Multivariate Polynomial Regression of Euclidean Degree Extends the Stability for Fast Approximations of Trefethen Functions

TL;DR

This paper introduces a multivariate polynomial regression method based on Euclidean degree that enhances stability in fast approximations of Trefethen functions, supported by an adaptive domain decomposition technique.

Contribution

It proposes a novel regression scheme using Euclidean degree and an adaptive domain decomposition to improve stability in multivariate polynomial approximation.

Findings

Euclidean degree resists instability better than total or maximum degree.

The method extends stability for analytic Trefethen functions.

Adaptive domain decomposition further enhances approximation stability.

Abstract

We address classic multivariate polynomial regression tasks from a novel perspective resting on the notion of general polynomial $l_p$-degree, with total, Euclidean, and maximum degree being the centre of considerations. While ensuring stability is a theoretically known and empirically observable limitation of any computational scheme seeking for fast function approximation, we show that choosing Euclidean degree resists the instability phenomenon best. Especially, for a class of analytic functions, we termed Trefethen functions, we extend recent argumentations that suggest this result to be genuine. We complement the novel regression scheme, presented herein, by an adaptive domain decomposition approach that extends the stability for fast function approximation even further.