Parameter Choice Strategies for Least-squares Approximation of Noisy Smooth Functions on the Sphere

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), pp. 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.