CAREER: Large-Scale Bayesian Inverse Problems Governed by Differential and Differential-Algebraic Equations

职业：微分方程和微分代数方程控制的大规模贝叶斯逆问题

• 批准号: 1654311
• 负责人: Noemi Petra
• 金额: $ 40万
• 依托单位: University of California - Merced
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1654311&HistoricalAwards=false
• 关键词: CAREER; Large; Scale; Bayesian; Inverse

Nontechnical explanation of the project's broader significance and importance: Model-based projections of real life applications will play a central role in prediction and decision-making, in environment and climate change applications, for instance, to anticipate ice sheet contribution to sea level rise, or in the context of energy applications, to predict faults and assess dynamic stability in a power grid. However, models are typically subject to considerable uncertainties stemming from uncertain inputs to the model (e.g., coefficient fields, constitutive laws, source terms, geometries, and initial and/or boundary conditions) as well as from noisy and limited observations. While many of these input quantities cannot be directly observed or measured, they can be inferred from observations, such as those of ice surface velocities in ice sheets. This typically leads to an extremely challenging mathematical problem. This project aims to enable the propagation of uncertainties from data/observations through inference to prediction and increase predictability of complex physical systems. The selected driving application (i.e., the ice sheet model) for research and education activities, capture important general and complex algorithmic challenges such as large-scale, nonlinearity, time-dependence, and ill-posedness. The research will be, therefore, applicable to a broader spectrum of problems. The algorithms, mathematical findings and open source codes will be shared through peer reviewed journal papers, and presentations at conferences and workshops. Technical description of the project: Bayesian inversion facilitates the integration of data with complex physics-based models to quantify and reduce uncertainties in model predictions. This opens the door to more advanced capabilities for prediction and decision-making under uncertainty. However, the algorithmic developments for Bayesian inversion are subject to several challenges. For instance, characterizing the posterior distributions of parameters or predictions inevitably requires repeated evaluations of (possibly) large-scale and complex forward models governed by differential equations. In addition, the posterior distribution has a complex structure stemming from the presence of possibly nonlinear forward models and heterogeneous sources of data. To overcome these computational challenges, it is essential to exploit problem structure (e.g., derivatives and local sensitivity of the data with respect to parameters). The objectives of this proposal is to conduct exploratory work in addressing the mathematical and computational barriers in solving large-scale Bayesian inverse problems governed by differential equations. Developing mathematically rigorous and computationally efficient and robust methods in the context of statistical inference has the potential of transformative research in the field of modern computational inverse problems. In particular, the PI and her student will work on the following vertically-integrated research areas: (i) scalable algorithms for large-scale inverse problems (here the focus will be on second derivative (i.e., Hessian) approximations for inverse problems and on developing efficient preconditioners for inexact Newton-Krylov systems to increase the computational efficiency of inverse solvers), and (ii) uncertainty quantification in high dimensions (here the focus will be on building Hessian- and reduced order model-based methods for efficient posterior exploration in high dimensions). The proposed research requires an interdisciplinary perspective, namely it brings together applied mathematics, scientific computing and statistics.

该项目更广泛意义和重要性的非技术解释：基于模型的现实生活应用预测将在环境和气候变化应用的预测和决策中发挥核心作用，例如，预测冰盖对海平面上升的贡献，或者在能源应用中，预测故障并评估电网的动态稳定性。 然而，模型通常会受到相当大的不确定性的影响，这些不确定性源于模型的不确定输入（例如系数场、本构定律、源项、几何形状以及初始和/或边界条件）以及噪声和有限的观测。虽然许多这些输入量无法直接观察或测量，但可以从观察中推断出它们，例如冰盖中的冰表面速度。这通常会导致极具挑战性的数学问题。该项目旨在通过推理到预测来实现数据/观测的不确定性传播，并提高复杂物理系统的可预测性。 用于研究和教育活动的所选驱动应用程序（即冰盖模型）捕获了重要的通用和复杂算法挑战，例如大规模、非线性、时间依赖性和不适定性。因此，这项研究将适用于更广泛的问题。算法、数学发现和开源代码将通过同行评审的期刊论文以及会议和研讨会上的演示来共享。项目的技术描述：贝叶斯反演促进了数据与复杂的基于物理的模型的集成，以量化和减少模型预测的不确定性。这为在不确定性下进行更先进的预测和决策能力打开了大门。 然而，贝叶斯反演的算法开发面临着一些挑战。例如，表征参数或预测的后验分布不可避免地需要对（可能）由微分方程控制的大规模且复杂的前向模型进行重复评估。 此外，由于可能存在非线性前向模型和异构数据源，后验分布具有复杂的结构。为了克服这些计算挑战，必须利用问题结构（例如，数据相对于参数的导数和局部敏感性）。该提案的目标是开展探索性工作，解决解决由微分方程控制的大规模贝叶斯逆问题的数学和计算障碍。在统计推断的背景下开发数学严谨、计算高效且鲁棒的方法具有现代计算逆问题领域变革性研究的潜力。特别是，PI和她的学生将致力于以下垂直整合的研究领域：（i）大规模反问题的可扩展算法（这里的重点将是反问题的二阶导数（即Hessian）近似以及为不精确的 Newton-Krylov 系统开发高效的预处理器，以提高逆求解器的计算效率），以及 (ii) 高维不确定性量化（这里的重点是构建 Hessian 阶和降阶阶）基于模型的方法，用于高维度的有效后验探索）。拟议的研究需要跨学科的视角，即它将应用数学、科学计算和统计学结合在一起。

项目成果

Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field

• DOI: 10.1088/1361-6420/aad91e
• 发表时间: 2018-01
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: R. Nicholson;N. Petra;J. Kaipio

Optimal Design of Large-scale Bayesian Linear Inverse Problems Under Reducible Model Uncertainty: Good to Know What You Don't Know

可约模型不确定性下的大规模贝叶斯线性逆问题的优化设计：了解你不知道的知识是有好处的

• DOI: 10.1137/20m1347292
• 发表时间: 2021
• 期刊: SIAM/ASA Journal on Uncertainty Quantification
• 作者: Alexanderian, Alen;Petra, Noemi;Stadler, Georg;Sunseri, Isaac
• 通讯作者: Sunseri, Isaac

On the implementation of a quasi-Newton interior-point method for PDE-constrained optimization using finite element discretizations

• DOI: 10.1080/10556788.2022.2117354
• 发表时间: 2022-11
• 期刊: Optimization Methods and Software
• 影响因子: 2.2
• 作者: C. Petra;M. Troya;N. Petra;Youngsoo Choi;G. Oxberry;D. Tortorelli

Statistical Treatment of Inverse Problems Constrained by Differential Equations-Based Models with Stochastic Terms

• DOI: 10.1137/18m122073x
• 发表时间: 2018-10
• 期刊: SIAM/ASA J. Uncertain. Quantification
• 作者: E. Constantinescu;N. Petra;J. Bessac;C. Petra

Hierarchical off-diagonal low-rank approximation of Hessians in inverse problems, with application to ice sheet model initialization

反演问题中 Hessians 的分层非对角低秩逼近，及其在冰盖模型初始化中的应用

• DOI: 10.1088/1361-6420/acd719
• 发表时间: 2023
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: Hartland, Tucker;Stadler, Georg;Perego, Mauro;Liegeois, Kim;Petra, Noémi
• 通讯作者: Petra, Noémi