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Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs

Georgios Karagiannis, Guang Lin
2014 Journal of Computational Physics  
2014) 'Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs.  ...  We propose the Bayesian calibration of computer model mixture method which relies on the idea of representing the real system output as a mixture of the available computer model outputs with unknown input  ...  A computational highlight of the procedure is that it builds the unknown mixture weight functions via a stochastic bases selection from a pool of basis functions in a datadriven manner.  ... 
doi:10.1016/j.jcp.2013.11.016 fatcat:6w3nha6i6bdgllc5mc5tun3vpq

Surrogate and Reduced-Order Modeling: A Comparison of Approaches for Large-Scale Statistical Inverse Problems [chapter]

M. Frangos, Y. Marzouk, K. Willcox, B. van Bloemen Waanders
2010 Large-Scale Inverse Problems and Quantification of Uncertainty  
Though some of the methods we review exploit prior information, they largely focus on simplifying or accelerating evaluations of a stochastic model for the data, and thus are also applicable in a frequentist  ...  We then present a detailed comparison of reduced-order modeling and stochastic spectral approximations in Sections 0.4 and 0.5, respectively.  ...  While a survey of polynomial chaos methods for solving ODEs and PDEs with random inputs is beyond the scope of this chapter (see for instance Najm (2009) ; Xiu (2009)), we highlight two broad classes  ... 
doi:10.1002/9780470685853.ch7 fatcat:oloehpfm7zesfg6lcdzw3zprga

A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems

Youssef Marzouk, Dongbin Xiu
2009 Communications in Computational Physics  
Stochastic collocation methods, based on generalized polynomial chaos (gPC), are used to construct a polynomial approximation of the forward solution over the support of the prior distribution.  ...  Combined with high accuracy of the gPC-based forward solver, the new algorithm can provide great efficiency in practical applications.  ...  Acknowledgments The work of Y. Marzouk  ... 
doi:10.4208/cicp.2009.v6.p826 fatcat:77g2x4ufbzcj5fqc65n2srimga

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

Georgios Karagiannis, Bledar A. Konomi, Guang Lin
2015 Journal of Computational Physics  
2015) 'A Bayesian mixed shrinkage prior procedure for spatialstochastic basis selection and evaluation of gPC expansions : applications to elliptic SPDEs.  ...  Yet, it inherits all the advantages of Bayesian model uncertainty methods, e.g. accounts for uncertainty about basis significance and provides interval estimation through posterior distributions.  ...  Generalized polynomial chaos We consider a stochastic system with solution u(x; ξ) that depends on a d x -dimensional vector of spatial input variables x ∈ X and a d ξ -dimensional vector of random input  ... 
doi:10.1016/j.jcp.2014.12.034 fatcat:hz32ujwdp5frdcscoiwwhzmvom

Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates

Ahmed H. Elsheikh, Ibrahim Hoteit, Mary F. Wheeler
2014 Computer Methods in Applied Mechanics and Engineering  
An efficient Bayesian calibration method based on the nested sampling (NS) algorithm and non-intrusive polynomial chaos method is presented.  ...  The proposed algorithm is applied for calibration and model selection of subsurface flow models. Crown  ...  Doostan and Owhadi [28] combined the idea of sparse calibration and gPC to approximate the solution of PDEs with stochastic coefficients.  ... 
doi:10.1016/j.cma.2013.11.001 fatcat:qkbt3jieejaepjkytpb2dcshga

Low-rank separated representation surrogates of high-dimensional stochastic functions: Application in Bayesian inference

AbdoulAhad Validi
2014 Journal of Computational Physics  
The performance of the method is studied through its application to three numerical examples including a 41-dimensional elliptic PDE and a 21-dimensional cavity flow.  ...  Furthermore, this vector valued separated representation-based model, in comparison to the available scalar-valued case, leads to a significant reduction in the cost of approximation by an order of magnitude  ...  The Janus supercomputer is a joint effort of the University of Colorado Boulder, the University of Colorado Denver and the National Center for Atmospheric Research.  ... 
doi:10.1016/j.jcp.2013.12.024 fatcat:mo654rjdqnhnrobqc4nqnquxsm

Development and Realization of Validation Benchmarks [article]

Farid Mohammadi
2021 arXiv   pre-print
Additionally, to accelerate the analysis for computationally demanding flow and transport models in porous media, the framework is equipped with a model reduction technique, namely Bayesian Sparse Polynomial  ...  We demonstrate the capabilities of the aforementioned Bayesian validation framework by applying it to an application for validation as well as uncertainty quantification of fluid flow in fractured porous  ...  Polynomial Chaos Expansion In a probabilistic framework, uncertainties in input parameters are modeled via random variables.  ... 
arXiv:2011.13216v2 fatcat:bzipici7jffvxmpghhsk334vqi

Solution of the 3D density-driven groundwater flow problem with uncertain porosity and permeability

