Optimizing the shape parameters of radial basis functions: an application to automobile crashworthiness

Radial basis functions (RBFs) are approximate mathematical models that can mimic the behaviour of rapidly changing and computationally expensive simulations, such as finite element simulations for predicting automobile crash responses. The most popular way of selecting optimal RBF shape parameters is based on minimizing the global cross-validation error (CVE). Solving this optimization problem may lead to the construction of globally accurate RBF models, but the shape parameters are assumed to be constant over the entire design space. On the other hand, having flexible shape parameters that can change over the design space may allow the local behaviour to be captured better, thereby improving the accuracy. Thus, optimizing the RBF shape parameters based on minimization of the pointwise CVE rather than the global CVE is proposed in this paper. Three benchmark mathematical functions followed by an automobile crash problem are used to evaluate the effectiveness of the proposed method. It is found that the RBF models based on the minimum pointwise CVE outperform the RBF models based on the minimum global CVE.