Abstract
We consider discrete penalized least-squares approximation on the unit sphere S-d, where the minimizer is sought in the space P-L(S-d) of spherical polynomials of degree <= L. The penalized least-squares functional is the sum of two terms both involving the weights and the nodes of a positive weight quadrature rule with polynomial degree of exactness at least 2L. The first one is a discrete least-squares functional measuring the squared weighted l(2) discrepancy between the noisy data and the approximation. The second term is the product of a regularization parameter lambda >= 0 times a penalization term that can be interpreted as an approximation of a squared semi-norm in the Sobolev (Hilbert) space H-s(S-d). The approximation can be computed directly via a summation process and does not require the solving of a linear system. For lambda = 0 (which is only appropriate if there is almost no noise), our approximation becomes a case of hyperinterpolation. As lambda > 0 increases, less weight is given to data fitting and more weight is given to keeping the approximation smooth. We derive L-2(S-d) error estimates for the approximation of functions from the Sobolev Hilbert space H-s(S-d), where s > d/2, from noisy data for the regularization parameter lambda chosen (i) as lambda = 0 (hyperinterpolation), (ii) for general lambda > 0, and (iii) with Morozov's discrepancy principle (an a posteriori parameter choice strategy). The L-2(S-d) error estimates in case (i) and (iii) are in a sense order-optimal. Numerical experiments explore the approximation method and illustrate the theoretical results. (C)& nbsp;2022 Elsevier B.V. All rights reserved.
NSTL
Elsevier