Solving interpolation problems on surfaces stochastically and greedily

Choosing suitable shape parameters in the kernel-based interpolation problems is an open question, whose solutions can guarantee accuracy and numerical stability. In this paper, we study various ways to select Kernel's shape parameters for interpolation problems on surfaces. In particular, we use exact and stochastically approximated cross validation approaches to select the shape parameters. When we solve the resultant matrix systems, we also deploy a greedy trial subspace selection algorithm to improve robustness. Numerical experiments are inserted along our discussion to demonstrate the feasibility and robustness of our proposed methods.