An Optimal Transport Based Multiscale Method for Inverse Problems

自从计算能力出现以来，反问题理论的应用已经扩展到几乎所有使用数学方法的科学和工程领域。逆问题的例子可以在医学成像的各个领域、地球物理学的几个领域（包括震源反演和碳氢化合物勘探）以及数据科学中的许多机器学习应用中找到。本研究将最优运输这一经典分析课题与数据驱动问题中许多广泛使用的方法结合起来。本文的研究结果将有助于更好地理解现有的数值方法，并促进求解高精度、快速收敛的反问题新技术的发展。广泛的应用也将加强学术界和工业界之间的伙伴关系和合作。学生将有很多机会参与这项研究，将最优运输的理论特性转化为现代科学和工程的各种应用。本研究分析了基于最优输运的地震反演固有的多尺度特征，建立了求解一般非线性大尺度反演问题的鲁棒算法。重点研究了约束局部优化中目标函数的设计。测量模型预测和数据之间最小二乘不匹配的标准方法是使用频率推进和加权方法，其中不同的频率被分开处理；首先消除低频误差，然后消除高频误差。这种基于多尺度反演方案的特殊排序通过减轻基于梯度的优化中的局部最小问题和加速收敛，解决了反演中的两个最大挑战。PI最近的工作引入了一个使用Wasserstein距离作为目标函数的地震逆问题框架。利用最优传输理论，PI证明了该度量提供了一个凸优化景观，并且PI的数值实验证明了在最小二乘范数有困难的情况下收敛到全局最小化。提出的研究将探讨基于最优输运的反演与现有的频率推进和加权方法之间的联系，以将最优输运技术扩展到地震学以外的非线性反演问题。特别是，PI将为定量光声断层扫描（QPAT）和低温电子显微镜（cryo-EM）制定最佳的基于输运的反演。这项工作中的方法将使用迭代方法和动态系统的现有框架进行收敛分析。本研究的理论结果将揭示各种数据驱动逆问题和迭代方法中数据拟合（残差约简）和模型拟合（解误差）之间的关系。将开发用于地震成像、医学成像和生物学反演的计算算法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.