Optimal Adaptive Ridgelet Schemes for Linear Transport Equations

TL;DR

This paper introduces an optimal ridgelet-based numerical scheme for linear transport equations, effectively handling singularities with proven system matrix properties and demonstrating promising numerical results.

Contribution

It develops a novel ridgelet-based method with proven optimal complexity and well-conditioned system matrices for solving linear transport equations.

Findings

System matrix is uniformly well-conditioned and compressible.

The scheme achieves optimal complexity even with line singularities.

Numerical experiments confirm effective approximation and singularity localization.

Abstract

In this paper we present a novel method for the numerical solution of linear transport equations, which is based on ridgelets. Such equations arise for instance in radiative transfer or in phase contrast imaging. Due to the fact that ridgelet systems are well adapted to the structure of linear transport operators, it can be shown that our scheme operates in optimal complexity, even if line singularities are present in the solution. The key to this is showing that the system matrix (with diagonal preconditioning) is uniformly well-conditioned and compressible -- the proof for the latter represents the main part of the paper. We conclude with some numerical experiments about $N$-term approximations and how they are recovered by the solver, as well as localisation of singularities in the ridgelet frame.