Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2023045369
Miguel Angel Alvarez Ballesteros, Neil K. Chada, Ajay Jasra
In this paper, we consider the development of unbiased estimators for the ensemble Kalman-Bucy filter (EnKBF). The EnKBF is a continuous-time filtering methodology, which can be viewed as a continuous-time analog of the famous discrete-time ensemble Kalman filter. Our unbiased estimators will be motivated from recent work (Rhee and Glynn, Oper. Res., 63:1026-1053, 2015) which introduces randomization as a means to produce unbiased and finite variance estimators. The randomization enters through both the level of discretization and through the number of samples at each level. Our unbiased estimator will be specific to models that are linear and Gaussian. This is due to the fact that the EnKBF itself is consistent, in the large particle limit N → ∞, with the Kalman-Bucy filter, which allows us one derive theoretical insights. Specifically, we introduce two unbiased EnKBF estimators that will be applied to two particular variants of the EnKBF, which are the deterministic and vanilla EnKBF. Numerical experiments are conducted on a linear Ornstein-Uhlenbeck process, which includes a high-dimensional example. Our unbiased estimators will be compared to the multilevel. We also provide a proof of the multilevel deterministic EnKBF, which provides a guideline for some of the unbiased methods.
{"title":"UNBIASED ESTIMATION OF THE VANILLA AND DETERMINISTIC ENSEMBLE KALMAN-BUCY FILTERS","authors":"Miguel Angel Alvarez Ballesteros, Neil K. Chada, Ajay Jasra","doi":"10.1615/int.j.uncertaintyquantification.2023045369","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2023045369","url":null,"abstract":"In this paper, we consider the development of unbiased estimators for the ensemble Kalman-Bucy filter (EnKBF). The EnKBF is a continuous-time filtering methodology, which can be viewed as a continuous-time analog of the famous discrete-time ensemble Kalman filter. Our unbiased estimators will be motivated from recent work (Rhee and Glynn, Oper. Res., 63:1026-1053, 2015) which introduces randomization as a means to produce unbiased and finite variance estimators. The randomization enters through both the level of discretization and through the number of samples at each level. Our unbiased estimator will be specific to models that are linear and Gaussian. This is due to the fact that the EnKBF itself is consistent, in the large particle limit N → ∞, with the Kalman-Bucy filter, which allows us one derive theoretical insights. Specifically, we introduce two unbiased EnKBF estimators that will be applied to two particular variants of the EnKBF, which are the deterministic and vanilla EnKBF. Numerical experiments are conducted on a linear Ornstein-Uhlenbeck process, which includes a high-dimensional example. Our unbiased estimators will be compared to the multilevel. We also provide a proof of the multilevel deterministic EnKBF, which provides a guideline for some of the unbiased methods.","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135636630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2023045300
Isaac Sunseri, Alen Alexanderian, Joseph Hart, Bart Van Bloemen Waanders
We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by PDEs with infinite-dimensional parameters. In previous works, HDSA has been used to assess the sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on a model inverse problem governed by a PDE for heat conduction.
{"title":"Hyper-differential sensitivity analysis for nonlinear Bayesian inverse problems","authors":"Isaac Sunseri, Alen Alexanderian, Joseph Hart, Bart Van Bloemen Waanders","doi":"10.1615/int.j.uncertaintyquantification.2023045300","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2023045300","url":null,"abstract":"We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by PDEs with infinite-dimensional parameters. In previous works, HDSA has been used to assess the sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on a model inverse problem governed by a PDE for heat conduction.","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136299088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2023047256
Niklas Miska, S. Freitag, D. Balzani
{"title":"Nested optimal uncertainty quantification for an efficient incorporation of random fields - Application to sheet metal forming","authors":"Niklas Miska, S. Freitag, D. Balzani","doi":"10.1615/int.j.uncertaintyquantification.2023047256","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2023047256","url":null,"abstract":"","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67531474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2023045687
M. Ganesh, S. Hawkins, A. Tartakovsky, R. Tipireddy
{"title":"A stochastic domain decomposition and post-processing algorithm for epistemic uncertainty quantification","authors":"M. Ganesh, S. Hawkins, A. Tartakovsky, R. Tipireddy","doi":"10.1615/int.j.uncertaintyquantification.2023045687","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2023045687","url":null,"abstract":"","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67531870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2023041042
Jeferson Osmar de Almeida, F. Rochinha
{"title":"Uncertainty Quantification of water-flooding in oil reservoirs computational simulations using a probabilistic learning approach","authors":"Jeferson Osmar de Almeida, F. Rochinha","doi":"10.1615/int.j.uncertaintyquantification.2023041042","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2023041042","url":null,"abstract":"","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67531316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2023044543
Andrew White, S. Mahadevan
{"title":"Discrepancy modeling for model calibration with multivariate output","authors":"Andrew White, S. Mahadevan","doi":"10.1615/int.j.uncertaintyquantification.2023044543","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2023044543","url":null,"abstract":"","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67531628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2023043457
