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

2015/01/09 by Pereverzyev, Sergei. V., Sloan, Ian. H., Tkachenko, Pavlo · 1 citation
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1501.02090

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.

Cited by

• Quasi-Monte Carlo hyperinterpolation

Related

• Parameter choice strategies for regularized least squares approximation of noisy continuous functions on the unit circle
• Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions
• Uniform approximation on the sphere by least squares polynomials
• Truncated Kernel Stochastic Gradient Descent on Spheres
• Polynomial approximation of noisy functions