A Multiscale RBF Method for Severely Ill-Posed Problems on Spheres

We propose and analyze the support vector approach to approximating the solution of a severely ill-posed problem Au = f on the sphere, in which A is an ill-posed map from the unit sphere to a concentric larger sphere. The Vapnik's epsilon-intensive function is adopted in the regularization technique to reduce the error induced by noisy data. The method is then extended to a multiscale algorithm by varying the support radius of the radial basis functions at each scale. We discuss the convergence of the multiscale support vector approach and provide strategies for choosing both regularization parameters and cut-off parameters at each level. Numerical examples are constructed to verify the efficiency of the multiscale support vector approach.