Residual-Error Cross-Validation method for selecting a suitable shape parameter for RBF interpolation

Abstract

Radial Basis Functions (RBFs) have shown the potential to be a universal mesh-free method for solving interpolation and differential equations with highly accurate results. However, the trade-off principle states that while deciding the shape parameter’s value for an RBF such as Multiquadric (MQ) or Gaussian (GA), a compromise must be made between achieving accuracy and stability because of the resultant ill-conditioned matrix.This study focus on the behaviors between the maximum and residual errors for the RBF interpolation. Based on the error behaviors, we propose a new approach, Residual-Error Cross Validation (RECV), to quickly select a suitable c value for an interpolant using an RBF containing a shape parameter. The numerical results showed that an RBF interpolant could yield high accuracy with the RECV c and a sufficiently small fill distance. Combining the RECV method and LOOCV method, we can easily avoid the local optimum issue when applying an optimization algorithm.