Applications of Mathematics, Vol. 70, No. 6, pp. 941-991, 2025
Efficient Karhunen-Loève expansions via Legendre-Galerkin discretization and tensor structure
Michal Béreš
Received July 13, 2025. Published online December 5, 2025.
Abstract: We develop an efficient framework for Karhunen-Loève expansions of isotropic Gaussian random fields on hyper-rectangular domains. The approach approximates the covariance 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.
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Affiliations: Michal Béreš, Institute of Geonics of the CAS, Studentská 1768/9, 708 00 Ostrava, Czech Republic; Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava, Czech Republic, e-mail: michal.beres@vsb.cz
