Efficient Karhunen-Loève expansions via Legendre-Galerkin discretization and tensor structure

Abstract

We develop an efficient framework for Karhunen-Loève expansions of isotropic Gaussian random fields on hyper-rectangular domains. The approach approximates the co-variance kernel by a positive mixture of squared-exponentials, fitted via Newton optimization with a theoretically informed initialization; we also provide convergence estimates for this Gaussian-mixture approximation. The resulting separable kernel enables a Legendre-Galerkin discretization with a Kronecker product structure across dimensions, together with submatrices exhibiting even/odd parity. For assembly, we employ a Duffy-type transformation followed by Gaussian quadrature. These structural properties substantially reduce memory usage and arithmetic cost compared with naive formulations. All algorithms and numerical experiments are released in an open-source repository that reproduces every figure and table. For completeness, a concise derivation of the three-term recurrence for Legendre polynomials is included in appendix.