Partial Differential Equations with random coefficients and Inverse Problems

该项目涉及分析具有非均匀系数的偏微分方程（例如描述波或粒子在模拟为随机介质的复杂介质中的传播）和反问题理论。许多解决方案的微分方程与非均匀系数是我们无法访问，因为他们太昂贵，以获得即使与今天的计算能力。宏观模型的推导，以某种方式平均小尺度的不均匀性，然后是有序的。在许多应用中，人们不仅对解的（确定性）系综平均感兴趣，对此存在许多理论，而且还对其随机波动的定量描述感兴趣，即，无法以确定性方式建模的部分。了解后者的波动是该项目的第一个目标。一旦提出了模型，第二个问题涉及从可用的测量值重建方程中的系数，通常在感兴趣的域的边界处执行。该项目的第二个主要目标是定量地了解这些反问题如何受到解中随机波动的影响。具有随机（高度异质）系数的方程在应用科学中无处不在。应用包括地质盆地和核反应堆的建模，复合材料的制造，探测波或粒子的传播，因为它们被用于遥感，医学成像和地球物理成像。 该项目将使人们更好地了解这些应用中现有测量的质量，然后回答以下类型的问题：我们可以从现有测量中了解我们的介质（例如，医学成像中的人体，大气成像中的污染物浓度）？ 哪些尺度是我们可以理解的，哪些尺度是数学上无法重建的？如何最佳地减轻数据中不可避免的噪声的影响？

The project concerns the analysis of partial differential equations with heterogeneous coefficients (for instance describing the propagation of waves or particles in complex media modeled as random media) and the theory of inverse problems. Many solutions to differential equations with heterogeneous coefficients are not accessible to us because they are too expensive to obtain even with today's computational capabilities. The derivation of macroscopic models, which average the small-scale heterogeneities one way or another, is then in order. In many applications, one is interested not only in the (deterministic) ensemble average of the solution, for which many theories exist, but also in a quantitative description of its random fluctuations, i.e., the part that cannot be modeled in a deterministic manner. Understanding the latter fluctuations is the first goal of the project. Once a model has been proposed, the second question pertains to the reconstruction of the coefficients in the equation from available measurements, typically performed at the boundary of a domain of interest. A quantitative understanding of how these inverse problems are affected by the random fluctuations in the solution is the second major objective of the project. Equations with random (highly heterogeneous) coefficients are ubiquitous in applied sciences. Applications include the modeling of geological basins and of nuclear reactors, the manufacturing of composite materials, the propagation of probing waves or particles as they are used in remote sensing, medical imaging, and geophysical imaging. The project will provide a better understanding of the quality of available measurements in these applications and then provide answers to the following type of questions: what is it we can learn about our medium (e.g. a human body in medical imaging, a concentration of pollutants in atmospheric imaging) from available measurements? What are the scales that we can understand and those that mathematically cannot be reconstructed? How does one optimally mitigate the influence of unavoidable noise in the data?