{"title":"Deep Surrogate Accelerated Delayed-Acceptance Hamiltonian Monte Carlo for Bayesian Inference of Spatio-Temporal Heat Fluxes in Rotating Disc Systems","authors":"Teo Deveney, Eike H. Mueller, T. Shardlow","doi":"10.1137/22m1513113","DOIUrl":"https://doi.org/10.1137/22m1513113","url":null,"abstract":"","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47046368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Simple, Bias-free Approximation of Covariance Functions by the Multilevel Monte Carlo Method Having Nearly Optimal Complexity","authors":"A. Chernov, Erik Marc Schetzke","doi":"10.1137/22m1506845","DOIUrl":"https://doi.org/10.1137/22m1506845","url":null,"abstract":"","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43116576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christophe Audouze, Aaron E. Klein, Adrian Butscher, Nigel Morris, P. Nair, M. Yano
. We present a non-intrusive approach to robust structural topology optimization. Specifically, we consider optimization of mean- and variance-based robustness metrics of a linear functional output associated with the linear elasticity equation in the presence of probabilistic un- certainties in the loading and material properties. To provide an efficient approximation of higher-dimensional problems, we approximate the solution to the governing stochastic partial differential equations using the anchored ANOVA Petrov-Galerkin (AAPG) projection scheme. We then develop a non-intrusive quadrature-based formulation to evaluate the robustness metric and the associated shape derivative. The formulation is non-intrusive in the sense that it works with any level-set-based topology optimization code that can provide deterministic displacements, outputs, and shape deriva- tives for selected stochastic parameter values. We demonstrate the effectiveness of the proposed approach on various problems under loading and material uncertainties.
{"title":"Robust Level-Set-Based Topology Optimization Under Uncertainties Using Anchored ANOVA Petrov–Galerkin Method","authors":"Christophe Audouze, Aaron E. Klein, Adrian Butscher, Nigel Morris, P. Nair, M. Yano","doi":"10.1137/22m1524722","DOIUrl":"https://doi.org/10.1137/22m1524722","url":null,"abstract":". We present a non-intrusive approach to robust structural topology optimization. Specifically, we consider optimization of mean- and variance-based robustness metrics of a linear functional output associated with the linear elasticity equation in the presence of probabilistic un- certainties in the loading and material properties. To provide an efficient approximation of higher-dimensional problems, we approximate the solution to the governing stochastic partial differential equations using the anchored ANOVA Petrov-Galerkin (AAPG) projection scheme. We then develop a non-intrusive quadrature-based formulation to evaluate the robustness metric and the associated shape derivative. The formulation is non-intrusive in the sense that it works with any level-set-based topology optimization code that can provide deterministic displacements, outputs, and shape deriva- tives for selected stochastic parameter values. We demonstrate the effectiveness of the proposed approach on various problems under loading and material uncertainties.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43039097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 3, Page 788-813, September 2023. Abstract. We propose a method for the accurate estimation of rare event or failure probabilities for expensive-to-evaluate numerical models in high dimensions. The proposed approach combines ideas from large deviation theory and adaptive importance sampling. The importance sampler uses a cross-entropy method to find an optimal Gaussian biasing distribution, and reuses all samples made throughout the process for both the target probability estimation and for updating the biasing distributions. Large deviation theory is used to find a good initial biasing distribution through the solution of an optimization problem. Additionally, it is used to identify a low-dimensional subspace that is most informative of the rare event probability. This subspace is used for the cross-entropy method, which is known to lose efficiency in higher dimensions. The proposed method does not require smoothing of indicator functions nor does it involve numerical tuning parameters. We compare the method with a state-of-the-art cross-entropy-based importance sampling scheme using three examples: a high-dimensional failure probability estimation benchmark, a problem governed by a diffusion equation, and a tsunami problem governed by the time-dependent shallow water system in one spatial dimension.
