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?
Intellectual Merit; waves in random media and equations with random coefficients. The first outcome of the grant pertains to the imaging of buried inclusions in cluttered environments and the analysis of stochastic partial differential equations. For the reconstruction of buried inclusions, we obtain a precise statistical characterization of the random fluctuations in the measurements of scattered wave fields. This allows us to devise reconstruction procedures with minimal statistical variance. In the analysis of partial differential equations with random coefficients oscillating at a microscopic scale, two competing forces are in play. In some settings, stochasticity is overwhelming and survives at the level of macroscopic description of the phenomenon. In other settings, stochasticity averages out by application of a law of large numbers. In the framework of elliptic equations with random potential, we have obtained a precise transition from a regime where stochasticity wins to a regime where averaging (homogenization) wins. Intellectual Merit; hybrid inverse problems in medical imaging. The second outcome of the grant relates to medical imaging modalities referred to as hybrid modalities. Most successful imaging modalities rely on two properties; they are high-contrast, to provide good discrimination between healthy and non-healthy tissues, and they are high-resolution, to allow us to detect anomalies at an early stage. Hybrid modalities come from the physical coupling of a high-resolution low-contrast modality with a high-contrast, low-resolution, modality. From the mathematical point of view, these modalities often involve redundant systems of partial differential equations. Several such systems have been solved explicity. Explicit reconstruction procedures have been obtained for imaging modalities such as Photo-Acoustic tomography and Ultrasound-Modulated Electrical Impedance Tomography (see figure). When no explicit reconstruction procedure was found, several methodologies have been introduced to analyze the mathematical problem. In particular, stabilily estimates have been obtained in many setting of hybrid inverse problems. Stability estimates give precise descriptions of the amplification of noise in available data during the reconstruction of the parameters of interest. It is directly related to the resolution one expects in such reconstructions. Broader impact in scientific community. The results presented above have been published or submitted to international journals that are read by the mathematical and physical communities. They have also been disseminated during conference, workshops, and seminars attended by mathematicians, physicists, and biomedical engineers. Two workshops were organized at the Banff International Research Station in Banff, Alberta, during the timespan of the grant. The first one was devoted to novel methods in medical imaging and was attended by mathematicians and bio-engineers from the Americas, Europe, and Asia. The second workshop aimed at bridging the gap between what is understood mathematically and what is pursued in the Engineering communities in the field of Uncertainty Quantification. It was also attended by experts from both the mathematical and the engineering communities. The explicit reconstruction procedures were presented to several Engineering groups, for instance in the field of Photo-acoustic tomography. They have been tested by several engineering groups in the US and Europe. Broader Impact in Education. Several undergraduate and graduate students were partially funded by the grant. At the graduate level, two students have been partially funded to work on hybrid inverse problems and other inverse problems. These are Nick Hoell, who obtained a postdoctoral position in the mathematics department at the University of Toronto after graduating in 2011; and Francois Monard, who joined the mathematics department at the University of Washington as an Acting Assistant Professor after graduating in 2012. Two students have been partially funded to work on the analysis of equations with random coefficients. These are Ningyao Zhang and Yu Gu, who are expected to graduate in the following two years. At the undergraduate level, several exchange students came from Europe to work on the topics of the grant. They are Clement Ray, Thomas Boulier, Xavier Arhan, and Tristan Agaesse. A course on inverse problems including the research results mentioned above and open to both undergraduate and graduate students was also developed and so far taught twice at Columbia University.
