A Primer on the Karhunen-Loève Expansion

This article provides a primer on the spectral representation of random fields via the Karhunen-Loève 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.