Stability Results for Scattered Data Interpolation by Trigonometric Polynomials

TL;DR

This paper introduces a fast, reliable algorithm for interpolating scattered data on the torus using multivariate trigonometric polynomials, achieving near-linear complexity with numerical validation.

Contribution

It presents a novel conjugate gradient-based algorithm combined with FFTs for efficient scattered data interpolation on the torus, with complexity of order O(M log M).

Findings

Algorithm achieves O(M log M) complexity.

Numerical examples demonstrate high efficiency.

Applicable to multivariate data on the torus.

Abstract

A fast and reliable algorithm for the optimal interpolation of scattered data on the torus by multivariate trigonometric polynomials is presented. The algorithm is based on a variant of the conjugate gradient method in combination with the fast Fourier transforms for nonequispaced nodes. The main result is that under mild assumptions the total complexity for solving the interpolation problem at M arbitrary nodes is of order O(M logM). This result is obtained by the use of localised trigonometric kernels where the localisation is chosen in accordance to the spatial dimension d. Numerical examples show the efficiency of the new algorithm.