Resolution-optimal exponential and double-exponential transform methods for functions with endpoint singularities

TL;DR

The paper presents a novel numerical method using specialized exponential and double-exponential transforms for efficiently approximating functions with endpoint singularities, outperforming standard methods in resolving oscillations with fewer degrees of freedom.

Contribution

It introduces a new class of variable transforms that achieve near-optimal resolution for functions with endpoint singularities, surpassing existing exponential methods.

Findings

Achieves near-machine epsilon accuracy with fewer degrees of freedom.

Requires 4-10 times fewer degrees of freedom than existing techniques.

Theoretically proven to outperform standard exponential transforms.

Abstract

We introduce a numerical method for the approximation of functions which are analytic on compact intervals, except at the endpoints. This method is based on variable transforms using particular parametrized exponential and double-exponential mappings, in combination with Fourier-like approximation in a truncated domain. We show theoretically that this method is superior to variable transform techniques based on the standard exponential and double-exponential mappings. In particular, it can resolve oscillatory behaviour using near-optimal degrees of freedom, whereas the standard mappings require degrees of freedom that grow superlinearly with the frequency of oscillation. We highlight these results with several numerical experiments. Therein it is observed that near-machine epsilon accuracy is achieved using a number of degrees of freedom that is between four and ten times smaller than those of existing techniques.