Comparative Study On Multiple Width-Defining Methods For Radial Basis Functions

Abstract

In structural problems, numerical methods such as the Finite Element Method are often used due to the scarce and limited applicability of analytical methods. In these cases, the design optimization may become computationally costly and the time consumed starts to be a hindrance. To overcome this problem, a significant effort has been made by researchers to understand and improve the so-called surrogate models. Surrogate models provide computational efficiency by using a few samples from the true function to build an approximated response surface to predict points in the design space not yet evaluated during the optimization process. This approximated surface may also be improved at each generation with the addition of new samples in regions of interest on a methodology known as Sequential Approximate Optimization (SAO). In this context, the Radial Basis Functions (RBF) are a powerful and robust surrogate model while keeping implementation simple. The Gaussian function is often chosen as the basis function despite uncertainty on the definition of one of its main parameters: the kernel width ( ). This paper performed a comparative study on different methods to estimate the width parameter using two types of solutions: closed-form expressions proposed by different researchers in the last few years and direct search methods. The efficiency of each of these approaches is assessed using metrics such as the number of high fidelity model evaluations and the error at the end of each optimization.