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.

In this paper we present a novel method for the numerical solution of linear advection 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.

In [GO15], the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the last remaining requirement to achieve this goal -by showing that ridgelets, on which the solver is based, approximate functions with line singularities (which may appear as solutions to the advection equation) with the best possible approximation rate.Structurally, the proof resembles [Can01], where a similar result was proved for a different ridgelet construction, which is however not well-suited for use in a PDE solver (and in particular, not suitable for the CDD-schemes [CDD01] we are interested in). Due to the differences between the two ridgelet constructions, we have to deal with quite a different set of issues, but are also able to relax the (support) conditions on the function being approximated. Finally, the proof employs a new convolution-type estimate that could be of independent interest due to its sharpness.

No abstract

No abstract