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.
The research component of the project had two main contributions: (i) the solution of hybrid inverse problems, which find applications in several novel medical imaging modalities; and (ii) the propagation of stochasticity from coefficients in differential equations to solutions of such equations, which finds numerous applications, for instance the propagation of seismic waves through the complex Earth structure, or the propagation of light through a turbulent atmosphere. The main results of the first component are explicit reconstruction procedures and an analysis of the stability of such reconstruction in the inevitable presence of noise in available measurements, for several medical imaging modalities including Elastography, Photo-acoustic Tomography, and Ultrasound Modulated Tomography. These theoretical results were validated by several numerical simulations. The main results of the second component are the macroscopic understanding of such a propagation in a large class of parabolic models when the stochasticity has a Gaussian structure, and in a smaller class of models for more general structures of stochasticity. Such mathematically difficult questions are important in many practical settings: for instance in understanding how some available measurements (solutions of equations) are affected by unknown, random, parameters in these equations. These results were disseminated to the applied mathematics and engineering communities via numerous publications in peer-reviewed international journals and through presentations at domestic and international conferences. The project also had an important educational component. Four PhD graduate students were trained during the course of the project. Three have graduated. Two pursued academic careers as assistant professors at the University of Washington and Stanford University. The third student went to the financial industry in New York. Three postdoctoral fellows were also mentored during the project. They all have academic positions now, two in France at the CNRS and INRIA, and one at Michigan State University.