Partial Differential Equations with random coefficients and Inverse Problems

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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?