Parameter choice strategies for least-squares approximation of noisy smooth functions on the sphere

TL;DR

This paper develops and analyzes a regularized least-squares method for reconstructing smooth functions on the sphere from noisy data, providing error bounds, parameter selection strategies, and numerical validation.

Contribution

It introduces a novel parameter choice strategy for regularized least-squares approximation of noisy spherical data, with theoretical error bounds and practical algorithms.

Findings

Derived reconstruction error bounds depending on regularization parameters.

Proposed a priori and a posteriori parameter selection strategies.

Numerical experiments confirm the theoretical error estimates.

Abstract

We consider a polynomial reconstruction of smooth functions from their noisy values at discrete nodes on the unit sphere by a variant of the regularized least-squares method of An et al., SIAM J. Numer. Anal. 50 (2012), 1513--1534. As nodes we use the points of a positive-weight cubature formula that is exact for all spherical polynomials of degree up to $2M$, where $M$ is the degree of the reconstructing polynomial. We first obtain a reconstruction error bound in terms of the regularization parameter and the penalization parameters in the regularization operator. Then we discuss a priori and a posteriori strategies for choosing these parameters. Finally, we give numerical examples illustrating the theoretical results.