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1. 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. 

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2. 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. 

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3. Polynomial Chaos-Based Controller Design for Uncertain Linear Systems With State and Control Constraints

   https://doi.org/10.1115/1.4038800  

   Nandi, Souransu ; Migeon, Victor ; Singh, Tarunraj ; Singla, Puneet ( July 2018 , Journal of Dynamic Systems, Measurement, and Control)

   For linear dynamic systems with uncertain parameters, design of controllers which drive a system from an initial condition to a desired final state, limited by state constraints during the transition is a nontrivial problem. This paper presents a methodology to design a state constrained controller, which is robust to time invariant uncertain variables. Polynomial chaos (PC) expansion, a spectral expansion, is used to parameterize the uncertain variables permitting the evolution of the uncertain states to be written as a polynomial function of the uncertain variables. The coefficients of the truncated PC expansion are determined using the Galerkin projection resulting in a set of deterministic equations. A transformation of PC polynomial space to the Bernstein polynomial space permits determination of bounds on the evolving states of interest. Linear programming (LP) is then used on the deterministic set of equations with constraints on the bounds of the states to determine the controller. Numerical examples are used to illustrate the benefit of the proposed technique for the design of a rest-to-rest controller subject to deformation constraints and which are robust to uncertainties in the stiffness coefficient for the benchmark spring-mass-damper system. 

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4. Stochastic Sensitivities across Scales and Physics

   Wang, Z. ; Ghanem, R. ( July 2019 , EMI 2019)

   The polynomial chaos expansions (PCE) provide stochastic representations of quantities of interest (QoI) in terms of a vector of standardized random variables that represent all uncertainties influencing the QoI. These uncertainties could reflect statistical scatter in estimated probabilistic model (of which the mean, variance, or PCE coefficients are but examples), or errors in the underlying functional model between input and output (e.g. physics models). In this paper, we show how PCE permit the evaluation of sensitivities with respect to all these uncertainties, and provide a rational paradigm for resource allocation aimed at model validation. We will demonstrate the methodologies on examples drawn across science and engineering. 

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5. Structure exploiting methods for fast uncertainty quantification in multiphase flow through heterogeneous media

   https://doi.org/s10596-021-10085-8  

   Cleaves, Helen ; Alexanderian, Alen ; Saad, Bilal ( December 2021 , Computational geosciences)

   We present a computational framework for dimension reduction and surrogate modeling to accelerate uncertainty quantification in computationally intensive models with high-dimensional inputs and function-valued outputs. Our driving application is multiphase flow in saturated-unsaturated porous media in the context of radioactive waste storage. For fast input dimension reduction, we utilize an approximate global sensitivity measure, for function-valued outputs, motivated by ideas from the active subspace methods. The proposed approach does not require expensive gradient computations. We generate an efficient surrogate model by combining a truncated Karhunen-Loeve (KL) expansion of the output with polynomial chaos expansions, for the output KL modes, constructed in the reduced parameter space. We demonstrate the effectiveness of the proposed surrogate modeling approach with a comprehensive set of numerical experiments, where we consider a number of function-valued (temporally or spatially distributed) QoIs. 

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