Moving least squares via orthogonal polynomials

TL;DR

This paper introduces a robust and efficient moving least squares method using orthogonal polynomials, improving stability and accuracy in derivative estimation on discrete points in multiple dimensions.

Contribution

It presents a novel approach combining orthogonal polynomials with moving least squares to enhance stability and accuracy, avoiding singularities caused by node configurations.

Findings

Method is robust, accurate, and convergent.

Effective in estimating derivatives on random point distributions.

Improves stability over traditional moving least squares methods.

Abstract

A method for moving least squares interpolation and differentiation is presented in the framework of orthogonal polynomials on discrete points. This yields a robust and efficient method which can avoid singularities and breakdowns in the moving least squares method caused by particular configurations of nodes in the system. The method is tested by applying it to the estimation of first and second derivatives of test functions on random point distributions in two and three dimensions and by examining in detail the evaluation of second derivatives on one selected configuration. The accuracy and convergence of the method are examined with respect to length scale (point separation) and the number of points used. The method is found to be robust, accurate and convergent.