{"title":"Large Deviation Theory-based Adaptive Importance Sampling for Rare Events in High Dimensions","authors":"Shanyin Tong, Georg Stadler","doi":"10.1137/22m1524758","DOIUrl":"https://doi.org/10.1137/22m1524758","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 3, Page 788-813, September 2023. <br/> Abstract. We propose a method for the accurate estimation of rare event or failure probabilities for expensive-to-evaluate numerical models in high dimensions. The proposed approach combines ideas from large deviation theory and adaptive importance sampling. The importance sampler uses a cross-entropy method to find an optimal Gaussian biasing distribution, and reuses all samples made throughout the process for both the target probability estimation and for updating the biasing distributions. Large deviation theory is used to find a good initial biasing distribution through the solution of an optimization problem. Additionally, it is used to identify a low-dimensional subspace that is most informative of the rare event probability. This subspace is used for the cross-entropy method, which is known to lose efficiency in higher dimensions. The proposed method does not require smoothing of indicator functions nor does it involve numerical tuning parameters. We compare the method with a state-of-the-art cross-entropy-based importance sampling scheme using three examples: a high-dimensional failure probability estimation benchmark, a problem governed by a diffusion equation, and a tsunami problem governed by the time-dependent shallow water system in one spatial dimension.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Claudia Schillings, Claudia Totzeck, Philipp Wacker
SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 3, Page 757-787, September 2023. Abstract. We propose an approach based on function evaluations and Bayesian inference to extract higher-order differential information of objective functions from a given ensemble of particles. Pointwise evaluation of some potential V in an ensemble contains implicit information about first- or higher-order derivatives, which can be made explicit with little computational effort (ensemble-based gradient inference). We suggest using this information for the improvement of established ensemble-based numerical methods for optimization and sampling such as consensus-based optimization and Langevin-based samplers. Numerical studies indicate that the augmented algorithms are often superior to their gradient-free variants; in particular, the augmented methods help the ensembles to escape their initial domain, to explore multimodal, non-Gaussian settings, and to speed up the collapse at the end of optimization dynamics. The code for the numerical examples in this manuscript can be found in the paper’s Github repository.
{"title":"Ensemble-Based Gradient Inference for Particle Methods in Optimization and Sampling","authors":"Claudia Schillings, Claudia Totzeck, Philipp Wacker","doi":"10.1137/22m1533281","DOIUrl":"https://doi.org/10.1137/22m1533281","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 3, Page 757-787, September 2023. <br/> Abstract. We propose an approach based on function evaluations and Bayesian inference to extract higher-order differential information of objective functions from a given ensemble of particles. Pointwise evaluation of some potential V in an ensemble contains implicit information about first- or higher-order derivatives, which can be made explicit with little computational effort (ensemble-based gradient inference). We suggest using this information for the improvement of established ensemble-based numerical methods for optimization and sampling such as consensus-based optimization and Langevin-based samplers. Numerical studies indicate that the augmented algorithms are often superior to their gradient-free variants; in particular, the augmented methods help the ensembles to escape their initial domain, to explore multimodal, non-Gaussian settings, and to speed up the collapse at the end of optimization dynamics. The code for the numerical examples in this manuscript can be found in the paper’s Github repository.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Physics-based covariance models provide a systematic way to construct covariance models that are consistent with the underlying physical laws in Gaussian process analysis. The unknown parameters in the covariance models can be estimated using maximum likelihood estimation, but direct construction of the covariance matrix and classical strategies of computing with it require physical model runs, storage complexity, and computational complexity. To address such challenges, we propose to approximate the discretized covariance function using hierarchical matrices. By utilizing randomized range sketching for individual off-diagonal blocks, the construction process of the hierarchical covariance approximation requires physical model applications and the maximum likelihood computations require effort per iteration. We propose a new approach to compute exactly the trace of products of hierarchical matrices which results in the expected Fisher information matrix being computable in as well. The construction is totally matrix-free and the derivatives of the covariance matrix can then be approximated in the same hierarchical structure by differentiating the whole process. Numerical results are provided to demonstrate the effectiveness, accuracy, and efficiency of the proposed method for parameter estimations and uncertainty quantification.
{"title":"Scalable Physics-Based Maximum Likelihood Estimation Using Hierarchical Matrices","authors":"Yian Chen, Mihai Anitescu","doi":"10.1137/21m1458880","DOIUrl":"https://doi.org/10.1137/21m1458880","url":null,"abstract":"Physics-based covariance models provide a systematic way to construct covariance models that are consistent with the underlying physical laws in Gaussian process analysis. The unknown parameters in the covariance models can be estimated using maximum likelihood estimation, but direct construction of the covariance matrix and classical strategies of computing with it require physical model runs, storage complexity, and computational complexity. To address such challenges, we propose to approximate the discretized covariance function using hierarchical matrices. By utilizing randomized range sketching for individual off-diagonal blocks, the construction process of the hierarchical covariance approximation requires physical model applications and the maximum likelihood computations require effort per iteration. We propose a new approach to compute exactly the trace of products of hierarchical matrices which results in the expected Fisher information matrix being computable in as well. The construction is totally matrix-free and the derivatives of the covariance matrix can then be approximated in the same hierarchical structure by differentiating the whole process. Numerical results are provided to demonstrate the effectiveness, accuracy, and efficiency of the proposed method for parameter estimations and uncertainty quantification.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135703572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note we solve a general statistical inverse problem under absence of knowledge of both the noise level and the noise distribution via application of the (modified) heuristic discrepancy principle. Hereby the unbounded (non-Gaussian) noise is controlled via introducing an auxiliary discretization dimension and choosing it in an adaptive fashion. We first show convergence for completely arbitrary compact forward operator and ground solution. Then the uncertainty of reaching the optimal convergence rate is quantified in a specific Bayesian-like environment. We conclude with numerical experiments.
