Propagation of Stochasticity in PDEs and Hybrid Inverse Problems

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Equations with highly heterogeneous, poorly resolved, coefficients abound in applied sciences. The project proposes to analyze the influence of the statistical properties of these coefficients on the quantities of interest in many applied areas such as the uncertainty quantifications in climate predictions, the study of geological basins, and the manufacturing of composite materials. Novel methods in medical imaging often referred to as coupled-physics imaging modalities (such as Elastography or Photo-Acoustic tomography) have the potential to strongly enhance our capabilities to detect unhealthy tissues at a very early stage. The proposal will further the mathematical understanding of such problems and in close collaboration with medical physicists and bio-engineers provide robust reconstruction procedures for such modalities.This project focuses on the theoretical and mathematical understanding of: (i) the propagation of uncertainty from heterogeneous coefficients to the solutions of differential equations; and (ii) coupled-physics inverse problems, which find applications in novel modalities in medical and geophysical imaging. The latter inverse problem aims to understand the coefficients that can or cannot be reconstructed in a given experimental environment, and the limitations in the accuracy with which such reconstructions are possible based on a given noise level. The former problem aims, among other things, to propose physics-based models for the statistical properties of the measurement noise that limit the reconstruction's accuracy of many inverse problems.