Parameter choice strategies for least-squares approximation of noisy smooth functions on the sphere
This work addresses the practical challenge of selecting regularization parameters for spherical function approximation, offering both a priori and a posteriori strategies for users in geophysics or related fields.
The paper develops parameter choice strategies for regularized least-squares polynomial reconstruction of noisy smooth functions on the sphere, providing error bounds and demonstrating effectiveness through numerical examples.
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.