Numerical solutions of stochastic problems involving random processes π(π‘), which constitutes infinite families
of random variables, require to represent these processes by finite dimensional (FD) models ππ (π‘), i.e.,
deterministic functions of time depending on finite numbers π of random variables. Most available FD models
match the mean, correlation, and other global properties of π(π‘). They provide useful information to a broad
range of problems, but cannot be used to estimate extremes or other sample properties of π(π‘). We develop FD
models ππ (π‘) for processes π(π‘) with continuous samples and establish conditions under which these models
converge weakly to π(π‘) in the space of continuous functions as π β β. These theoretical results are illustrated
by numerical examples which show that, under the conditions established in this study, samples and extremes
of π(π‘) can be approximated by samples and extremes of ππ (π‘) and that the discrepancy between samples and
extremes of these processes decreases with π.
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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.
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 Award ID(s):
 2013697
 NSFPAR ID:
 10338007
 Date Published:
 Journal Name:
 Theory of probability and mathematical statistics
 Volume:
 106
 ISSN:
 15477363
 Page Range / eLocation ID:
 121142
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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