Propagation of Stochasticity in PDEs and Hybrid Inverse Problems

在应用科学中，系数具有高度异质性、分辨率差的方程比比皆是。该项目建议分析这些系数的统计特性对许多应用领域中感兴趣的数量的影响，如气候预测中的不确定性量化、地质盆地研究和复合材料制造。医学成像中的新方法通常被称为耦合物理成像模式(如弹性成像或光声断层成像)，它有可能在非常早期的阶段极大地增强我们检测不健康组织的能力。该提案将进一步加深对此类问题的数学理解，并与医学物理学家和生物工程师密切合作，为此类模型提供稳健的重建过程。本项目侧重于以下方面的理论和数学理解：(I)从非均匀系数到微分方程解的不确定性传播；以及(Ii)耦合物理反问题，其在医学和地球物理成像的新模式中得到应用。后一个逆问题的目的是了解在给定的实验环境中可以或不能重建的系数，以及基于给定的噪声水平进行此类重建的精度限制。前一个问题的目的之一是为测量噪声的统计特性提出基于物理的模型，这些统计特性限制了许多逆问题的重建精度。

Equations with highly heterogeneous, poorly resolved, coefficients abound in applied sciences. The project proposes to analyze the influence of the statistical properties of these coefficients on the quantities of interest in many applied areas such as the uncertainty quantifications in climate predictions, the study of geological basins, and the manufacturing of composite materials. Novel methods in medical imaging often referred to as coupled-physics imaging modalities (such as Elastography or Photo-Acoustic tomography) have the potential to strongly enhance our capabilities to detect unhealthy tissues at a very early stage. The proposal will further the mathematical understanding of such problems and in close collaboration with medical physicists and bio-engineers provide robust reconstruction procedures for such modalities.This project focuses on the theoretical and mathematical understanding of: (i) the propagation of uncertainty from heterogeneous coefficients to the solutions of differential equations; and (ii) coupled-physics inverse problems, which find applications in novel modalities in medical and geophysical imaging. The latter inverse problem aims to understand the coefficients that can or cannot be reconstructed in a given experimental environment, and the limitations in the accuracy with which such reconstructions are possible based on a given noise level. The former problem aims, among other things, to propose physics-based models for the statistical properties of the measurement noise that limit the reconstruction's accuracy of many inverse problems.