{"title":"Noise Level Free Regularization of General Linear Inverse Problems under Unconstrained White Noise","authors":"Tim Jahn","doi":"10.1137/22m1506675","DOIUrl":"https://doi.org/10.1137/22m1506675","url":null,"abstract":"In this note we solve a general statistical inverse problem under absence of knowledge of both the noise level and the noise distribution via application of the (modified) heuristic discrepancy principle. Hereby the unbounded (non-Gaussian) noise is controlled via introducing an auxiliary discretization dimension and choosing it in an adaptive fashion. We first show convergence for completely arbitrary compact forward operator and ground solution. Then the uncertainty of reaching the optimal convergence rate is quantified in a specific Bayesian-like environment. We conclude with numerical experiments.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136284948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A crucial role in the theory of uncertainty quantification (UQ) of PDEs is played by the regularity of the solution with respect to the stochastic parameters; indeed, a key property one seeks to establish is that the solution is holomorphic with respect to (the complex extensions of) the parameters. In the context of UQ for the high-frequency Helmholtz equation, a natural question is therefore: how does this parametric holomorphy depend on the wavenumber ? The recent paper [35] showed for a particular nontrapping variable-coefficient Helmholtz problem with affine dependence of the coefficients on the stochastic parameters that the solution operator can be analytically continued a distance into the complex plane. In this paper, we generalize the result in [35] about -explicit parametric holomorphy to a much wider class of Helmholtz problems with arbitrary (holomorphic) dependence on the stochastic parameters; we show that in all cases the region of parametric holomorphy decreases with and show how the rate of decrease with is dictated by whether the unperturbed Helmholtz problem is trapping or nontrapping. We then give examples of both trapping and nontrapping problems where these bounds on the rate of decrease with of the region of parametric holomorphy are sharp, with the trapping examples coming from the recent results of [31]. An immediate implication of these results is that the -dependent restrictions imposed on the randomness in the analysis of quasi-Monte Carlo methods in [35] arise from a genuine feature of the Helmholtz equation with large (and not, for example, a suboptimal bound).
{"title":"Wavenumber-Explicit Parametric Holomorphy of Helmholtz Solutions in the Context of Uncertainty Quantification","authors":"E. A. Spence, J. Wunsch","doi":"10.1137/22m1486170","DOIUrl":"https://doi.org/10.1137/22m1486170","url":null,"abstract":"A crucial role in the theory of uncertainty quantification (UQ) of PDEs is played by the regularity of the solution with respect to the stochastic parameters; indeed, a key property one seeks to establish is that the solution is holomorphic with respect to (the complex extensions of) the parameters. In the context of UQ for the high-frequency Helmholtz equation, a natural question is therefore: how does this parametric holomorphy depend on the wavenumber ? The recent paper [35] showed for a particular nontrapping variable-coefficient Helmholtz problem with affine dependence of the coefficients on the stochastic parameters that the solution operator can be analytically continued a distance into the complex plane. In this paper, we generalize the result in [35] about -explicit parametric holomorphy to a much wider class of Helmholtz problems with arbitrary (holomorphic) dependence on the stochastic parameters; we show that in all cases the region of parametric holomorphy decreases with and show how the rate of decrease with is dictated by whether the unperturbed Helmholtz problem is trapping or nontrapping. We then give examples of both trapping and nontrapping problems where these bounds on the rate of decrease with of the region of parametric holomorphy are sharp, with the trapping examples coming from the recent results of [31]. An immediate implication of these results is that the -dependent restrictions imposed on the randomness in the analysis of quasi-Monte Carlo methods in [35] arise from a genuine feature of the Helmholtz equation with large (and not, for example, a suboptimal bound).","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135717187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
E. Spiller, R. Wolpert, Pablo Tierz, Taylor G. Asher
{"title":"The Zero Problem: Gaussian Process Emulators for Range-Constrained Computer Models","authors":"E. Spiller, R. Wolpert, Pablo Tierz, Taylor G. Asher","doi":"10.1137/21m1467420","DOIUrl":"https://doi.org/10.1137/21m1467420","url":null,"abstract":"","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86767355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nonparametric Posterior Learning for Emission Tomography","authors":"F. Goncharov, E. Barat, T. Dautremer","doi":"10.1137/21m1463367","DOIUrl":"https://doi.org/10.1137/21m1463367","url":null,"abstract":"","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72367575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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