GP-Recipe: Gaussian Process approximation to linear operations in numerical methods

Abstract

We introduce new Gaussian Process (GP) high-order approximations to linear operations that are frequently used in various numerical methods. Our method employs the kernel-based GP regression modeling, a non-parametric Bayesian approach to regression that operates on the probability distribution over all admissible functions that fit observed data. We begin in the first part with discrete data approximations to various linear operators applied to smooth data using the most popular squared exponential kernel function. In the second part, we discuss data interpolation across discontinuities with sharp gradients, for which we introduce a new GP kernel that fits discontinuous data without oscillations. The current study extends our previous GP work on polynomial-free shock-capturing methods in finite difference and finite volume methods to a suite of linear operator approximations on smooth data. The formulations introduced in this paper can be readily adopted in daily practices in numerical methods, including numerical approximations of finite differences, quadrature rules, interpolations, and reconstructions, which are most frequently used in numerical modeling in modern science and engineering applications. In the test problems, we demonstrate that the GP approximated solutions feature improved solution accuracy compared to the conventional finite-difference counterparts.