Collaborative Research: Stochastic Approximations for the Solution and Uncertainty Analysis of Data-Intensive Inverse Problems

在从地球物理学和大气科学到医学成像和网络通信等科学领域，数据正在以惊人的速度产生。这些数据通常与感兴趣的数量间接相关，并且数据集在许多情况下是动态增长的。然后，从这些数据中提取所需的信息需要解决非常大的数据密集型反问题，可能是重复的和实时的。由于验证和不确定性分析的要求，获得这样的解决方案的计算挑战变得更加复杂，这很容易使计算变得令人望而却步。该项目将开发数学/统计方法和计算工具，用于解决数据密集型逆问题。该方法的核心是对此类问题的随机重新表述，旨在显著降低计算成本，同时适应现代硬件架构。将开发一个框架来解决大数据、逆问题、数据分析和不确定性量化之间的接口所产生的挑战。首先，将介绍求解线性和非线性逆问题的随机方法，以便有效的随机优化方法可以克服当前算法的硬件限制，并在近实时的情况下生成解和不确定性评估。新的理论和可扩展的方法将在随机框架内发展，从而确保解决方案的准确性，可靠性和鲁棒性。其次，将开发用于模型验证、误差分析和不确定度量化的先进工具。通过与应用科学家合作（例如在大气遥感方面），本项目开发的方法将对科学家和工程师具有直接的实际效用。

In scientific fields ranging from geophysics and atmospheric science to medical imaging and network communication, data are being generated at remarkable rates. Such data are typically indirectly related to quantities of interest and the data sets are in many cases dynamically growing. Extracting desired information from these data then requires the solution of very large data-intensive inverse problems, perhaps repeatedly and in real time. The computational challenges of obtaining such a solution are compounded by the demands of validation and uncertainty analysis, which can easily become computationally prohibitive. This project will develop mathematical/statistical methods and computational tools for the solution of data-intensive inverse problems. The core of this approach is a stochastic reformulation of such problems that aims to significantly reduce the computational costs while adapting to modern hardware architectures.A framework will be developed to address the challenges arising at the interface between big data, inverse problems, data analysis, and uncertainty quantification. First, randomized methods for the solution of linear and nonlinear inverse problems will be introduced, so that efficient stochastic optimization methods can be used to overcome the hardware limitations of current algorithms and to generate solutions and uncertainty assessments in near-real time. New theory and scalable methods will be developed within the stochastic framework, thereby ensuring solution accuracy, reliability, and robustness. Second, advanced tools will be developed for model validation, error analysis, and uncertainty quantification. By partnering with application scientists (e.g., in atmospheric remote sensing), methods developed in this project will be of immediate practical utility for scientists and engineers.