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

对该项目更广泛的意义和重要性的非技术性解释：基于模型的实际应用预测将在预测和决策中发挥核心作用，例如，在环境和气候变化应用中，预测冰盖对海平面上升的贡献，或在能源应用中，预测故障和评估电网的动态稳定性。然而，由于模型的不确定输入(例如，系数场、本构定律、源项、几何以及初始和/或边界条件)以及噪声和有限的观测，模型通常受到相当大的不确定性。虽然这些输入量中的许多不能直接观察或测量，但它们可以从观测中推断出来，例如冰盖中的冰面速度。这通常会导致一个极具挑战性的数学问题。该项目旨在通过推断来传播来自数据/观测的不确定性，以进行预测，并提高复杂物理系统的可预测性。选定的用于研究和教育活动的驾驶应用程序(即冰盖模型)捕捉到了重要的一般性和复杂的算法挑战，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。这些算法、数学发现和开放源代码将通过同行评议的期刊论文以及在会议和研讨会上的演讲来分享。项目技术描述：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中的不确定性。这为不确定情况下的预测和决策打开了更先进的能力之门。然而，贝叶斯反演的算法发展面临着几个挑战。例如，表征参数或预测的后验分布不可避免地需要重复评估(可能)由微分方程式控制的大规模和复杂的正演模型。此外，后验分布具有复杂的结构，这源于可能存在的非线性正演模型和不同来源的数据。为了克服这些计算挑战，必须利用问题结构(例如，数据对参数的导数和局部敏感度)。这项提议的目的是在解决由微分方程控制的大规模贝叶斯反问题的数学和计算障碍方面进行探索性工作。在统计推理的背景下发展数学上严格的、计算上有效的和健壮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和她的学生将致力于以下垂直整合的研究领域：(I)大规模反问题的可伸缩算法(这里的重点将是反问题的二阶导数(即，海森)近似，以及为不精确的牛顿-克里洛夫系统开发有效的预条件，以提高逆求解器的计算效率)，以及(Ii)高维的不确定性量化(这里的重点将是建立基于海森和降阶模型的方法，用于高效的高维后验探索)。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。

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.