Finite dimensional models for random microstructures

Finite dimensional (FD) models, i.e., deterministic functions of space depending on finite sets of random variables, are used extensively in applications to generate samples of random fields $Z(x)$ and construct approximations of solutions $U(x)$ of ordinary or partial differential equations whose random coefficients depend on $Z(x)$. FD models of $Z(x)$ and $U(x)$ constitute surrogates of these random fields which target various properties, e.g., mean/correlation functions or sample properties. We establish conditions under which samples of FD models can be used as substitutes for samples of $Z(x)$ and $U(x)$ for two types of random fields $Z(x)$ and a simple stochastic equation. Some of these conditions are illustrated by numerical examples.