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Prior-informed Uncertainty Modelling with Bayesian Polynomial Approximations [article]

Chun Yui Wong, Pranay Seshadri, Andrew B. Duncan, Ashley Scillitoe, Geoffrey Parks
2022 arXiv   pre-print
We show that through a Bayesian framework, such prior knowledge can be leveraged to produce orthogonal polynomial approximations with enhanced predictive accuracy.  ...  In this paper, we describe a Bayesian formulation of polynomial approximations capable of incorporating uncertainties in input data.  ...  Using 105 training inputs (different for each output), we fit two models with σ m = 0.001: • Two Bayesian polynomial models, with prior on the coefficients p(α) = N (0, 0.001I) 1 . • A coregional polynomial  ... 
arXiv:2203.03508v2 fatcat:kjfqwppg4rhhpimbmxhwy5gd24

Uncertainty reduction and characterization for complex environmental fate and transport models: An empirical Bayesian framework incorporating the stochastic response surface method

Suhrid Balakrishnan, Amit Roy, Marianthi G. Ierapetritou, Gregory P. Flach, Panos G. Georgopoulos
2003 Water Resources Research  
In this work, a computationally efficient Bayesian framework for the reduction and characterization of parametric uncertainty in computationally demanding environmental 3-D numerical models has been developed  ...  Input parameter uncertainty, based initially on expert opinion, was found to decrease in all variables of the posterior distribution.  ...  In a Bayesian context, the distribution of interest is the joint posterior distribution of the model parameters P(θ|d), which incorporates both prior information on the model parameters θ, as well as additional  ... 
doi:10.1029/2002wr001810 fatcat:ukxdywc6jfdwzjpcbmfuc52ajy

Computationally efficient uncertainty propagation and reduction using the stochastic response surface method

S.S. Isukapalli, S. Balakrishnan, P.G. Georgopoulos
2004 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601)  
Furthermore, SRSM can be used in conjunction with Bayesian methods such as Markov Chain Monte Carlo (MCMC) methods to reduce uncertainties by incorporating observational information in estimates of model  ...  ADIFOR provides estimates of partial derivatives from a single model run, and this partial derivative information is used in the determination of the coefficients of the polynomial chaos expansions.  ...  USING THE REDUCED SRSM MODEL IN CONJUNCTION WITH BAYESIAN METHODS FOR UNCERTAINTY REDUCTION For many environmental problems, prior information on physically reasonable values of model parameters is often  ... 
doi:10.1109/cdc.2004.1430381 fatcat:4bbqkeuy4fbcdkdehrzbqm74z4

A Generalized Polynomial Chaos-Based Method for Efficient Bayesian Calibration of Uncertain Computational Models [article]

Piyush Tagade, Han-Lim Choi
2012 arXiv   pre-print
Bayesian inference is used to update the prior probability distribution of the generalized Polynomial Chaos basis, which along with the chaos expansion coefficients represent the posterior probability  ...  This paper addresses the Bayesian calibration of dynamic models with parametric and structural uncertainties, in particular where the uncertain parameters are unknown/poorly known spatio-temporally varying  ...  The prior uncertainty is projected on a Hermite polynomial chaos basis.  ... 
arXiv:1211.0158v1 fatcat:sbdtvbk4ifbh5iwlooskzpoo4m

Chaos Expansion based Bootstrap Filter to Calibrate CO2 Injection Models

Sergey Oladyshkin, Patrick Schröder, Holger Class, Wolfgang Nowak
2013 Energy Procedia  
The current work deals with an advanced framework for history matching of underground carbon dioxide (CO 2 ) storage based on the arbitrary polynomial chaos expansion (aPC).  ...  We will combine the aPC with Bootstrap filtering in order to match the model to past observations.  ...  The authors would like to express their thanks to Lena Walter at the Department of Hydromechanics and Modeling of Hydrosystems, University of Stuttgart, and Thomas Kempka at the Helmholtz Centre Potsdam  ... 
doi:10.1016/j.egypro.2013.08.046 fatcat:t7zldxmcevb5hmqbs5gztq44kq

Efficient Bayesian inversion for shape reconstruction of lithography masks

Nando Farchmin, Martin Hammerschmidt, Philipp-Immanuel Schneider, Matthias Wurm, Bernd Bodermann, Markus Bär, Sebastian Heidenreich
2020 Journal of Micro/Nanolithography  
Approach: We use a non-intrusive polynomial chaos based approximation of the forward model which increases speed and thus enables the exploration of the posterior through direct Bayesian inference.  ...  Conclusion: The use of a polynomial chaos surrogate allows to quantify both parameter influences and reconstruction uncertainties.  ...  the forward model with a global polynomial. 17, 18 We additionally show how this surrogate allows for a Bayesian approach to the inverse problem.  ... 
doi:10.1117/1.jmm.19.2.024001 fatcat:7qwl2jfajbgxvlrorbpten5rxq

Learning Model Discrepancy of an Electric Motor with Bayesian Inference

David N. John, Michael Schick, Vincent Heuveline
2019 Proceedings (MDPI)  
A polynomial expansion is used to approximate and learn model discrepancy.  ...  This work investigates the influence of model discrepancies onto the calibration of physical model parameters and further considers a Bayesian inference framework including an attempt to correct for model  ...  of the unknowns a prior containing some information about the reference.  ... 
doi:10.3390/proceedings2019033011 fatcat:qxhgda7cpna5tbfe7x7f5luk4i

The sparse Polynomial Chaos expansion: a fully Bayesian approach with joint priors on the coefficients and global selection of terms [article]

