Propagation of Stochasticity in PDEs and Hybrid Inverse Problems

在应用科学中，具有高度非均质、难解系数的方程比比皆是。该项目建议分析这些系数的统计性质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、地质盆地的研究和复合材料的制造。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扫描），具有在早期阶段强烈增强我们检测不健康组织的能力的潜力。该提案将进一步从数学角度理解这些问题，并与医学物理学家和生物工程师密切合作，为这些模式提供可靠的重建程序。该项目侧重于对以下方面的理论和数学理解：(i)从非均质系数到微分方程解的不确定性传播；（ii）耦合物理逆问题，在医学和地球物理成像的新模式中得到应用。后一个反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，以及基于给定噪声水平的这种重建可能的精度的限制。前一个问题的目的是，除其他外，提出基于物理的模型，用于测量噪声的统计特性，这些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.