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
Full-text PDF
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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.
References
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- H. Cramér and M. R. Leadbetter, Stationary and related stochastic processes, reprint of the 1967 original, Dover Publications, Inc., Mineola, NY, 2004. MR 2108670
- W. B. Davenport. Probability and random processes, McGraw-Hill Book Company, New York, 1970.
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- E. J. Hannan, Multiple time series, John Wiley & Sons, Inc., New York, 1970. MR 0279952
- D. B. Hernández, Lectures on probability and second order random fields, Series on Advances in Mathematics for Applied Sciences, 30, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1412573
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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