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.

中文翻译：

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圆形域中光线追踪的传递算子方法

中高频范围内波能分布的计算可以简化为射线追踪计算。根据能量密度的算子方程解决光线追踪问题会导致非齐次方程，该方程涉及在合适的 Sobolev 空间上定义的 Perron-Frobenius 算子。即使对于相当简单的几何形状，更不用说现实场景，例如室内声学中的典型边界值问题或机械振动，数值近似也是必要的。在这里，我们通过严格的方法研究近似方案的收敛性。对于圆形台球，我们证明使用傅立叶基础的有限秩近似的收敛遵循幂律，其中幂取决于驱动系统的源分布的平滑度。