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1. An Efficient Polynomial Chaos Expansion Method for Uncertainty Quantification in Dynamic Systems

   https://doi.org/10.3390/applmech2030026  

   Son, Jeongeun; Du, Yuncheng (September 2021, Applied Mechanics)

   null (Ed.)

   Uncertainty is a common feature in first-principles models that are widely used in various engineering problems. Uncertainty quantification (UQ) has become an essential procedure to improve the accuracy and reliability of model predictions. Polynomial chaos expansion (PCE) has been used as an efficient approach for UQ by approximating uncertainty with orthogonal polynomial basis functions of standard distributions (e.g., normal) chosen from the Askey scheme. However, uncertainty in practice may not be represented well by standard distributions. In this case, the convergence rate and accuracy of the PCE-based UQ cannot be guaranteed. Further, when models involve non-polynomial forms, the PCE-based UQ can be computationally impractical in the presence of many parametric uncertainties. To address these issues, the Gram–Schmidt (GS) orthogonalization and generalized dimension reduction method (gDRM) are integrated with the PCE in this work to deal with many parametric uncertainties that follow arbitrary distributions. The performance of the proposed method is demonstrated with three benchmark cases including two chemical engineering problems in terms of UQ accuracy and computational efficiency by comparison with available algorithms (e.g., non-intrusive PCE). 

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2. Modified Polynomial Chaos Expansion for Efficient Uncertainty Quantification in Biological Systems

   https://doi.org/10.3390/applmech1030011  

   Son, Jeongeun; Du, Dongping; Du, Yuncheng (September 2020, Applied Mechanics)

   null (Ed.)

   Uncertainty quantification (UQ) is an important part of mathematical modeling and simulations, which quantifies the impact of parametric uncertainty on model predictions. This paper presents an efficient approach for polynomial chaos expansion (PCE) based UQ method in biological systems. For PCE, the key step is the stochastic Galerkin (SG) projection, which yields a family of deterministic models of PCE coefficients to describe the original stochastic system. When dealing with systems that involve nonpolynomial terms and many uncertainties, the SG-based PCE is computationally prohibitive because it often involves high-dimensional integrals. To address this, a generalized dimension reduction method (gDRM) is coupled with quadrature rules to convert a high-dimensional integral in the SG into a few lower dimensional ones that can be rapidly solved. The performance of the algorithm is validated with two examples describing the dynamic behavior of cells. Compared to other UQ techniques (e.g., nonintrusive PCE), the results show the potential of the algorithm to tackle UQ in more complicated biological systems. 

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3. 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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4. Efficiency of Uncertainty Propagation Methods for Estimating Output Moments

   https://doi.org/10.1016/B978-0-12-818597-1.50078-3  

   Mohammadi, Samira; Cremaschi, Selen (July 2019, Computeraided Chemical Engineering)

   Uncertainty propagation methods are used to estimate the distribution of model outputs resulting from a set of uncertain model outputs. There are a number of uncertainty propagation methods available in literature. This paper compares six non-intrusive uncertainty propagation methods, Latin Hypercube Sampling, Full Factorial Integration, Univariate Dimension Reduction, Halton series, Sobol series, and Polynomial Chaos Expansion, in terms of their efficiency for estimating the first four moments of the output distribution using computational experiments. The results suggest employing FFNI if there are few uncertain inputs, up to three. Uncertainty propagation methods that utilize Halton and Sobol series are found to be robust for estimating output moments as the number of uncertain inputs increased. In general, higher order polynomial chaos expansion approximations (3rd-5th order) obtained accurate estimates of model outputs with fewer model evaluations. 

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5. Inverse Modeling of Hydrologic Parameters in CLM4 via Generalized Polynomial Chaos in the Bayesian Framework

   https://doi.org/10.3390/computation10050072  

   Karagiannis, Georgios; Hou, Zhangshuan; Huang, Maoyi; Lin, Guang (May 2022, Computation)

   In this work, generalized polynomial chaos (gPC) expansion for land surface model parameter estimation is evaluated. We perform inverse modeling and compute the posterior distribution of the critical hydrological parameters that are subject to great uncertainty in the Community Land Model (CLM) for a given value of the output LH. The unknown parameters include those that have been identified as the most influential factors on the simulations of surface and subsurface runoff, latent and sensible heat fluxes, and soil moisture in CLM4.0. We set up the inversion problem in the Bayesian framework in two steps: (i) building a surrogate model expressing the input–output mapping, and (ii) performing inverse modeling and computing the posterior distributions of the input parameters using observation data for a given value of the output LH. The development of the surrogate model is carried out with a Bayesian procedure based on the variable selection methods that use gPC expansions. Our approach accounts for bases selection uncertainty and quantifies the importance of the gPC terms, and, hence, all of the input parameters, via the associated posterior probabilities. 

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