A fast surrogate cross validation algorithm for meshfree RBF collocation approaches

Cross-validation is an important tool in the Radial Basis Function (RBF) collocation setting, especially for the crucial tuning of the shape parameter related to the radial basis function. In this paper, we define a new efficient surrogate cross-validation algorithm, which computes an accurate approximation of the true validation error with much less computational effort with respect to a standard implementation. The proposed scheme is first analyzed and described in detail and then tested in various numerical experiments that confirm its efficiency and effectiveness. •Cross-validation is an important tool for tuning parameters in kernel-based interpolation and collocation frameworks.•The extended Rippa's algorithm is a well-established strategy to reduce the computational cost required by cross-validation in the interpolation setting. However, in literature, the proposed adaptations of this approach to the collocation context consist of empirical adjustments with limited theoretical insights.•We propose and analyze a surrogate cross-validation algorithm that approximates the true validation error, and can be considered as the natural extension of Rippa's scheme into the collocation setting.•The efficiency of the proposed surrogate scheme, as well as its effectiveness when compared to the empirical adaptations proposed in literature, are proved by means of various numerical experiments involving various PDE collocation settings.