A Unified Framework for Efficient Kernel and Polynomial Interpolation

TL;DR

This paper introduces a unified interpolation framework combining kernels and polynomials, with efficient algorithms for computation, applicable to Euclidean domains and manifolds, outperforming polynomial least squares especially near boundaries.

Contribution

The authors develop a novel unified interpolation scheme that integrates kernels and polynomials, along with specialized linear algebra techniques for efficient computation on complex domains.

Findings

Unified interpolation scheme successfully combines kernels and polynomials.

Efficient linear algebra procedures enable practical implementation.

Outperforms polynomial least squares in boundary-rich settings.

Abstract

We present a unified interpolation scheme that combines compactly-supported positive-definite kernels and multivariate polynomials. This unified framework generalizes interpolation with compactly-supported kernels and also classical polynomial least squares approximation. To facilitate the efficient use of this unified interpolation scheme, we present specialized numerical linear algebra procedures that leverage standard matrix factorizations. These procedures allow for efficient computation and storage of the unified interpolant. We also present a modification to the numerical linear algebra that allows us to generalize the application of the unified framework to target functions on manifolds with and without boundary. Our numerical experiments on both Euclidean domains and manifolds indicate that the unified interpolant is superior to polynomial least squares for the interpolation of target functions in settings with boundaries.