Numerical Considerations for the Construction of Karhunen-Lo\`{e}ve Expansions

TL;DR

This paper investigates numerical methods for constructing Karhunen-Loève expansions of stochastic processes, analyzing discretization, eigenvalue estimation, and convergence in various dimensions and domain topologies.

Contribution

It establishes the algebraic equivalence between Fredholm integral eigenproblems and SVD of sample matrices, providing benchmarks and insights for numerical KLE construction.

Findings

Eigenvalue estimates converge with increasing sample size.

Numerical solutions match analytical benchmarks for specific kernels.

Discretization and domain topology significantly affect KLE accuracy.

Abstract

This report examines numerical aspects of constructing Karhunen-Lo\`{e}ve expansions (KLEs) for second-order stochastic processes. The KLE relies on the spectral decomposition of the covariance operator via the Fredholm integral equation of the second kind, which is then discretized on a computational grid, leading to an eigendecomposition task. We derive the algebraic equivalence between this Fredholm-based eigensolution and the singular value decomposition of the weight-scaled sample matrix, yielding consistent solutions for both model-based and data-driven KLE construction. Analytical eigensolutions for exponential and squared-exponential covariance kernels serve as reference benchmarks to assess numerical consistency and accuracy in 1D settings. The convergence of SVD-based eigenvalue estimates and of the empirical distributions of the KL coefficients to their theoretical $\mathcal{N}(0,1)$ target are characterized as a function of sample count. Higher-dimensional configurations include a two-dimensional irregular domain discretized by unstructured triangular meshes with two refinement levels, and a three-dimensional toroidal domain whose non-simply-connected topology motivates a comparison between Euclidean and shortest interior path distances between the grid points. The numerical results highlight the interplay between the discretization strategy, quadrature rule, and sample count, and their impact on the KLE results.