Approximation of Karhunen-Loève Decomposition of Isotropic Gaussian Random Fields Using Orthogonal Polynomials and Gaussian Quadratures

Abstract

The goal of this thesis is to define and implement an efficient method to approximate the KarhunenLoève decomposition. The thesis first provides a review of the theory related to orthogonal polynomials, weighted inner products, and Gaussian quadrature rules. Key ideas are explained in a clear and accessible way. These include the three-term recurrence relation for generating orthogonal polynomials and the Golub-Welsch algorithm for computing quadrature nodes and weights from the associated Jacobi matrix. Fundamental concepts from probability theory are introduced, focusing on covariance functions and Gaussian random fields, to establish the necessary background for the primary application explored in this work. The core application demonstrates the use of the Galerkin method, together with orthogonal polynomial bases and Gaussian quadrature rules, to achieve an efficient numerical approximation of the Karhunen-Loève (KL) expansion for stochastic processes. All methods discussed are implemented in Python, and the corresponding code is available at the specified repository https://github.com/SED0280/SED0280ThesisCode.