Radial Basis Functions for Solving PDEs: Advances Over The Last 25 Years

This talk will be given in layman’s terms. Radial Basis Functions (RBFs) were introduced in the 1970s, first as a tool for interpolating scattered 2-D data. Since then, both our knowledge about RBFs and their range of applications have grown tremendously, with the numerical solution of partial differential equations becoming a particularly important one. We will start off by introducing what an RBF is and why they are so special compared to other basis functions. We will then discuss how they can be seen as a generalization of finite difference and pseudospectral methods. Next, we demonstrate how to calculate discrete derivative operators with RBFs. Lastly, with a variety of examples, we will illustrate how RBF methodologies provide a promising innovative approach to numerically solving PDEs to high-order accuracy in arbitrary geometries, demonstrating both performance and scalability on highly parallel computer architectures.