# On the Numerical Approximation of the Karhunen-Lo\`{e}ve Expansion for Lognormal Random Fields

## TL;DR

This paper provides a detailed error analysis for the numerical approximation of the Karhunen-Loève expansion applied to lognormal random fields, crucial for PDEs with random coefficients, balancing truncation, sampling, and discretization errors.

## Contribution

It introduces new error estimates that optimize the approximation of the KL expansion for lognormal fields, integrating truncation, sampling, and numerical errors.

## Key findings

- Derived error bounds for KL expansion approximation
- Quantified the impact of truncation, sampling, and discretization errors
- Validated theoretical results with numerical experiments in multiple stochastic dimensions

## Abstract

The Karhunen-Lo\`{e}ve (KL) expansion is a popular method for approximating random fields by transforming an infinite-dimensional stochastic domain into a finite-dimensional parameter space. Its numerical approximation is of central importance to the study of PDEs with random coefficients. In this work, we analyze the approximation error of the Karhunen-Lo\`eve expansion for lognormal random fields. We derive error estimates that allow the optimal balancing of the truncation error of the expansion, the Quasi Monte-Carlo error for sampling in the stochastic domain and the numerical approximation error in the physical domain. The estimate is given in the number $M$ of terms maintained in the KL expansion, in the number of sampling points $N$, and in the discretization mesh size $h$ in the physical domain employed in the numerical solution of the eigenvalue problems during the expansion. The result is used to quantify the error in PDEs with random coefficients. We complete the theoretical analysis with numerical experiments in one and multiple stochastic dimensions.