Dongjin Lee, S. Rahman
{"title":"HIGH-DIMENSIONAL STOCHASTIC DESIGN OPTIMIZATION UNDER DEPENDENT RANDOM VARIABLES BY A DIMENSIONALLY DECOMPOSED GENERALIZED POLYNOMIAL CHAOS EXPANSION","authors":"Dongjin Lee, S. Rahman","doi":"10.1615/int.j.uncertaintyquantification.2023043457","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2023043457","url":null,"abstract":"","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67531888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2022039993
Jigar Parekh, Roel Verstappen
Undeterred by its inherent limitations, Reynolds-averaged Navier-Stokes (RANS) based modeling is still considered the most recognized approach for several computational fluid dynamics (CFD) applications. Recently, in the turbulence modeling community, quantification of model-form uncertainties in RANS has attracted significant interest. We present a stochastic RANS solver with an efficient implementation of the intrusive polynomial chaos (IPC) method in OpenFOAM. The stochastic solver quantifies and propagates the uncertainties associated with the output of the RANS model (eddy viscosity or Reynolds stress tensor). Two distinct high-dimensional variants of the uncertainties are considered, namely, the random eddy viscosity field (REVF) and the random Reynolds stress tensor field (RRSTF). The randomness is introduced in the approximated eddy viscosity field and the Reynolds stress tensor, while asserting the realizability. The stochastic RANS solver has been tested on various benchmark problems for RANS turbulence modeling. In this study, we discuss two important problems where the stochastic RANS solver shows significantly better performance than the traditional uncertainty quantification (UQ) methods. The first problem analyzed is the flow over periodic hills with a REVF, while the second stochastic problem considered is the flow in a square duct with a RRSTF. Along with the comparison for three different RANS turbulence models, a detailed analysis of the stochastic solver based on various influential model parameters is also presented. The IPC based stochastic solver demonstrated the potential to be used in the UQ analysis of further complex CFD applications, especially when a large number of deterministic simulations is not feasible, e.g., wind farm CFD simulations.
{"title":"QUANTIFICATION AND PROPAGATION OF MODEL-FORM UNCERTAINTIES IN RANS TURBULENCE MODELING VIA INTRUSIVE POLYNOMIAL CHAOS","authors":"Jigar Parekh, Roel Verstappen","doi":"10.1615/int.j.uncertaintyquantification.2022039993","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2022039993","url":null,"abstract":"Undeterred by its inherent limitations, Reynolds-averaged Navier-Stokes (RANS) based modeling is still considered the most recognized approach for several computational fluid dynamics (CFD) applications. Recently, in the turbulence modeling community, quantification of model-form uncertainties in RANS has attracted significant interest. We present a stochastic RANS solver with an efficient implementation of the intrusive polynomial chaos (IPC) method in OpenFOAM. The stochastic solver quantifies and propagates the uncertainties associated with the output of the RANS model (eddy viscosity or Reynolds stress tensor). Two distinct high-dimensional variants of the uncertainties are considered, namely, the random eddy viscosity field (REVF) and the random Reynolds stress tensor field (RRSTF). The randomness is introduced in the approximated eddy viscosity field and the Reynolds stress tensor, while asserting the realizability. The stochastic RANS solver has been tested on various benchmark problems for RANS turbulence modeling. In this study, we discuss two important problems where the stochastic RANS solver shows significantly better performance than the traditional uncertainty quantification (UQ) methods. The first problem analyzed is the flow over periodic hills with a REVF, while the second stochastic problem considered is the flow in a square duct with a RRSTF. Along with the comparison for three different RANS turbulence models, a detailed analysis of the stochastic solver based on various influential model parameters is also presented. The IPC based stochastic solver demonstrated the potential to be used in the UQ analysis of further complex CFD applications, especially when a large number of deterministic simulations is not feasible, e.g., wind farm CFD simulations.","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135235913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2023046861
Michael Griebel, Uta Seidler
We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic PDE where the diffusion coefficient is parametrized by means of a Karhunen-Lo`eve expansion. The approximation of the equivalent parametric problem requires a restriction of the countably infinite-dimensional parameter space to a finite-dimensional parameter set, a spatial discretization and an approximation in the parametric variables. We consider a sparse grid approach between these approximation directions in order to reduce the computational effort and propose a dimension-adaptive combination technique. In addition, a sparse grid quadrature for the high-dimensional parametric approximation is employed and simultaneously balanced with the spatial and stochastic approximation. Our adaptive algorithm constructs a sparse grid approximation based on the benefit-cost ratio such that the regularity and thus the decay of the Karhunen-Lo`eve coefficients is not required beforehand. The decay is detected and exploited as the algorithm adjusts to the anisotropy in the parametric variables. We include numerical examples for the Darcy problem with a lognormal permeability field, which illustrate a good performance of the algorithm: For sufficiently smooth random fields, we essentially recover the spatial order of convergence as asymptotic convergence rate with respect to the computational cost.
