On applying the maximum volume principle to a basis selection problem in multivariate polynomial interpolation

TL;DR

This paper explores using the maximum volume principle to select polynomial bases in multivariate interpolation, aiming to improve stability and accuracy by bounding the Lebesgue constant through Vandermonde matrix properties.

Contribution

It introduces a novel approach linking maximum volume selection to stability and error bounds in multivariate polynomial interpolation.

Findings

Large Vandermonde matrix volume correlates with interpolation accuracy.

The method effectively bounds the Lebesgue constant, enhancing stability.

Numerical examples confirm practical effectiveness.

Abstract

The maximum volume principle is investigated as a means to solve the following problem: Given a set of arbitrary interpolation nodes, how to choose a set of polynomial basis functions for which the Lagrange interpolation problem is well-defined with reasonable interpolation error? The interpolation error is controlled by the Lebesgue constant of multivariate polynomial interpolation and it is proven that the Lebesgue constant can effectively be bounded by the reciprocals of the volume (i.e., determinant in modulus) and the minimal singular value of the multidimensional Vandermonde matrix associated with the interpolation problem. This suggests that a large volume of the Vandermonde system can be used as an indicator of accuracy and stability of the resulting interpolating polynomial. Numerical examples demonstrate that the approach outlined in this paper works remarkably well in practical computations.