Alexander Litvinenko, Dmitry Logashenko, Raul Tempone, Gabriel Wittum, David Keyes
2020 GEM - International Journal on Geomathematics  
For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute the mean, variance, and exceedance probabilities for the mass fraction.  ...  Thus, accurate modeling of water pollution at the surface and in groundwater aquifers is vital. Here, we consider a density-driven groundwater flow problem with uncertain porosity and permeability.  ...  We describe the stochastic modeling, integration methods, and the generalized polynomial chaos expansion technique in Sect. 3.  ... 
doi:10.1007/s13137-020-0147-1 fatcat:zahtjfyxvjcljh2t476n65gtx4

Propagation of Uncertainties in Density-Driven Flow [article]

Alexander Litvinenko, Dmitry Logashenko, Raul Tempone, Gabriel Wittum, David Keyes
2019 arXiv   pre-print
Here, we demonstrate a parallel method to quantify the propagation of the uncertainty in the dispersal of pollution in subsurface flow.  ...  We construct a low-cost generalized polynomial chaos expansion (gPC) surrogate model, where the gPC coefficients are computed by projection on sparse and full tensor grids.  ...  We used the resources of the Supercomputing Laboratory at KAUST, under the development project k1051. We would like to thank the KAUST core lab for the assistance with Shaheen II supercomputer.  ... 
arXiv:1905.01770v1 fatcat:xixzcmxsmjhc3e3typcu2v52da

A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems [article]

Katiana Kontolati, Dimitrios Loukrezis, Dimitrios D. Giovanis, Lohit Vandanapu, Michael D. Shields
2022 arXiv   pre-print
to construct a mapping to quantities of interest via polynomial chaos expansions (PCE).  ...  Constructing surrogate models for uncertainty quantification (UQ) on complex partial differential equations (PDEs) having inherently high-dimensional 𝒪(10^≥ 2) stochastic inputs (e.g., forcing terms,  ...  Acknowledgements This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under Award Number DE-SC0020428. D.  ... 
arXiv:2202.04648v1 fatcat:ze7fjffnpbfglcbmia7rzwmz2e

High-dimensional Stochastic Inversion via Adjoint Models and Machine Learning [article]

Charanraj A. Thimmisetty, Wenju Zhao, Xiao Chen, Charles H. Tong, Joshua A. White
2018 arXiv   pre-print
In this paper, we propose a novel Bayesian stochastic inversion methodology, characterized by a tight coupling between a gradient-based Langevin Markov Chain Monte Carlo (LMCMC) method and a kernel principal  ...  for models with Gaussian assumptions.  ...  differentiation-based discretized adjoint model of the KPCA-based and PCE-based ICDF transformation, and couple the discretized adjoint model with the high-fidelity adjoint PDE model.  ... 
arXiv:1803.06295v1 fatcat:5he57texdjfxdl3kp5sydd3zzy

Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark [article]

Nora Lüthen, Stefano Marelli, Bruno Sudret
2021 arXiv   pre-print
sparse regression solvers to approximate computer models with many input parameters, relying on only few model evaluations.  ...  Sparse polynomial chaos expansions are a popular surrogate modelling method that takes advantage of the properties of polynomial chaos expansions (PCE), the sparsity-of-effects principle, and powerful  ...  Surrogate modeling of high-dimensional problems via data-driven polynomial chaos expansions and sparse partial least square. Comput. Methods Appl. Mech. Engrg. 364, 112906. Zhu, X. and B.  ... 
arXiv:2002.01290v3 fatcat:bawss37oarczvdg5btxlmy2ng4

Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Zheng Zhang, Kim Batselier, Haotian Liu, Luca Daniel, Ngai Wong
2017 IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems  
This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further  ...  The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation.  ...  In [18] , a sparse generalized polynomial-chaos expansion was first computed as a stochastic model for the MEMS capacitor y(ξ).  ... 
doi:10.1109/tcad.2016.2618879 fatcat:4li26hkadvex5c3xs3eb2ijwk4

Learning and Meta-Learning of Stochastic Advection-Diffusion-Reaction Systems from Sparse Measurements [article]

Xiaoli Chen, Jinqiao Duan, George Em Karniadakis
2019 arXiv   pre-print
Subsequently, we attempt to optimize the hyper-parameters of sPINN by using the Bayesian optimization method (meta-learning), and compare the results with the empirically selected hyper-parameters of sPINN  ...  sparse measurements of the concentration field at random or pre-selected locations.  ...  In addition, we would like to thank Dr. Guofei Pang, Dr. Lu Lu, Dr. Xuhui Meng and Dr. Dongkun Zhang in the Division of Applied Mathematics at Brown University for their helpful suggestions.  ... 
arXiv:1910.09098v1 fatcat:shhzatkwhvgnppnl4gwrtggnbq

Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems [article]

Dongkun Zhang, Lu Lu, Ling Guo, George Em Karniadakis
2018 arXiv   pre-print
Multiple DNNs are designed to learn the modal functions of the arbitrary polynomial chaos (aPC) expansion of its solution by using stochastic data from sparse sensors.  ...  Here, we propose a new method with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty.  ...  In addition, we would like to thank Liu Yang for his generous advice.  ... 
arXiv:1809.08327v1 fatcat:5zimcd7pkjej5n3fba5qhlnpsi
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