Adaptive and anisotropic piecewise polynomial approximation

TL;DR

This paper reviews the theory of adaptive, anisotropic piecewise polynomial approximation, focusing on optimal partition properties, smoothness conditions for convergence, and algorithms for generating near-optimal adaptive partitions.

Contribution

It surveys recent advances in adaptive anisotropic approximation theory, highlighting developments in multivariate cases and refinement algorithms for near-optimal partitions.

Findings

Established theory for univariate adaptive approximation

Recent progress in multivariate anisotropic approximation

Development of fast algorithms for adaptive partitioning

Abstract

We survey the main results of approximation theory for adaptive piecewise polynomial functions. In such methods, the partition on which the piecewise polynomial approximation is defined is not fixed in advance, but adapted to the given function f which is approximated. We focus our discussion on (i) the properties that describe an optimal partition for f, (ii) the smoothness properties of f that govern the rate of convergence of the approximation in the Lp-norms, and (iii) fast refinement algorithms that generate near optimal partitions. While these results constitute a fairly established theory in the univariate case and in the multivariate case when dealing with elements of isotropic shape, the approximation theory for adaptive and anisotropic elements is still building up. We put a particular emphasis on some recent results obtained in this direction.