On the Approximation of Functions with Line Singularities by Ridgelets
For researchers in numerical PDEs, this work provides the missing theoretical guarantee for ridgelet-based solvers to achieve optimal computational complexity.
The paper proves that ridgelets achieve the optimal approximation rate for functions with line singularities, completing the theoretical foundation for optimal-complexity PDE solvers. The proof introduces a sharp convolution-type estimate.
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.