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; 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.

对该项目更广泛的重要性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预测和决策，环境和气候变化应用中发挥核心作用，例如，预测冰盖对海平面上升的贡献，或者在能源应用程序的背景下，预测故障并在电网中预测故障并评估动态稳定性。 但是，由于模型的不确定投入（例如，系数字段，本构定律，源术语，几何以及初始和/或边界条件）以及噪音和有限的观察结果，模型通常受到大量不确定性的影响。尽管无法直接观察或测量这些输入量中的许多，但可以从观察结果（例如冰层中的冰表面速度的观测值）推断出来。这通常会导致一个极具挑战性的数学问题。该项目旨在使数据/观察值通过推断到预测并提高复杂物理系统的可预测性来传播不确定性。 用于研究和教育活动的选定驾驶应用程序（即冰盖模型），捕获重要的一般和复杂的算法挑战，例如大规模，非线性，时间依赖性和不良性。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讲习班的演示共享。项目的技术描述：贝叶斯反演有助于将数据与基于复杂的物理模型的模型集成，以量化和减少模型预测中的不确定性。这为在不确定性下的预测和决策方面打开了更高级的能力。 但是，贝叶斯倒置的算法发展受到了一些挑战。例如，表征参数的后验分布或预测不可避免地需要重复评估（可能是）由微分方程控制的（可能）大规模和复杂的正向模型。 此外，后验分布具有复杂的结构，该结构可能是由于可能非线性正向模型和数据源的存在。为了克服这些计算挑战，必须利用问题结构（例如，数据对参数的衍生物和局部敏感性）至关重要。该提案的目标是在解决由微分方程控制的大规模贝叶斯逆问题时进行探索性工作。在统计推断的背景下，开发数学上严格且在计算上有效且鲁棒的方法具有在现代计算反问题领域进行变革性研究的潜力。特别是，PI和她的学生将研究以下垂直综合的研究领域：（i）大规模反问题的可伸缩算法（此处，重点将放在第二个导数（即Hessian）临界问题上，以及针对反向问题的近似问题，以及在牛顿量化的量化量不足的量化量的有效高度较高的II级别的高效高度效应，并增加计算效应的量相位效应，并增加了相反的效应。 （在这里，重点将是建立基于订单模型的Hessian和减少订单模型的方法，以在高维度中有效后探测）。拟议的研究需要跨学科的观点，即它汇集了应用数学，科学计算和统计数据。