标题： 关于随机离散数据的随机场Karhunen-Loève展开的数值逼近 显示英文标题

标题： On the Numerical Approximation of the Karhunen-Loève Expansion for Random Fields with Random Discrete Data

摘要： 在许多应用中，随机场反映了不确定参数，并且它们的矩通常作为建模过程的一部分，因此是已知的。 然而，存在实际情况下并非如此。 因此，我们不假设知道随机场的矩或展开项，而只是拥有它们的离散样本。 本文的主要贡献在于从这些有限测量中近似真实的协方差算子。 我们推导了显式的误差估计，包括协方差算子的有限秩逼近误差、在随机域中采样的蒙特卡洛型误差以及在物理域中的数值离散化误差。 为此，我们使用了适用于高维应用的现代截断协方差估计器，其中维度由测量过程的分辨率引入。 这使我们能够对三个离散化参数给出充分条件，以保证误差保持在预定精度$\varepsilon$以下。 显示英文摘要

摘要： In many applications, random fields reflect uncertain parameters, and often their moments are part of the modeling process and thus well known. However, there are practical situations where this is simply not the case. Therefore, we do not assume that we know moments or expansion terms of the random fields, but only have discretized samples of them. The main contribution of this paper concerns the approximation of the true covariance operator from these finite measurements. We derive explicit error estimates that include the finite-rank approximation error of the covariance operator, the Monte Carlo-type error for sampling in the stochastic domain, and the numerical discretization error in the physical domain. For this purpose, we use modern tapering covariance estimators adapted to high-dimensional applications, where the dimension is introduced by the resolution of the measurement process. This allows us to give sufficient conditions on the three discretization parameters to guarantee that the error is kept below a prescribed accuracy $\varepsilon$.