A two-stage approximation strategy for piecewise smooth functions in two and three dimensions

Abstract

Abstract Given values of a piecewise smooth function $f$ on a square grid within a domain $[0,1]^d$, $d=2,3$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. The insight used here is that the behavior near the boundaries, or near a singularity curve, is fully characterized and identified by the values of certain differences of the data across the boundary and across the singularity curve. We refer to these values as the signature of $f$. In this paper, we aim at using these values in order to define the approximation. That is, we look for an approximation whose signature is matched to the signature of $f$. Given function data on a grid, assuming the function is piecewise smooth, first, the singularity structure of the function is identified. For example, in the two-dimensional case, we find an approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously, we find the approximations to the different segments of $f$. A system of equations derived from the principle of matching the signature of the approximation and the function with respect to the given grid defines a first stage approximation. A second stage improved approximation is constructed using a global approximation to the error obtained in the first stage approximation.