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 Karhunen-Loève expansion of random fields using only discretized samples, providing explicit error estimates that combine spatial discretization, sampling, and covariance approximation errors.

Contribution

It introduces a novel approach to approximate the covariance operator of random fields from discrete samples without prior knowledge of moments or expansion terms, with explicit error bounds.

Findings

Explicit error estimates combining multiple discretization errors.

Conditions for achieving prescribed accuracy in approximation.

Effective low-rank covariance approximations for smooth covariance functions.

Abstract

Many physical and mathematical models involve random fields in their input data. Examples are ordinary differential equations, partial differential equations and integro--differential equations with uncertainties in the coefficient functions described by random fields. They also play a dominant role in problems in machine learning. In this article, we do not assume to have knowledge of the moments or expansion terms of the random fields but we instead have only given discretized samples for them. We thus model some measurement process for this discrete information and then approximate the covariance operator of the original random field. Of course, the true covariance operator is of infinite rank and hence we can not assume to get an accurate approximation from a finite number of spatially discretized observations. On the other hand, smoothness of the true (unknown) covariance function results in effective low rank approximations to the true covariance operator. We derive explicit error estimates that involve the finite rank approximation error of the covariance operator, the Monte-Carlo-type errors for sampling in the stochastic domain and the numerical discretization error in the physical domain. This permits to give sufficient conditions on the three discretization parameters to guarantee that an error below a prescribed accuracy $\varepsilon$ is achieved.