Adaptive Approximation of Functions with Discontinuities

TL;DR

This paper introduces an adaptive approximation algorithm that efficiently approximates functions with discontinuities by dividing the domain into subregions, computing local approximations, and merging them to avoid Gibbs phenomena.

Contribution

The paper proposes a novel adaptive method for approximating discontinuous functions by partitioning the domain and constructing local approximations that are then combined globally.

Findings

Effective handling of discontinuities without Gibbs phenomenon

Parallelizable subdomain approximation approach

Works with fixed scattered data, no resampling needed

Abstract

One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in those subdomains, and these {\em sub-approximations} can possibly be calculated efficiently in parallel, as long as the subdomains do not overlap. This paper proposes a class of algorithms that first calculate sub-approximations on non-overlapping subdomains, then extend the subdomains as much as possible and finally produce a global solution on the given domain by letting the subdomains fill the whole domain. Consequently, there will be no Gibbs phenomenon along the boundaries of the subdomains. Throughout, the algorithm works for fixed scattered input data of the function itself, not on spectral data, and it does not resample.