Novel Approaches to Wave Propagation in Random Media

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This project addresses mathematical challenges associated with imaging and sensing in complicated environments. Remote sensing data can be used for environmental monitoring of atmospheric pollution, for evaluation of health of forests and crops, for detecting fine details on the earth's surface, and for tracking of satellites and other flying objects. As ever more, finer-resolution data becomes available, a better understanding of how these are affected by clutter in the environment is important. Such understanding is also needed in the context of designing schemes for imaging and communication through turbulent media, of the type found for instance in the turbulent (cluttered) atmosphere, as well as in the context of biomedical imaging. Further developments of such techniques require a deep understanding of how the probing wave field is affected by the local medium or tissue fabric, the clutter. The project also involves techniques for better understanding and modeling of financial and economic time series, which is useful for monitoring and stabilization of markets. Important questions relate then to how one can design schemes for early detection of when the fabric, the local clutter statistics, of the market is changing and we are entering a new market regime. Similar challenges relate to analysis of internet traffic, such as detecting changes to normal traffic flow caused by an intruder. The project aims to develop fundamental mathematical frameworks that allow one to deal with such important challenges, moreover, to use modern tools associated with machine learning in the detection of fabric changes. Graduate students will be trained through involvement in the research.The challenge of a rigorous characterization of laser beams propagating through multiscale media and interfaces is of fundamental importance both from the physical and mathematical perspectives. This project studies new approaches that allow one to model wave propagation in complicated non-classic media, particularly through so called non-Kolmogorov turbulence. Such modeling is becoming increasingly important in the context of propagation through the atmosphere and furthermore in biomedical imaging. The analysis will involve asymptotic analysis of statistical moments of fields governed by partial differential equations with random coefficients and exploit scale separations in the system. Novel imaging techniques increasingly make use of wave spectral information, or information in the covariance structure for the polarization modes of the wave field itself or its intensity, that is, speckle correlations. To evaluate such imaging techniques based on wave spectral information, one needs information about higher-order moments to assess image stability, or noise level, and to optimize the schemes. A main objective of the project is to further the theory that describes this statistical structure and, moreover, to use it in the context of classic and deep learning imaging modalities. Specifically, novel approaches for modeling wave propagation in multiscale and complex media will be used in the design of imaging approaches that can detect changes in the local tissue fabric in the context of biomedical imaging. The investigator will also study modeling and estimation in the context of complex economic time series with a multi-fractal character, with applications to detection of market regime changes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.