Numerical Methods for Multiscale Inverse Problems and Applications to Sonar Imaging

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The principal investigator aims to develop mathematical and computational tools for data-driven research by using theoretically-sound prediction models and adaptive numerical methods that capture intrinsic features of complex physical processes. This research project is intended to provide new guarantees for the solvability of a class of inverse problems important in mathematics, commercial industries, and defense operations. The project will involve undergraduate and graduate students, who will receive training in numerical methods, analysis, and scientific applications. The principal investigator will pursue an original strategy for extracting details from large scale datasets using a new class of efficient methods that exploit problem-dependent features of processes occurring on multiple scales. The design of numerical schemes is adaptable to qualitative scientific knowledge and has the potential to significantly enhance the accessibility and performance of current inversion methods. The basis of the approach is the selection of a low-dimensional parameter that describes key microscopic details and the development of numerical methods that retain an intrinsic knowledge of parameter values while solving large-scale models, substantially reducing computational costs. Deliverables of the project will be the new methodology, as well as scientific applications to large scale inverse problems in sonar imaging, where the main challenge is to capture the appropriate physics while maintaining computational time and memory demands acceptable for current computer architectures.