Equations with random coefficients and Inverse Problems

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The first main thrust of the proposal is the analysis of the random fluctuations of solutions to equations with random coefficients. The investigator and his collaborators will develop asymptotic models for random fluctuations for several classes of partial differential equations and will use these models as Benchmarks to assess the validity of upscaling algorithms. These models will also be applied to the understanding of the random fluctuations of high frequency waves propagating in heterogeneous media. Such fluctuations control the accuracy of reconstructions of buried inclusions in cluttered media. The second main thrust of the proposal is the theoretical and numerical analysis of hybrid inverse problems as they appear in medical imaging. Such inverse problems, which are actively researched in the biomedical imaging community, promise to devise novel imaging modalities that combine high resolution with high contrast.Partial differential equations with random coefficients and their numerical solutions are ubiquitous in applied sciences and engineering and find direct applications, e.g., in the quantification of uncertainties in geological basins and other large scale environments, in atmospheric effects that participate in the analysis of global warming, and in the manufacturing of composite materials. Photo-acoustic Tomography and other novel medical imaging methods in the class of hybrid inverse problems offer to revolutionize the practice of medical imaging by providing an unprecedented combination of high contrast and high resolution capabilities.