发表机构
Department of Stochastic Simulation and Safety Research for Hydrosystems, University of Stuttgart（斯图加特大学水系统随机模拟与安全研究系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出利用FFT加速的Toeplitz算子实现快速Karhunen-Loève展开，显著降低协方差矩阵特征求解的计算与存储开销，适用于大规模高斯随机场模拟。
AI 中文摘要
高斯随机场是随机偏微分方程、不确定性量化和地质统计模拟等领域中广泛使用的工具。获取高斯随机场的一种方法是使用截断的Karhunen-Loève展开（KLE）。计算该展开需要求解一个$N \times N$协方差矩阵的主特征对，其中$N$是网格单元的总数。通常使用Krylov特征求解器计算这些特征对，该求解器仅在矩阵-向量乘积中依赖协方差算子。若矩阵密集存储，则需$\mathcal{O}(N^{2})$内存，每次乘积需$\mathcal{O}(N^{2})$时间。对于等距网格上的平稳核，协方差矩阵变为（块）Toeplitz矩阵，利用基于FFT的循环嵌入可在$\mathcal{O}(N\log N)$时间内完成乘积，且无需组装密集矩阵。在匹配的单线程比较中，特征求解的中位加速比从$N = 4096$时的$18 \times$增长到$N = 2^{15}$时的$183 \times$。这使得计算原本在标准公式下不可行的离散化场成为可能。我们证明相同的构造可推广到非可分离核以及$d$维情形，使用块Toeplitz矩阵。我们将其扩展到任意域上分段常数场，这些域是张量网格的子集。计算节省随问题规模增加而增长，存储量从$\mathcal{O}(N^{2})$降至$\mathcal{O}(2^{d}N)$。
英文摘要
Gaussian random fields are a versatile tool used in the fields of stochastic PDEs, uncertainty quantification, and geostatistical simulation. One way to obtain them is to use a truncated Karhunen-Loève expansion (KLE). Computing the expansion requires the leading eigenpairs of an $N \times N$ covariance matrix, where $N$ is the total number of grid cells. These are usually computed with a Krylov eigensolver, which relies on the covariance operator only within matrix-vector products. Stored densely, the matrix takes $\mathcal{O}(N^{2})$ memory and each product $\mathcal{O}(N^{2})$ time. For a stationary kernel on an equispaced grid, the covariance matrix becomes (block-) Toeplitz and the product evaluates in $\mathcal{O}(N\log N)$ time using FFT-based circulant embedding, without the need to assemble the dense matrix. In a matched single-threaded comparison, the median speedup of the eigensolve grows from $18 \times$ at $N = 4096$ to $183 \times$ at $N = 2^{15}$. This makes it possible to compute discretized fields that would otherwise be infeasible to compute in the standard formulation. We show that the same construction carries over to non-separable kernels as well as $d$ dimensions, using block-Toeplitz matrices. We extend it to piecewise-constant fields on arbitrary domains given as subsets of a tensor grid. Computational savings grow with problem size, and storage drops from $\mathcal{O}(N^{2})$ to $\mathcal{O}(2^{d}N)$.
