More Like this

1. Finite dimensional models for extremes of Gaussian and non-Gaussian processes

   H, Xu; M. Grigoriu (January 2022, 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
2. FINITE DIMENSIONAL MODELS FOR RANDOM MICROSTRUCTURES

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

   M. Grigoriu (May 2022, 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
3. Finite dimensional surrogates for extreme events

   Hui Xu, Mircea D. (July 2023, 14th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP14 Dublin, Ireland, July 9-13, 2023)

   Numerical solutions of stochastic problems require the representation of random functions in their definitions by finite dimensional (FD) models, i.e., deterministic functions of time and finite sets of random variables. It is common to represent the coefficients of these FD surrogates by polynomial chaos (PC) models. We propose a novel model, referred to as the polynomial chaos translation (PCT) model, which matches exactly the marginal distributions of the FD coefficients and approximately their dependence. PC- and PCT- based FD models are constructed for a set of test cases and a wind pressure time series recorded at the boundary layer wind tunnel facility at the University of Florida. The PCT-based models capture the joint distributions of the FD coefficients and the extremes of target times series accurately while PC-based FD models do not have this capability. 

   more » « less
4. Are seismic fragility curves fragile?

   https://doi.org/doi.org/10.1016/j.probengmech.2020.103115  

   Grigoriu, M. and (January 2021, Probabilistic engineering mechanics)

   null (Ed.)

   A hypothetical seismic site is constructed for which the probability law of the seismic ground acceleration process 𝑋(𝑡) is specified. Since the seismic hazard is known, the performance of the incremental dynamic analysis- (IDA) and multiple stripe analysis- (MSA) based fragilities, which are used extensively in Earthquake Engineering, can be assessed without ambiguity. It is shown that the IDA- and MSA-based fragilities are unsatisfactory for moderate and large seismic events, are sensitive to the particular parameters used for their construction, and may or may not improve with the sample size. Also, the usefulness of the optimization algorithms for selecting ground motions records is questionable. 

   more » « less
5. Finite dimensional models for random functions

   https://doi.org/https://doi.org/10.1016/j.jcp.2018.09.029  

   Grigoriu, Mircea (September 2018, Journal of computational physics)

   Truncated Karhunen–Loève (KL) representations are used to construct finite dimensional (FD) models for non-Gaussian functions with finite variances. The second moment specification of the random coefficients of these representations are enhanced to full probabilistic characterization by using translation, polynomial chaos, and translated polynomial chaos models, referred to as T, PC, and PCT models. Following theoretical considerations on KL representations and T, PC, and PCT models, three numerical examples are presented to illustrate the implementation and performance of these models. The PCT models inherit the desirable features of both T and PC models. It approximates accurately all quantities of interest considered in these examples. 

   more » « less