SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 182-211, March 2024. Abstract.Filtering problems are derived from a sequential minimization of a quadratic function representing a compromise between the model and data. In this paper, we use the Perron–Frobenius operator in a stochastic process to develop a Perron–Frobenius operator filter. The proposed method belongs to Bayesian filtering and works for non-Gaussian distributions for nonlinear stochastic dynamical systems. The recursion of the filtering can be characterized by the composition of the Perron–Frobenius operator and likelihood operator. This gives a significant connection between the Perron–Frobenius operator and Bayesian filtering. We numerically fulfill the recursion by approximating the Perron–Frobenius operator by Ulam’s method. In this way, the posterior measure is represented by a convex combination of the indicator functions in Ulam’s method. To get a low-rank approximation for the Perron–Frobenius operator filter, we take a spectral decomposition for the posterior measure by using the eigenfunctions of the discretized Perron–Frobenius operator. The Perron–Frobenius operator filter employs data instead of flow equations to model the evolution of underlying stochastic dynamical systems. In contrast, standard particle filters require explicit equations or transition probability density for sampling. A few numerical examples are presented to illustrate the advantage of the Perron–Frobenius operator filter over the particle filter and extended Kalman filter.
{"title":"Perron–Frobenius Operator Filter for Stochastic Dynamical Systems","authors":"Ningxin Liu, Lijian Jiang","doi":"10.1137/23m1547391","DOIUrl":"https://doi.org/10.1137/23m1547391","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 182-211, March 2024. <br/>Abstract.Filtering problems are derived from a sequential minimization of a quadratic function representing a compromise between the model and data. In this paper, we use the Perron–Frobenius operator in a stochastic process to develop a Perron–Frobenius operator filter. The proposed method belongs to Bayesian filtering and works for non-Gaussian distributions for nonlinear stochastic dynamical systems. The recursion of the filtering can be characterized by the composition of the Perron–Frobenius operator and likelihood operator. This gives a significant connection between the Perron–Frobenius operator and Bayesian filtering. We numerically fulfill the recursion by approximating the Perron–Frobenius operator by Ulam’s method. In this way, the posterior measure is represented by a convex combination of the indicator functions in Ulam’s method. To get a low-rank approximation for the Perron–Frobenius operator filter, we take a spectral decomposition for the posterior measure by using the eigenfunctions of the discretized Perron–Frobenius operator. The Perron–Frobenius operator filter employs data instead of flow equations to model the evolution of underlying stochastic dynamical systems. In contrast, standard particle filters require explicit equations or transition probability density for sampling. A few numerical examples are presented to illustrate the advantage of the Perron–Frobenius operator filter over the particle filter and extended Kalman filter.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140152281","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}
Chih-Li Sung, Yi (Irene) Ji, Simon Mak, Wenjia Wang, Tao Tang
SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 157-181, March 2024. Abstract. In an era where scientific experiments can be very costly, multifidelity emulators provide a useful tool for cost-efficient predictive scientific computing. For scientific applications, the experimenter is often limited by a tight computational budget, and thus wishes to (i) maximize predictive power of the multifidelity emulator via a careful design of experiments, and (ii) ensure this model achieves a desired error tolerance with some notion of confidence. Existing design methods, however, do not jointly tackle objectives (i) and (ii). We propose a novel stacking design approach that addresses both goals. A multilevel reproducing kernel Hilbert space (RKHS) interpolator is first introduced to build the emulator, under which our stacking design provides a sequential approach for designing multifidelity runs such that a desired prediction error of [math] is met under regularity assumptions. We then prove a novel cost complexity theorem that, under this multilevel interpolator, establishes a bound on the computation cost (for training data simulation) needed to achieve a prediction bound of [math]. This result provides novel insights on conditions under which the proposed multifidelity approach improves upon a conventional RKHS interpolator which relies on a single fidelity level. Finally, we demonstrate the effectiveness of stacking designs in a suite of simulation experiments and an application to finite element analysis.
{"title":"Stacking Designs: Designing Multifidelity Computer Experiments with Target Predictive Accuracy","authors":"Chih-Li Sung, Yi (Irene) Ji, Simon Mak, Wenjia Wang, Tao Tang","doi":"10.1137/22m1532007","DOIUrl":"https://doi.org/10.1137/22m1532007","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 157-181, March 2024. <br/> Abstract. In an era where scientific experiments can be very costly, multifidelity emulators provide a useful tool for cost-efficient predictive scientific computing. For scientific applications, the experimenter is often limited by a tight computational budget, and thus wishes to (i) maximize predictive power of the multifidelity emulator via a careful design of experiments, and (ii) ensure this model achieves a desired error tolerance with some notion of confidence. Existing design methods, however, do not jointly tackle objectives (i) and (ii). We propose a novel stacking design approach that addresses both goals. A multilevel reproducing kernel Hilbert space (RKHS) interpolator is first introduced to build the emulator, under which our stacking design provides a sequential approach for designing multifidelity runs such that a desired prediction error of [math] is met under regularity assumptions. We then prove a novel cost complexity theorem that, under this multilevel interpolator, establishes a bound on the computation cost (for training data simulation) needed to achieve a prediction bound of [math]. This result provides novel insights on conditions under which the proposed multifidelity approach improves upon a conventional RKHS interpolator which relies on a single fidelity level. Finally, we demonstrate the effectiveness of stacking designs in a suite of simulation experiments and an application to finite element analysis.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140107583","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}
Guillaume Chennetier, Hassane Chraibi, Anne Dutfoy, Josselin Garnier
SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 128-156, March 2024. Abstract. Piecewise deterministic Markov processes (PDMPs) can be used to model complex dynamical industrial systems. The counterpart of this modeling capability is their simulation cost, which makes reliability assessment untractable with standard Monte Carlo methods. A significant variance reduction can be obtained with an adaptive importance sampling method based on a cross-entropy procedure. The success of this method relies on the selection of a good family of approximations of the committor function of the PDMP. In this paper original families are proposed. Their forms are based on reliability concepts related to fault tree analysis: minimal path sets and minimal cut sets. They are well adapted to high-dimensional industrial systems. The proposed method is discussed in detail and applied to academic systems and to a realistic system from the nuclear industry.
{"title":"Adaptive Importance Sampling Based on Fault Tree Analysis for Piecewise Deterministic Markov Process","authors":"Guillaume Chennetier, Hassane Chraibi, Anne Dutfoy, Josselin Garnier","doi":"10.1137/22m1522838","DOIUrl":"https://doi.org/10.1137/22m1522838","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 128-156, March 2024. <br/> Abstract. Piecewise deterministic Markov processes (PDMPs) can be used to model complex dynamical industrial systems. The counterpart of this modeling capability is their simulation cost, which makes reliability assessment untractable with standard Monte Carlo methods. A significant variance reduction can be obtained with an adaptive importance sampling method based on a cross-entropy procedure. The success of this method relies on the selection of a good family of approximations of the committor function of the PDMP. In this paper original families are proposed. Their forms are based on reliability concepts related to fault tree analysis: minimal path sets and minimal cut sets. They are well adapted to high-dimensional industrial systems. The proposed method is discussed in detail and applied to academic systems and to a realistic system from the nuclear industry.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140053756","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 12, Issue 1, Page 101-127, March 2024. Abstract. In this work, we develop a multifidelity Bayesian experimental design framework to efficiently quantify the rare-event statistics of an input-to-response (ItR) system with given input probability and expensive function evaluations. The key idea here is to leverage low-fidelity samples whose responses can be computed with a cost of a certain fraction of that for high-fidelity samples, in an optimized configuration to reduce the total computational cost. To accomplish this goal, we employ a multifidelity Gaussian process as the surrogate model of the ItR function and develop a new acquisition based on which the optimized next sample can be selected in terms of its location in the sample space and the fidelity level. In addition, we develop an inexpensive analytical evaluation of the acquisition and its derivative, avoiding numerical integrations that are prohibitive for high-dimensional problems. The new method is mainly tested in a bifidelity context for a series of synthetic problems with varying dimensions, low-fidelity model accuracy, and computational costs. Compared with the single-fidelity method and the bifidelity method with a predefined fidelity hierarchy, our method consistently shows the best (or among the best) performance for all the test cases. Finally, we demonstrate the superiority of our method in solving an engineering problem of estimating rare-event statistics of ship motion in irregular waves, using computational fluid dynamics with two different grid resolutions as the high- and low-fidelity models.
{"title":"Multifidelity Bayesian Experimental Design to Quantify Rare-Event Statistics","authors":"Xianliang Gong, Yulin Pan","doi":"10.1137/22m1503956","DOIUrl":"https://doi.org/10.1137/22m1503956","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 101-127, March 2024. <br/> Abstract. In this work, we develop a multifidelity Bayesian experimental design framework to efficiently quantify the rare-event statistics of an input-to-response (ItR) system with given input probability and expensive function evaluations. The key idea here is to leverage low-fidelity samples whose responses can be computed with a cost of a certain fraction of that for high-fidelity samples, in an optimized configuration to reduce the total computational cost. To accomplish this goal, we employ a multifidelity Gaussian process as the surrogate model of the ItR function and develop a new acquisition based on which the optimized next sample can be selected in terms of its location in the sample space and the fidelity level. In addition, we develop an inexpensive analytical evaluation of the acquisition and its derivative, avoiding numerical integrations that are prohibitive for high-dimensional problems. The new method is mainly tested in a bifidelity context for a series of synthetic problems with varying dimensions, low-fidelity model accuracy, and computational costs. Compared with the single-fidelity method and the bifidelity method with a predefined fidelity hierarchy, our method consistently shows the best (or among the best) performance for all the test cases. Finally, we demonstrate the superiority of our method in solving an engineering problem of estimating rare-event statistics of ship motion in irregular waves, using computational fluid dynamics with two different grid resolutions as the high- and low-fidelity models.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140002037","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 12, Issue 1, Page 69-100, March 2024. Abstract. Variational inference is an approximation framework for Bayesian inference that seeks to improve quantified uncertainty in predictions by optimizing a simplified distribution over parameters to stand in for the full posterior. Capturing model variations that remain consistent with training data enables more robust predictions by reducing parameter sensitivity. This work introduces a fixed-point optimization for variational inference that is applicable when every feasible log density can be expressed as a linear combination of functions from a given basis. In such cases, the optimizer becomes a fixed-point of projective integral updates. When the basis spans univariate quadratics in each parameter, the feasible distributions are Gaussian mean-fields and the projective integral updates yield quasi-Newton variational Bayes (QNVB). Other bases and updates are also possible. Since these updates require high-dimensional integration, this work begins by proposing an efficient quasirandom sequence of quadratures for mean-field distributions. Each iterate of the sequence contains two evaluation points that combine to correctly integrate all univariate quadratic functions and, if the mean-field factors are symmetric, all univariate cubics. More importantly, averaging results over short subsequences achieves periodic exactness on a much larger space of multivariate polynomials of quadratic total degree. The corresponding variational updates require four loss evaluations with standard (not second-order) backpropagation to eliminate error terms from over half of all multivariate quadratic basis functions. This integration technique is motivated by first proposing stochastic blocked mean-field quadratures, which may be useful in other contexts. A PyTorch implementation of QNVB allows for better control over model uncertainty during training than competing methods. Experiments demonstrate superior generalizability for multiple learning problems and architectures.
{"title":"Projective Integral Updates for High-Dimensional Variational Inference","authors":"Jed A. Duersch","doi":"10.1137/22m1529919","DOIUrl":"https://doi.org/10.1137/22m1529919","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 69-100, March 2024. <br/> Abstract. Variational inference is an approximation framework for Bayesian inference that seeks to improve quantified uncertainty in predictions by optimizing a simplified distribution over parameters to stand in for the full posterior. Capturing model variations that remain consistent with training data enables more robust predictions by reducing parameter sensitivity. This work introduces a fixed-point optimization for variational inference that is applicable when every feasible log density can be expressed as a linear combination of functions from a given basis. In such cases, the optimizer becomes a fixed-point of projective integral updates. When the basis spans univariate quadratics in each parameter, the feasible distributions are Gaussian mean-fields and the projective integral updates yield quasi-Newton variational Bayes (QNVB). Other bases and updates are also possible. Since these updates require high-dimensional integration, this work begins by proposing an efficient quasirandom sequence of quadratures for mean-field distributions. Each iterate of the sequence contains two evaluation points that combine to correctly integrate all univariate quadratic functions and, if the mean-field factors are symmetric, all univariate cubics. More importantly, averaging results over short subsequences achieves periodic exactness on a much larger space of multivariate polynomials of quadratic total degree. The corresponding variational updates require four loss evaluations with standard (not second-order) backpropagation to eliminate error terms from over half of all multivariate quadratic basis functions. This integration technique is motivated by first proposing stochastic blocked mean-field quadratures, which may be useful in other contexts. A PyTorch implementation of QNVB allows for better control over model uncertainty during training than competing methods. Experiments demonstrate superior generalizability for multiple learning problems and architectures.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139766388","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 12, Issue 1, Page 30-68, March 2024. Abstract. This paper analyzes a popular computational framework for solving infinite-dimensional Bayesian inverse problems, discretizing the prior and the forward model in a finite-dimensional weighted inner product space. We demonstrate the benefit of working on a weighted space by establishing operator-norm bounds for finite element and graph-based discretizations of Matérn-type priors and deconvolution forward models. For linear-Gaussian inverse problems, we develop a general theory for characterizing the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and maximum a posteriori (MAP) estimators for nonlinear inverse problems. Our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh refinement.
{"title":"Analysis of a Computational Framework for Bayesian Inverse Problems: Ensemble Kalman Updates and MAP Estimators under Mesh Refinement","authors":"Daniel Sanz-Alonso, Nathan Waniorek","doi":"10.1137/23m1567035","DOIUrl":"https://doi.org/10.1137/23m1567035","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 30-68, March 2024. <br/> Abstract. This paper analyzes a popular computational framework for solving infinite-dimensional Bayesian inverse problems, discretizing the prior and the forward model in a finite-dimensional weighted inner product space. We demonstrate the benefit of working on a weighted space by establishing operator-norm bounds for finite element and graph-based discretizations of Matérn-type priors and deconvolution forward models. For linear-Gaussian inverse problems, we develop a general theory for characterizing the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and maximum a posteriori (MAP) estimators for nonlinear inverse problems. Our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh refinement.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139669988","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 12, Issue 1, Page 1-29, March 2024. Abstract. In this paper, we consider the numerical solution of a nonlinear Schrödinger equation with spatial random potential. The randomly shifted quasi-Monte Carlo (QMC) lattice rule combined with the time-splitting pseudospectral discretization is applied and analyzed. The nonlinearity in the equation induces difficulties in estimating the regularity of the solution in random space. By the technique of weighted Sobolev space, we identify the possible weights and show the existence of QMC that converges optimally at the almost-linear rate without dependence on dimensions. The full error estimate of the scheme is established. We present numerical results to verify the accuracy and investigate the wave propagation.
{"title":"Error Estimate of a Quasi-Monte Carlo Time-Splitting Pseudospectral Method for Nonlinear Schrödinger Equation with Random Potentials","authors":"Zhizhang Wu, Zhiwen Zhang, Xiaofei Zhao","doi":"10.1137/22m1525181","DOIUrl":"https://doi.org/10.1137/22m1525181","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 1-29, March 2024. <br/> Abstract. In this paper, we consider the numerical solution of a nonlinear Schrödinger equation with spatial random potential. The randomly shifted quasi-Monte Carlo (QMC) lattice rule combined with the time-splitting pseudospectral discretization is applied and analyzed. The nonlinearity in the equation induces difficulties in estimating the regularity of the solution in random space. By the technique of weighted Sobolev space, we identify the possible weights and show the existence of QMC that converges optimally at the almost-linear rate without dependence on dimensions. The full error estimate of the scheme is established. We present numerical results to verify the accuracy and investigate the wave propagation.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139580631","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}
Suraj Yerramilli, Akshay Iyer, Wei Chen, Daniel W. Apley
SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 4, Page 1357-1381, December 2023. Abstract. Real engineering and scientific applications often involve one or more qualitative inputs. Standard Gaussian processes (GPs), however, cannot directly accommodate qualitative inputs. The recently introduced latent variable Gaussian process (LVGP) overcomes this issue by first mapping each qualitative factor to underlying latent variables (LVs) and then uses any standard GP covariance function over these LVs. The LVs are estimated similarly to the other GP hyperparameters through maximum likelihood estimation and then plugged into the prediction expressions. However, this plug-in approach will not account for uncertainty in estimation of the LVs, which can be significant especially with limited training data. In this work, we develop a fully Bayesian approach for the LVGP model and for visualizing the effects of the qualitative inputs via their LVs. We also develop approximations for scaling up LVGPs and fully Bayesian inference for the LVGP hyperparameters. We conduct numerical studies comparing plug-in inference against fully Bayesian inference over a few engineering models and material design applications. In contrast to previous studies on standard GP modeling that have largely concluded that a fully Bayesian treatment offers limited improvements, our results show that for LVGP modeling it offers significant improvements in prediction accuracy and uncertainty quantification over the plug-in approach.
{"title":"Fully Bayesian Inference for Latent Variable Gaussian Process Models","authors":"Suraj Yerramilli, Akshay Iyer, Wei Chen, Daniel W. Apley","doi":"10.1137/22m1525600","DOIUrl":"https://doi.org/10.1137/22m1525600","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 4, Page 1357-1381, December 2023. <br/> Abstract. Real engineering and scientific applications often involve one or more qualitative inputs. Standard Gaussian processes (GPs), however, cannot directly accommodate qualitative inputs. The recently introduced latent variable Gaussian process (LVGP) overcomes this issue by first mapping each qualitative factor to underlying latent variables (LVs) and then uses any standard GP covariance function over these LVs. The LVs are estimated similarly to the other GP hyperparameters through maximum likelihood estimation and then plugged into the prediction expressions. However, this plug-in approach will not account for uncertainty in estimation of the LVs, which can be significant especially with limited training data. In this work, we develop a fully Bayesian approach for the LVGP model and for visualizing the effects of the qualitative inputs via their LVs. We also develop approximations for scaling up LVGPs and fully Bayesian inference for the LVGP hyperparameters. We conduct numerical studies comparing plug-in inference against fully Bayesian inference over a few engineering models and material design applications. In contrast to previous studies on standard GP modeling that have largely concluded that a fully Bayesian treatment offers limited improvements, our results show that for LVGP modeling it offers significant improvements in prediction accuracy and uncertainty quantification over the plug-in approach.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138567754","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}
Seif Ben Bader, Helmut Harbrecht, Rolf Krause, Michael D. Multerer, Alessio Quaglino, Marc Schmidlin
SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 4, Page 1329-1356, December 2023. Abstract. We present a novel approach which aims at high-performance uncertainty quantification for cardiac electrophysiology simulations. Employing the monodomain equation to model the transmembrane potential inside the cardiac cells, we evaluate the effect of spatially correlated perturbations of the heart fibers on the statistics of the resulting quantities of interest. Our methodology relies on a close integration of multilevel quadrature methods, parallel iterative solvers, and space-time finite element discretizations, allowing for a fully parallelized framework in space, time, and stochastics. Extensive numerical studies are presented to evaluate convergence rates and to compare the performance of classical Monte Carlo methods such as standard Monte Carlo (MC) and quasi-Monte Carlo (QMC), as well as multilevel strategies, i.e., multilevel Monte Carlo (MLMC) and multilevel quasi-Monte Carlo (MLQMC) on hierarchies of nested meshes. We especially also employ a recently suggested variant of the multilevel approach for nonnested meshes to deal with a realistic heart geometry.
{"title":"Space-time Multilevel Quadrature Methods and their Application for Cardiac Electrophysiology","authors":"Seif Ben Bader, Helmut Harbrecht, Rolf Krause, Michael D. Multerer, Alessio Quaglino, Marc Schmidlin","doi":"10.1137/21m1418320","DOIUrl":"https://doi.org/10.1137/21m1418320","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 4, Page 1329-1356, December 2023. <br/> Abstract. We present a novel approach which aims at high-performance uncertainty quantification for cardiac electrophysiology simulations. Employing the monodomain equation to model the transmembrane potential inside the cardiac cells, we evaluate the effect of spatially correlated perturbations of the heart fibers on the statistics of the resulting quantities of interest. Our methodology relies on a close integration of multilevel quadrature methods, parallel iterative solvers, and space-time finite element discretizations, allowing for a fully parallelized framework in space, time, and stochastics. Extensive numerical studies are presented to evaluate convergence rates and to compare the performance of classical Monte Carlo methods such as standard Monte Carlo (MC) and quasi-Monte Carlo (QMC), as well as multilevel strategies, i.e., multilevel Monte Carlo (MLMC) and multilevel quasi-Monte Carlo (MLQMC) on hierarchies of nested meshes. We especially also employ a recently suggested variant of the multilevel approach for nonnested meshes to deal with a realistic heart geometry.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512777","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}
Sébastien J. Petit, Julien Bect, Paul Feliot, Emmanuel Vazquez
SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 4, Page 1308-1328, December 2023. Abstract. This article revisits the fundamental problem of parameter selection for Gaussian process interpolation. By choosing the mean and the covariance functions of a Gaussian process within parametric families, the user obtains a family of Bayesian procedures to perform predictions about the unknown function and must choose a member of the family that will hopefully provide good predictive performances. We base our study on the general concept of scoring rules, which provides an effective framework for building leave-one-out selection and validation criteria and a notion of extended likelihood criteria based on an idea proposed by Fasshauer et al. [“Optimal” scaling and stable computation of meshfree kernel methods, 2009], which makes it possible to recover standard selection criteria, such as the generalized cross-validation criterion. Under this setting, we empirically show on several test problems of the literature that the choice of an appropriate family of models is often more important than the choice of a particular selection criterion (e.g., the likelihood versus a leave-one-out selection criterion). Moreover, our numerical results show that the regularity parameter of a Matérn covariance can be selected effectively by most selection criteria.
{"title":"Parameter Selection in Gaussian Process Interpolation: An Empirical Study of Selection Criteria","authors":"Sébastien J. Petit, Julien Bect, Paul Feliot, Emmanuel Vazquez","doi":"10.1137/21m1444710","DOIUrl":"https://doi.org/10.1137/21m1444710","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 4, Page 1308-1328, December 2023. <br/> Abstract. This article revisits the fundamental problem of parameter selection for Gaussian process interpolation. By choosing the mean and the covariance functions of a Gaussian process within parametric families, the user obtains a family of Bayesian procedures to perform predictions about the unknown function and must choose a member of the family that will hopefully provide good predictive performances. We base our study on the general concept of scoring rules, which provides an effective framework for building leave-one-out selection and validation criteria and a notion of extended likelihood criteria based on an idea proposed by Fasshauer et al. [“Optimal” scaling and stable computation of meshfree kernel methods, 2009], which makes it possible to recover standard selection criteria, such as the generalized cross-validation criterion. Under this setting, we empirically show on several test problems of the literature that the choice of an appropriate family of models is often more important than the choice of a particular selection criterion (e.g., the likelihood versus a leave-one-out selection criterion). Moreover, our numerical results show that the regularity parameter of a Matérn covariance can be selected effectively by most selection criteria.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512773","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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