MATLAB Programming for Kernel-Based Methods

Abstract

This paper presents an implementation of kernel--based methods using MATLAB, a powerful tool for numerical computation and data analysis. Kernel methods are pivotal in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations (PDEs), neural networks, and Machine Learning. Due to the connection of kernels to Hilbert spaces of functions, kernel-based methods often have optimality properties. The package allows to switch between different kernels and different scales without changing programs, just by changing parameters. In addition, it enables applications to partial differential equations to handle derivatives of kernels efficiently, without re-programming the derivatives if the kernel is changed. Special emphasis is placed on practical implementation, showcasing MATLAB code snippets and functions that facilitate the application of these algorithms to function approximation and numerical solution of PDEs.