FFRT: A Fast Finite Ridgelet Transform for Radiative Transport

Abstract

This paper introduces an FFT-based implementation of a fast finite ridgelet transform which we call FFRT. Inspired by recent work where it was shown that ridgelet discretizations of linear transport equations can be easily preconditioned by diagonal preconditioning we use the FFRT for the numerical solution of such equations. Combining this FFRT-based method with a sparse collocation scheme we construct a novel solver for the radiative transport equation which results in uniformly well-conditioned linear systems.

Optimal adaptive ridgelet schemes for linear advection equations

“…Such a solver can then be combined with an angular discretisation scheme (be it via collocation, tensor products etc.) to solve the full RTE (1.2), as done in [14].…”

“…Said otherwise, the optimal representation of functions with singularities in different directions is achieved within a single frame, which is crucial for solving the full radiative transport problem (1.2). • Ridgelets also have the favourable property of possessing a multiscale structure, which opens the door to employ sparse tensor frame constructions for mitigating the curse of dimensionality [18] when solving the full RTE (1.2), as has been done in [14].…”

“…In [14], we have implemented a version -the source iteration -based on a sparse collocation scheme to solve the full RTE efficiently. There, a key feature of the use of ridgelets is that collocation in angle leads to uniformly well-conditioned linear systems to be solved, independent of the spatial resolution -a key property for efficient parallelisation.…”

“…Following recent results [17], that ridgelets permit the construction of simple diagonal preconditioners for linear advection equations which arise in collocation-type discretisation methods for kinetic transport equations (such as radiative transport), we intend this paper (and its companion [14]) to be a first step towards establishing directional representation systems as a useful tool for solving PDEs.…”

On the approximation of functions with line singularities by ridgelets

“…One possibility to reintroduce the scattering term is via an iterative scheme -for example by evaluating the integral for the previous iterand and adding the result to the right-hand side. We refer to [EGO15], where an FFT-based ridgelet discretisation based on this "source iteration" has been implemented.…”