Smoothing approximation of noisy scattered data on spheres

Abstract:

We consider the hybrid approach to scattered data approximation on the sphere S2 with radial basis functions plus a polynomial. If the given data of the unknown function contain noise, then we need to use smoothing approximation, rather than interpolation, to approximate the unknown function. The smoothing approximation depends on the smoothing parameter which controls the balance between fitting the data and the smoothness of the approximant, and the choice of the smoothing parameter depending on the level of noise is a crucial question. In this talk we present L2(S2) and uniform error estimates for smoothing approximation in the hybrid approach, where the smoothing parameter is chosen with Morozov’s discrepancy principle, an a-posteriori parameter choice strategy. These results are generalizations of corresponding error estimates for smoothing approximation with thin plate splines on a bounded domain in the plane due to Wei, Hon, and Wang (Inverse Problems, 21 (2005), 657–672). This is joint work in progress with Ian Sloan and Rob Womersley.