Equations with random coefficients and Inverse Problems

该建议的第一个要点是分析具有随机系数的方程的解的随机波动。研究者和他的合作者将为几类偏微分方程开发随机波动的渐近模型，并将使用这些模型作为基准来评估升级算法的有效性。这些模型也将应用于理解在非均匀介质中传播的高频波的随机波动。这种波动控制了在杂乱介质中埋藏包裹体重建的准确性。该建议的第二个主要推力是混合逆问题的理论和数值分析，因为他们出现在医学成像。这类反问题在生物医学成像领域得到了积极的研究，有望设计出结合高分辨率和高对比度的新型成像方式。具有随机系数的偏微分方程及其数值解在应用科学和工程中无处不在，并有直接的应用，例如，在地质盆地和其他大尺度环境的不确定性的量化中，在参与全球变暖分析的大气效应中，以及在复合材料的制造中。光声断层成像和其他新型医学成像方法在混合反问题的类别提供了革命性的医学成像实践，提供了前所未有的高对比度和高分辨率能力的组合。

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.