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

TL;DR

This paper derives 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 from finite measurements of finite measurements of the covariance operator.

Abstract

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 $\varepsilon$.

Theorems & Definitions (20)

• Theorem 2.1
• Lemma 2.1: Regularity of the eigenfunctions $\{\phi_{\ell}\}_{\ell=1}^{\infty}$
• Lemma 3.1: Semi-discrete spatial approximation error estimate
• proof
• Lemma 3.2
• Proposition 3.1: Conforming Galerkin approximation estimate
• proof
• Theorem 3.1
• proof
• Proposition 3.2