Comparing Deterministic and Statistical Optimization Techniques for the Shape Parameter Selection in RBF Interpolation

In this paper, we compare two novel approaches for effectively determining the optimal value of the shape parameter in Radial Basis Function (RBF) interpolation, a crucial factor for numerical method accuracy. We analyze the results of applying the deterministic Leave-One-Out Cross Validation (LOOCV) method in combination with Lipschitz Global Optimization with Pessimistic Improvement (GOPI) and Optimistic Improvement (GOOI), contrasting them with the statistical Bayesian Optimization (BO). Both techniques yield similar validation errors, underlining their effectiveness in shape parameter search. However, the deciding factor in technique selection lies in computational time, which is contingent upon the cardinality of the interpolation set.