Applied Math And Analysis Seminar

Perspectives in Wave-based Inverse Problems: Large Scale Constrained Optimization and Machine Learning

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Speaker(s): Leonardo Andres Zepeda Nunez (University of Wisconsin, Mathematics)
Wave-based inverse problems have become ubiquitous and tightly ingrained within the fabric of modern life. From biomedical imaging to geophysical exploration and synthetic aperture radar (SAR), many technologies routinely harness waves' intrinsic property to transmit information great distances with little distortion. In most of these applications, resolution is often the most after sought property. Unfortunately, higher resolution in the output often means a significantly higher cost, which often becomes prohibitive, particularly for large-scale problems and problems in which the output is required in real-time. In this talk, we will provide two different approaches to reduce this cost while obtaining high-resolution images, and in some instances, even surpassing the resolution given by the diffraction limit. The first approach relies on solving the underlying PDE, the Helmholtz equation, in an efficient and scalable fashion, thus addressing the bottleneck of large-scale optimization-based methods. The second approach relies on the structure of the Fourier integral operator that models the data, which is then "neuralized" following the structure of the Cooley-Tukey FFT algorithm. This architecture allows us to bypass the full optimization loop and compute the output directly, provided that the scatterers are restricted to a relatively low-dimensional yet general class. We will describe both approaches, provide their rationale and range of applicability, and provide numerical evidence that showcases the methods' properties. Joint work with: Matt Li (MIT), Adrien Scheuer (MIT), Matthias Taus (TU-Wien), Russell Hewett (VT), and Laurent Demanet (MIT).

Virtual