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.

翻译：本报告探讨了为二阶随机过程构建Karhunen-Loève展开（KLE）的数值方面问题。KLE依赖于通过第二类Fredholm积分方程对协方差算子进行谱分解，随后在计算网格上离散化处理，最终转化为特征分解任务。我们推导了基于Fredholm的特征解与加权缩放样本矩阵奇异值分解之间的代数等价性，为基于模型和基于数据的KLE构建提供了一致解。指数核与平方指数协方差核的解析特征解作为参考基准，用于评估一维场景下的数值一致性与精度。我们刻画了基于SVD的特征值估计以及KL系数经验分布向其理论$\mathcal{N}(0,1)$目标收敛性随样本数量的变化特征。高维配置包括采用两种细化水平的非结构化三角网格离散化二维不规则区域，以及三维环面区域（其非单连通拓扑结构促使我们比较网格点间的欧几里得距离与最短内部路径距离）。数值结果揭示了离散化策略、求积规则与样本数量之间的相互作用及其对KLE结果的影响。