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Conditions under which samples of continuous stochastic processes 𝑋(𝑡) on bounded time intervals [0, 𝜏] can be represented by samples of finite dimensional (FD) processes 𝑋𝑑 (𝑡) are augmented such that samples of Slepian models 𝑆𝑑,𝑎(𝑡) of 𝑋𝑑 (𝑡) can be used as surrogates for samples of Slepian models 𝑆𝑎(𝑡) of 𝑋(𝑡). FD processes are deterministic functions of time and 𝑑 < ∞ random variables. The discrepancy between target and FD samples is quantified by the metric of the space 𝐶[0, 𝜏] of continuous functions. The numerical illustrations, which include Gaussian/nonGaussian FD processes and solutions of linear/nonlinear random vibration problems, are consistent with the theoretical findings in the paper.more » « less

M. Grigoriu ( , Theory of probability and mathematical statistics)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.more » « less

H, Xu ; M. Grigoriu ( , Probabilistic engineering mechanics)Numerical solutions of stochastic problems involving random processes 𝑋(𝑡), which constitutes infinite families of random variables, require to represent these processes by finite dimensional (FD) models 𝑋𝑑 (𝑡), i.e., deterministic functions of time depending on finite numbers 𝑑 of random variables. Most available FD models match the mean, correlation, and other global properties of 𝑋(𝑡). They provide useful information to a broad range of problems, but cannot be used to estimate extremes or other sample properties of 𝑋(𝑡). We develop FD models 𝑋𝑑 (𝑡) for processes 𝑋(𝑡) with continuous samples and establish conditions under which these models converge weakly to 𝑋(𝑡) in the space of continuous functions as 𝑑 → ∞. These theoretical results are illustrated by numerical examples which show that, under the conditions established in this study, samples and extremes of 𝑋(𝑡) can be approximated by samples and extremes of 𝑋𝑑 (𝑡) and that the discrepancy between samples and extremes of these processes decreases with 𝑑.more » « less