A Primer on the Karhunen-Lo\`eve Expansion

TL;DR

This paper explains the theoretical foundations and computational aspects of the Karhunen-Loève Expansion for representing random fields, bridging theory and practical application.

Contribution

It provides a comprehensive overview connecting operator theory, probability, and computational modeling in the context of KLE.

Findings

Analyzes convergence and optimality of KLE

Highlights computational considerations for KLE implementation

Bridges theoretical and practical aspects of spectral representation

Abstract

This article provides a primer on the spectral representation of random fields via the Karhunen-Lo\`eve Expansion (KLE). The goal is to bridge the gap between the theoretical foundations of the KLE and its application in computational modeling under uncertainty. We detail how tools from operator theory and probability are combined to analyze the convergence and optimality of the KLE. We also emphasize the associated computational and mathematical modeling considerations.