Paul-Christian Bürkner, Ilja Kröker, Sergey Oladyshkin, Wolfgang Nowak
2022 arXiv   pre-print
The suggested Bayesian PCE model directly aims to solve the two challenges named above: achieving a sparse PCE representation and estimating uncertainty of the PCE itself.  ...  Furthermore, the approximation error resembles an uncertainty that most existing PCE-based methods do not estimate.  ...  In addition to Standard PCE, we estimated Bayesian sparse PCE with the R2D2 prior (Bayesian-R2D2 ) and Bayesian PCE with improper flat priors on the coefficients (Bayesian-flat).  ... 
arXiv:2204.06043v1 fatcat:g66turnzrbdr3gwr6qndbgw4ae

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  ...  This metric shall provide a measure for comparison of the system response quantities of an experiment with the ones from a computational model, while accounting for uncertainties in both.  ...  Bayesian active learning The Bayesian framework provides a principled approach for incorporating prior information and/or uncertainties concerning the statistical model via a utility function which encapsulates  ... 
arXiv:2011.13216v2 fatcat:bzipici7jffvxmpghhsk334vqi

Cardiovascular Modeling With Adapted Parametric Inference

Didier Lucor, Olivier P. Le Maître, Muhammad Dauhoo, Laurent Dumas, Pierre Gabriel, Pauline Lafitte
2018 ESAIM Proceedings and Surveys  
Cette approche permet d'obtenir un gain numérique substantiel en termes d'efficacité et de précision par rapportà une approximation polynomiale directement basée sur unéchantillonnage selon la distribution  ...  We introduce a novel numerical pathway to construct a series of polynomial surrogate models, by regression, using samples drawn from a sequence of distributions likely to converge to the posterior distribution  ...  Dumas from the Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, France, for the organization and the invitation at the "Mathematical models in biology and medecine" school  ... 
doi:10.1051/proc/201862091 fatcat:pys3pdrpuvbzbplbiplx7opeoi

Bayesian Calibration and Validation of a Large‐scale and Time‐demanding Sediment Transport Model

Felix Beckers, Andrés Heredia, Markus Noack, Wolfgang Nowak, Silke Wieprecht, Sergey Oladyshkin
2020 Water Resources Research  
Since Bayesian methods require a significant number of simulation runs, this work proposes to construct a surrogate model, here with the arbitrary polynomial chaos technique.  ...  Key Points: • We reduce a time-demanding sediment transport model with a surrogate technique based on the arbitrary polynomial chaos expansion (aPC) • Bayesian model calibration and validation in a fraction  ...  With this, we can evaluate and quantify the prior and posterior uncertainties and assess the quality of Bayesian calibration and validation.  ... 
doi:10.1029/2019wr026966 fatcat:6fegslbmtbhb3ia57i4pwywihm

Bayesian Surrogate Analysis and Uncertainty Propagation

Sascha Ranftl, Wolfgang von der Linden
2021 Physical Sciences Forum  
The surrogate models are commonly chosen and deemed trustworthy based on heuristic measures, and substituted for the simulation in order to approximately propagate the simulation input uncertainties to  ...  For a Gaussian likelihood for the simulation data, with unknown surrogate variance and given a generalized linear surrogate model, the resulting formulas reduce to simple matrix multiplications.  ...  Bayesian Uncertainty Quantification We start with the general structure of uncertainty propagation problems based on surrogate models in Section 2.1.  ... 
doi:10.3390/psf2021003006 fatcat:xk5vo45f6bbjham6z3q46znc24

Bayesian model updating with consideration of modeling error

Erliang Zhang, Pierre Feissel, Jérôme Antoni
2010 European Journal of Computational Mechanics  
The polynomial chaos expansion is adopted as a surrogate model to propagate the parameter uncertainty and thus accelerate the evaluation of their posterior probability distribution.  ...  Moreover, one hybrid modal model is proposed by introducing some additional variables so as to deal with the modeling errors.  ...  The uncertainty of prior transfer function stems from the prior uncertainty of model parameters θ and is illustrated by using the Bayesian confidence interval of 90% on transfer function H 5 in Figure  ... 
doi:10.13052/ejcm.19.255-266 fatcat:ymrqagl4grdb3nkmkqgxqejrmm

A stochastic collocation approach for efficient integrated gear health prognosis

Fuqiong Zhao, Zhigang Tian, Yong Zeng
2013 Mechanical systems and signal processing  
Based on generalized polynomial chaos expansion, the approach is utilized to evaluate the uncertainty in gear remaining useful life prediction as well as the likelihood function in Bayesian inference.  ...  The collected condition monitoring data is incorporated into prognostics via Bayesian inference to update the distributions of uncertainties at certain inspection times.  ...  They will be updated through Bayesian framework using the measured condition data, in order to narrow down the prior distributions and make them converge or approximate to the real distributions.  ... 
doi:10.1016/j.ymssp.2013.03.004 fatcat:nc3qpjujyzfillpquyts52zo3q

Acceleration of uncertainty updating in the description of transport processes in heterogeneous materials

Anna Kučerová, Jan Sýkora, Bojana Rosić, Hermann G. Matthies
2012 Journal of Computational and Applied Mathematics  
Such an approximation of the FE model for the forward simulation perfectly suits the Bayesian updating.  ...  The presented uncertainty updating techniques are applied to the numerical model of coupled heat and moisture transport in heterogeneous materials with spatially varying coefficients defined by random  ...  Acknowledgments This outcome has been achieved with the financial support of the Czech Science Foundation, project No. 105/11/0411, the Czech Ministry of Education, Youth and Sports, project No.  ... 
doi:10.1016/j.cam.2012.02.003 fatcat:pn63f54axnfodifwuw5fbvwxny
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