An Optimal Transport Based Multiscale Method for Inverse Problems

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Since the advent of computing powers, the application of inverse problem theory has extended to almost all fields of science and engineering that use mathematical methods. Examples of inverse problems can be found in various fields within medical imaging, several areas of geophysics including earthquake source inversion and hydrocarbons exploration and many machine learning applications in data science. The proposed study will connect optimal transport, a classical analysis subject, with many widely used methods in data-driven problems. Results of this research will offer better understandings of existing numerical methods and promote the development of the new techniques for solving inverse problems with high accuracy and fast convergence. The wide range of applications will also increase partnerships and collaboration between academia and industry. Students will be offered many opportunities of joining this research in translating attractive theoretical properties of optimal transport onto various applications in modern science and engineering.The proposed research analyzes the intrinsic multiscale features in optimal transport-based seismic inversion to build robust algorithms for solving general nonlinear large-scale inverse problems. The focus is on designing objective functions in constrained local optimization. A standard approach of measuring the least-squares mismatch between model predictions and data is to use frequency marching and weighting methods in which different frequencies are treated separately; first the low-frequency errors are eliminated followed by high-frequency errors. This particular ordering based on the multiscale inversion scheme addresses two of the biggest challenges in inversion by mitigating problems with local minima in gradient-based optimization and accelerating convergence. The PI's recent work has introduced a framework for seismic inverse problems using the Wasserstein distance as the objective function. Using the theory of optimal transport, the PI proved that this metric offers a convex optimization landscape and the PI's numerical experiments demonstrate the convergence to global minimizers for cases where the least-squares norm has difficulties. The proposed research will investigate the connections between optimal transport-based inversion with existing frequency marching and weighting methods to extend the optimal transport techniques to nonlinear inverse problems beyond seismology. In particular, the PI will formulate optimal transport-based inversion for quantitative photoacoustic tomography (QPAT) and cryogenic electron microscopy (cryo-EM). Methods in this work will be developed using existing frameworks of iterative methods and dynamical systems for convergence analysis. Theoretical results from this research will shed light on the relationship between data fitting (residual reduction) and model fitting (solution error) in various data-driven inverse problems and iterative methods. Computational algorithms will be developed for inversion in seismic imaging, medical imaging, and biology.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.