On the Numerical Approximation of the Karhunen-Lo\`{e}ve Expansion for Random Fields with Random Discrete Data

TL;DR

This paper develops a method to approximate the covariance operator of random fields from discretized samples, providing explicit error bounds that account for sampling, discretization, and finite-rank approximation errors.

Contribution

It introduces a novel approach using tapering covariance estimators for high-dimensional data, with explicit error estimates and conditions for accuracy.

Findings

Explicit error bounds for covariance approximation

Conditions for controlling approximation errors

Applicability to high-dimensional measurement data

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 $\varepsilon$.