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Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



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
Published electronically: May 16, 2022
MathSciNet review: 4438447
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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

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