Numerical Considerations for the Construction of Karhunen-Loève Expansions

Abstract

This report examines numerical aspects of constructing Karhunen-Loève 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.

Figures (31)

• Figure 1: Eigenvalues corresponding to a 1D KL expansion with an exponential covariance kernel and a domain size $L=1$. The two dashed lines correspond to asymptotic trends evaluated via Eq. \ref{['eq:e1Dasymp']} for ${\ell_c}=0.02$ and ${\ell_c}=0.5$, respectively.
• Figure 2: Eigenvectors $f_1$, $f_5$, and $f_{15}$ corresponding to a 1D KL expansion with an exponential covariance kernel and a domain size $L=1$.
• Figure 3: Eigenvalues of a 1D KLE with a squared-exponential covariance kernel and $x\sim\mathcal{N}(0,\sigma_x)$ with $\sigma_x=1$. Note that correlation lengths are selected as ${\ell_c}=\{\sigma_x/4,\sigma_x,4\sigma_x\}$. The right frame shows a detail of the eigenvalue spectra up to the 5-th mode.
• Figure 4: Select eigenfunctions of a 1D KLE with a squared-exponential covariance kernel and a setup similar to Fig. \ref{['fig:sqexp1D_evals']}, ${\ell_c}=\{\sigma_x/4, \sigma_x, 4\sigma_x\}$ from left to right. Note the differences between these plots for the horizontal and the vertical scales.
• Figure 5: Numerical eigenvalues (solid lines, ${\mathrm{N}_{\mathrm{x}}} \in \{32,128,512\}$) versus analytical solution (dashed line) for each correlation length.