Constructing Approximations to Bivariate Piecewise-Smooth Functions

TL;DR

This paper presents a method to approximate bivariate piecewise-smooth functions, including those with discontinuities, using non-linear operations on smooth splines, applicable to noisy and non-uniform data.

Contribution

It introduces a novel approximation approach for piecewise-smooth functions with discontinuities, supported by a basic theorem for functions with jump discontinuities.

Findings

Effective approximation of functions with jump discontinuities.

Applicable to noisy and non-uniform data sets.

Provides a theoretical foundation for the approximation method.

Abstract

This paper demonstrates that the space of piecewise smooth functions can be well approximated by the space of functions defined by a set of simple (non-linear) operations on smooth uniform splines. The examples include bivariate functions with jump discontinuities or normal discontinuities across curves, and even across more involved geometries such as a 3-corner. The given data may be uniform or non-uniform, and noisy, and the approximation procedure involves non-linear least-squares minimization. Also included is a basic approximation theorem for functions with jump discontinuity across a smooth curve.