Adaptive multiresolution analysis based on anisotropic triangulations

TL;DR

This paper introduces a greedy refinement algorithm for adaptive multiresolution analysis using anisotropic triangulations, producing hierarchical data structures and wavelet bases with proven convergence and optimal triangle aspect ratios.

Contribution

It presents a novel greedy refinement procedure for anisotropic triangulations that adaptively minimizes local approximation errors and constructs multiresolution tools with convergence guarantees.

Findings

Refinement generates triangles with optimal aspect ratios.

Algorithm converges in the Lp norm for various approximations.

Numerical tests validate the effectiveness of the adaptive triangulation.

Abstract

A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions).