Ömer Deniz Akyildiz, Connor Duffin, Sotirios Sabanis, Mark Girolami
SIAM/ASA Journal on Uncertainty Quantification, Volume 10, Issue 4, Page 1560-1585, December 2022. Abstract. The recent statistical finite element method (statFEM) provides a coherent statistical framework to synthesize finite element models with observed data. Through embedding uncertainty inside of the governing equations, finite element solutions are updated to give a posterior distribution which quantifies all sources of uncertainty associated with the model. However to incorporate all sources of uncertainty, one must integrate over the uncertainty associated with the model parameters, the known forward problem of uncertainty quantification. In this paper, we make use of Langevin dynamics to solve the statFEM forward problem, studying the utility of the unadjusted Langevin algorithm (ULA), a Metropolis-free Markov chain Monte Carlo sampler, to build a sample-based characterization of this otherwise intractable measure. Due to the structure of the statFEM problem, these methods are able to solve the forward problem without explicit full PDE solves, requiring only sparse matrix-vector products. ULA is also gradient-based, and hence provides a scalable approach up to high degrees-of-freedom. Leveraging the theory behind Langevin-based samplers, we provide theoretical guarantees on sampler performance, demonstrating convergence, for both the prior and posterior, in the Kullback–Leibler divergence and in Wasserstein-2, with further results on the effect of preconditioning. Numerical experiments are also provided, to demonstrate the efficacy of the sampler, with a Python package also included.
{"title":"Statistical Finite Elements via Langevin Dynamics","authors":"Ömer Deniz Akyildiz, Connor Duffin, Sotirios Sabanis, Mark Girolami","doi":"10.1137/21m1463094","DOIUrl":"https://doi.org/10.1137/21m1463094","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 10, Issue 4, Page 1560-1585, December 2022. <br/> Abstract. The recent statistical finite element method (statFEM) provides a coherent statistical framework to synthesize finite element models with observed data. Through embedding uncertainty inside of the governing equations, finite element solutions are updated to give a posterior distribution which quantifies all sources of uncertainty associated with the model. However to incorporate all sources of uncertainty, one must integrate over the uncertainty associated with the model parameters, the known forward problem of uncertainty quantification. In this paper, we make use of Langevin dynamics to solve the statFEM forward problem, studying the utility of the unadjusted Langevin algorithm (ULA), a Metropolis-free Markov chain Monte Carlo sampler, to build a sample-based characterization of this otherwise intractable measure. Due to the structure of the statFEM problem, these methods are able to solve the forward problem without explicit full PDE solves, requiring only sparse matrix-vector products. ULA is also gradient-based, and hence provides a scalable approach up to high degrees-of-freedom. Leveraging the theory behind Langevin-based samplers, we provide theoretical guarantees on sampler performance, demonstrating convergence, for both the prior and posterior, in the Kullback–Leibler divergence and in Wasserstein-2, with further results on the effect of preconditioning. Numerical experiments are also provided, to demonstrate the efficacy of the sampler, with a Python package also included.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512784","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 10, Issue 4, Page 1629-1651, December 2022. Abstract. The numerical solution of risk-averse optimization problems constrained by PDEs requires substantial computational effort resulting from the discretization of the underlying PDE in both the physical and stochastic dimensions. To practically solve these challenging optimization problems, one must intelligently manage the individual discretization fidelities throughout the optimization iteration. In this work, we combine an inexact trust-region algorithm with the recently developed local reduced-basis approximation to efficiently solve risk-averse optimization problems with PDE constraints. The main contribution of this work is a numerical framework for systematically constructing surrogate models for the trust-region subproblem and the objective function using local reduced-basis approximations. We demonstrate the effectiveness of our approach through several numerical examples.
SIAM/ASA Journal on Uncertainty quantitation, vol . 10, Issue 4, Page 1629-1651, December 2022。摘要。受偏微分方程约束的风险规避优化问题的数值解需要大量的计算量，这是由于底层偏微分方程在物理和随机两个维度上的离散化造成的。为了实际解决这些具有挑战性的优化问题，必须在整个优化迭代过程中智能地管理单个离散化保真度。在这项工作中，我们将一种不精确的信任域算法与最近发展的局部约基近似相结合，以有效地解决具有PDE约束的风险规避优化问题。该工作的主要贡献是一个数值框架，用于系统地构建信任域子问题和目标函数的代理模型，并使用局部约基近似。通过几个数值算例证明了该方法的有效性。
{"title":"A Locally Adapted Reduced-Basis Method for Solving Risk-Averse PDE-Constrained Optimization Problems","authors":"Zilong Zou, Drew P. Kouri, Wilkins Aquino","doi":"10.1137/21m1411342","DOIUrl":"https://doi.org/10.1137/21m1411342","url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 10, Issue 4, Page 1629-1651, December 2022. <br/> Abstract. The numerical solution of risk-averse optimization problems constrained by PDEs requires substantial computational effort resulting from the discretization of the underlying PDE in both the physical and stochastic dimensions. To practically solve these challenging optimization problems, one must intelligently manage the individual discretization fidelities throughout the optimization iteration. In this work, we combine an inexact trust-region algorithm with the recently developed local reduced-basis approximation to efficiently solve risk-averse optimization problems with PDE constraints. The main contribution of this work is a numerical framework for systematically constructing surrogate models for the trust-region subproblem and the objective function using local reduced-basis approximations. We demonstrate the effectiveness of our approach through several numerical examples.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512755","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}
We propose a thorough comparison of polynomial chaos expansion (PCE) for indicator functions of the form 1 c ≤ X for some threshold parameter c ∈ R and a random variable X associated with classical orthogonal polynomials. We provide tight global and localized L 2 estimates for the resulting truncation of the PCE and numerical experiments support the tightness of the error estimates. We also compare the theoretical and numerical accuracy of PCE when extra quantile/probability transforms are applied, revealing different optimal choices according to the value of c in the center and the tails of the distribution of X .
{"title":"A Comparative Study of Polynomial-Type Chaos Expansions for Indicator Functions","authors":"Florian Bourgey, E. Gobet, C. Rey","doi":"10.1137/21m1413146","DOIUrl":"https://doi.org/10.1137/21m1413146","url":null,"abstract":"We propose a thorough comparison of polynomial chaos expansion (PCE) for indicator functions of the form 1 c ≤ X for some threshold parameter c ∈ R and a random variable X associated with classical orthogonal polynomials. We provide tight global and localized L 2 estimates for the resulting truncation of the PCE and numerical experiments support the tightness of the error estimates. We also compare the theoretical and numerical accuracy of PCE when extra quantile/probability transforms are applied, revealing different optimal choices according to the value of c in the center and the tails of the distribution of X .","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80794660","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":"Corrigendum: \"Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization\"","authors":"D. Kouri, T. Surowiec","doi":"10.1137/21m143251x","DOIUrl":"https://doi.org/10.1137/21m143251x","url":null,"abstract":"","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85580948","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}
. Spotlight mode airborne synthetic aperture radar (SAR) is a coherent imaging modality that is an 5 important tool in remote sensing. Existing methods for spotlight SAR image reconstruction from 6 phase history data typically produce a single image estimate which approximates the reflectivity 7 of an unknown ground scene, and therefore provide no quantification of the certainty with which 8 the estimate can be trusted. In addition, speckle affects all coherent imaging modalities causing a 9 degradation of image quality. Many point estimate image reconstruction methods incorrectly treat 10 speckle as additive noise resulting in an unnatural smoothing of the speckle that also reduces image 11 contrast. The purpose of this paper is to address the issues of speckle and uncertainty quantification 12 by introducing a sampling-based approach to SAR image reconstruction directly from phase history 13 data. In particular, a statistical model for speckle as well as a corresponding sparsity technique to 14 reduce it are directly incorporated into the model. Rather than a single point estimate, samples 15 of the resulting joint posterior density are efficiently obtained using a Gibbs sampler, which are in 16 turn used to derive estimates and other statistics which aid in uncertainty quantification. The latter 17 information is particularly important in SAR, where ground truth images even for synthetically-18 created examples are typically unknown. While similar methods have been deployed to process 19 formed images, this paper focuses on the integration of these techniques into image reconstruction 20 from phase history data. An example result using real-world data shows that, when compared with 21 existing methods, the sampling-based approach introduced provides parameter-free estimates with 22 improved contrast and significantly reduced speckle, as well as uncertainty quantification information. 23
{"title":"Sampling-based Spotlight SAR Image Reconstruction from Phase History Data for Speckle Reduction and Uncertainty Quantification","authors":"V. Churchill, A. Gelb","doi":"10.1137/20m1379721","DOIUrl":"https://doi.org/10.1137/20m1379721","url":null,"abstract":". Spotlight mode airborne synthetic aperture radar (SAR) is a coherent imaging modality that is an 5 important tool in remote sensing. Existing methods for spotlight SAR image reconstruction from 6 phase history data typically produce a single image estimate which approximates the reflectivity 7 of an unknown ground scene, and therefore provide no quantification of the certainty with which 8 the estimate can be trusted. In addition, speckle affects all coherent imaging modalities causing a 9 degradation of image quality. Many point estimate image reconstruction methods incorrectly treat 10 speckle as additive noise resulting in an unnatural smoothing of the speckle that also reduces image 11 contrast. The purpose of this paper is to address the issues of speckle and uncertainty quantification 12 by introducing a sampling-based approach to SAR image reconstruction directly from phase history 13 data. In particular, a statistical model for speckle as well as a corresponding sparsity technique to 14 reduce it are directly incorporated into the model. Rather than a single point estimate, samples 15 of the resulting joint posterior density are efficiently obtained using a Gibbs sampler, which are in 16 turn used to derive estimates and other statistics which aid in uncertainty quantification. The latter 17 information is particularly important in SAR, where ground truth images even for synthetically-18 created examples are typically unknown. While similar methods have been deployed to process 19 formed images, this paper focuses on the integration of these techniques into image reconstruction 20 from phase history data. An example result using real-world data shows that, when compared with 21 existing methods, the sampling-based approach introduced provides parameter-free estimates with 22 improved contrast and significantly reduced speckle, as well as uncertainty quantification information. 23","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73260717","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}
We propose to study quantile oriented sensitivity indices (QOSA indices) and quantile oriented Shapley effects (QOSE). Some theoretical properties of QOSA indices will be given and several calculations of QOSA indices and QOSE will allow to better understand the behaviour and the interest of these indices.
{"title":"Goal-Oriented Shapley Effects with Special Attention to the Quantile-Oriented Case","authors":"Kevin Elie-Dit-Cosaque, V. Maume-Deschamps","doi":"10.1137/21m1395247","DOIUrl":"https://doi.org/10.1137/21m1395247","url":null,"abstract":"We propose to study quantile oriented sensitivity indices (QOSA indices) and quantile oriented Shapley effects (QOSE). Some theoretical properties of QOSA indices will be given and several calculations of QOSA indices and QOSE will allow to better understand the behaviour and the interest of these indices.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75351884","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":"Objective Frequentist Uncertainty Quantification for Atmospheric (mathrm{CO}_2) Retrievals","authors":"Pratik V. Patil, Mikael Kuusela, J. Hobbs","doi":"10.1137/20m1356403","DOIUrl":"https://doi.org/10.1137/20m1356403","url":null,"abstract":"","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79285172","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":"Asymptotic Theory of (boldsymbol ell _1) -Regularized PDE Identification from a Single Noisy Trajectory","authors":"Yuchen He, Namjoon Suh, X. Huo, S. Kang, Y. Mei","doi":"10.1137/21m1398884","DOIUrl":"https://doi.org/10.1137/21m1398884","url":null,"abstract":"","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74831687","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}
As the scale of problems and data used for experimental design, signal processing and data assimilation grow, the oft-occuring least squares subproblems are correspondingly growing in size. As the scale of these least squares problems creates prohibitive memory movement costs for the usual incremental QR and Krylov-based algorithms, randomized least squares problems are garnering more attention. However, these randomized least squares solvers are difficult to integrate application algorithms as their uncertainty limits practical tracking of algorithmic progress and reliable stopping. Accordingly, in this work, we develop theoretically-rigorous, practical tools for quantifying the uncertainty of an important class of iterative randomized least squares algorithms, which we then use to track algorithmic progress and create a stopping condition. We demonstrate the effectiveness of our algorithm by solving a 0.78 TB least squares subproblem from the inner loop of incremental 4D-Var using only 195 MB of memory.
{"title":"Towards Practical Large-Scale Randomized Iterative Least Squares Solvers through Uncertainty Quantification","authors":"Nathaniel Pritchard, V. Patel","doi":"10.1137/22m1515057","DOIUrl":"https://doi.org/10.1137/22m1515057","url":null,"abstract":"As the scale of problems and data used for experimental design, signal processing and data assimilation grow, the oft-occuring least squares subproblems are correspondingly growing in size. As the scale of these least squares problems creates prohibitive memory movement costs for the usual incremental QR and Krylov-based algorithms, randomized least squares problems are garnering more attention. However, these randomized least squares solvers are difficult to integrate application algorithms as their uncertainty limits practical tracking of algorithmic progress and reliable stopping. Accordingly, in this work, we develop theoretically-rigorous, practical tools for quantifying the uncertainty of an important class of iterative randomized least squares algorithms, which we then use to track algorithmic progress and create a stopping condition. We demonstrate the effectiveness of our algorithm by solving a 0.78 TB least squares subproblem from the inner loop of incremental 4D-Var using only 195 MB of memory.","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42273518","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":"Risk-Adapted Optimal Experimental Design","authors":"D. Kouri, J. Jakeman, J. G. Huerta","doi":"10.1137/20m1357615","DOIUrl":"https://doi.org/10.1137/20m1357615","url":null,"abstract":"","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77042544","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号