-6544/ab9dca

• Journal Article Type: Article
• Acceptance Date: Jun 17, 2020
• Online Publication Date: Sep 30, 2020
• Publication Date: Nov 1, 2020
• Deposit Date: Sep 7, 2020
• Publicly Available Date: Oct 1, 2021
• Journal: Nonlinearity
• Print ISSN: 0951-7715
• Electronic ISSN: 1361-6544
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 33
• Issue: 11
• Pages: 5773-5790
• DOI: https://doi.org/10.1088/1361-6544/ab9dca
• Keywords: Dynamical systems and ergodic theory, Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov expone
• Public URL: https://nottingham-repository.worktribe.com/output/4888164
• Publisher URL: https://iopscience.iop.org/article/10.1088/1361-6544/ab9dca