{"title":"A Dimension-adaptive Combination Technique for Uncertainty Quantification","authors":"Michael Griebel, Uta Seidler","doi":"10.1615/int.j.uncertaintyquantification.2023046861","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2023046861","url":null,"abstract":"We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic PDE where the diffusion coefficient is parametrized by means of a Karhunen-Lo`eve expansion. The approximation of the equivalent parametric problem requires a restriction of the countably infinite-dimensional parameter space to a finite-dimensional parameter set, a spatial discretization and an approximation in the parametric variables. We consider a sparse grid approach between these approximation directions in order to reduce the computational effort and propose a dimension-adaptive combination technique. In addition, a sparse grid quadrature for the high-dimensional parametric approximation is employed and simultaneously balanced with the spatial and stochastic approximation. Our adaptive algorithm constructs a sparse grid approximation based on the benefit-cost ratio such that the regularity and thus the decay of the Karhunen-Lo`eve coefficients is not required beforehand. The decay is detected and exploited as the algorithm adjusts to the anisotropy in the parametric variables. We include numerical examples for the Darcy problem with a lognormal permeability field, which illustrate a good performance of the algorithm: For sufficiently smooth random fields, we essentially recover the spatial order of convergence as asymptotic convergence rate with respect to the computational cost.","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135445089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1615/int.j.uncertaintyquantification.2023047605
William Reese, Joseph Hart, Bart van Bloemen Waanders, Mauro Perego, John Jakeman, Arvind Saibaba
Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model which must be estimated. Although the Bayesian formulation is attractive for such problems, computational cost and high dimensionality frequently prohibit a thorough exploration of the parametric uncertainty. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to approximate properties of the Bayesian posterior distribution. For instance, the maximum a posteriori probability (MAP) and the Laplace approximation of the posterior covariance can be computed. In this article, we propose using hyper-differential sensitivity analysis (HDSA) to assess the sensitivity of the MAP point to changes in the auxiliary parameters. We establish an interpretation of HDSA as correlations in the posterior distribution. Our proposed framework is demonstrated on the inversion of bedrock topography for the Greenland ice sheet with uncertainties arising from the basal friction coefficient and climate forcing (ice accumulation rate). %Foundational assumptions for HDSA require satisfaction of the optimality conditions which are not always feasible or appropriate as a result of ill-posedness in the inverse problem. %We introduce novel theoretical and computational approaches to justify and enable HDSA for ill-posed inverse problems by projecting the sensitivities on likelihood informed subspaces and defining a posteriori updates.
{"title":"Hyper-differential sensitivity analysis in the context of Bayesian inference applied to ice-sheet problems","authors":"William Reese, Joseph Hart, Bart van Bloemen Waanders, Mauro Perego, John Jakeman, Arvind Saibaba","doi":"10.1615/int.j.uncertaintyquantification.2023047605","DOIUrl":"https://doi.org/10.1615/int.j.uncertaintyquantification.2023047605","url":null,"abstract":"Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model which must be estimated. Although the Bayesian formulation is attractive for such problems, computational cost and high dimensionality frequently prohibit a thorough exploration of the parametric uncertainty. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to approximate properties of the Bayesian posterior distribution. For instance, the maximum a posteriori probability (MAP) and the Laplace approximation of the posterior covariance can be computed. In this article, we propose using hyper-differential sensitivity analysis (HDSA) to assess the sensitivity of the MAP point to changes in the auxiliary parameters. We establish an interpretation of HDSA as correlations in the posterior distribution. Our proposed framework is demonstrated on the inversion of bedrock topography for the Greenland ice sheet with uncertainties arising from the basal friction coefficient and climate forcing (ice accumulation rate). %Foundational assumptions for HDSA require satisfaction of the optimality conditions which are not always feasible or appropriate as a result of ill-posedness in the inverse problem. %We introduce novel theoretical and computational approaches to justify and enable HDSA for ill-posed inverse problems by projecting the sensitivities on likelihood informed subspaces and defining a posteriori updates.","PeriodicalId":48814,"journal":{"name":"International Journal for Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135211134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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