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Transfer operator approach to ray-tracing in circular domains

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

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Authors

Julia Slipantschuk

MARTIN RICHTER MARTIN.RICHTER@NOTTINGHAM.AC.UK
Assistant Professor in Applied Mathematics

David J. Chappell

Profile image of GREGOR TANNER

GREGOR TANNER GREGOR.TANNER@NOTTINGHAM.AC.UK
Professor of Applied Mathematics

Wolfram Just

Oscar F. Bandtlow



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

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