The Karhunen-Lo\`eve series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise $L^2$-orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a $2d$ dimensional domain, where $d$, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix-vector product for iterative eigenvalue solvers. Two higher-order three-dimensional benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.

翻译：Karhunen- Lo ⁇ ⁇ éve 序列扩展( KLE) 解析一个随机随机变数,将一个随机过程分解成一系列无限的对称、不相干、不相干、随机随机的随机变数,对调值为$L2$2美元或正方方形函数。对于无限序列的任何给定的脱轨顺序,该基础是最佳的,因为平均正方形总错误最小化。正方形基函数的计算复杂性,以及直接解决方案技术的记忆要求,随着多元度、元素数量和自由度的增加而变得难以计算。首先,加勒金离异化需要数字集成一个2美元正向值的超正向值域域域域域域,其中美元表示空间维度。第二,离异的弱形主要系统矩阵密度,因此,经典定型元元元元元元元组和组程序以及直接解决方案的记忆要求变得难以快速计算。 这项工作的目标是大幅减少与双向值的正值直径直径直径直值域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域域的计算法(我们目前以一个基于基基基基基基基基基基基基基基基基基基基基基的基的基基基基的基的基的基的基的基的基度和基度和基底基底基底基底基度办法) 。