Finite dimensional models for random microstructures

Author:
M. Grigoriu

Journal:
Theor. Probability and Math. Statist. **106** (2022), 121-142

MSC (2020):
Primary 54C40, 14E20; Secondary 46E25, 20C20

DOI:
https://doi.org/10.1090/tpms/1168

Published electronically:
May 16, 2022

MathSciNet review:
4438447

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Additional Information

Abstract: 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.

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Additional Information

**M. Grigoriu**

Affiliation:
Department of Civil Engineering and Applied Mathematics, Cornell University, Ithaca, New York 18453

Email:
mdg12@cornell.edu

Keywords:
Material microstructures,
random fields,
space of continuous functions,
stochastic equations,
weak/almost sure convergence

Received by editor(s):
July 19, 2021

Accepted for publication:
September 30, 2021

Published electronically:
May 16, 2022

Additional Notes:
The work reported in this paper has been partially supported by the National Science Foundation under the grant CMMI-2013697. This support is gratefully acknowledged.

Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv