On the Approximation of Functions with Line Singularities by Ridgelets

TL;DR

This paper proves that ridgelets can optimally approximate functions with line singularities, which is crucial for developing efficient solvers for advection equations, advancing numerical PDE methods.

Contribution

It demonstrates that a specific ridgelet construction achieves optimal approximation rates for line singularities, improving upon previous methods and relaxing support conditions.

Findings

Ridgelets approximate line singularities with optimal rates.

The new convolution estimate is sharper and broadly applicable.

The results facilitate efficient PDE solvers for advection problems.

Abstract

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.