General adaptive rational interpolation with maximum order close to discontinuities

TL;DR

This paper introduces a generalized adaptive rational interpolation method that achieves near-optimal approximation order close to discontinuities, simplifying weight construction compared to WENO and demonstrating improved performance through experiments.

Contribution

It provides explicit formulas for weights in adaptive rational interpolation for any order, enhancing simplicity and effectiveness near discontinuities.

Findings

Achieves maximum order close to discontinuities

Simplifies weight construction compared to WENO

Demonstrates improved approximation through experiments

Abstract

Adaptive rational interpolation has been designed in the context of image processing as a new nonlinear technique that avoids the Gibbs phenomenon when we approximate a discontinuous function. In this work, we present a generalization to the method giving explicit expressions for all the weights for any order of the algorithm. It has a similar behavior to weighted essentially non oscillatory (WENO) technique, however because of the design of the weights in this case is more simple, we propose a new way to construct them obtaining the maximum order near the discontinuities. Some experiments are performed to demonstrate our results and to compare them with standard methods.