An Optimal Transport Based Multiscale Method for Inverse Problems

基于最优传输的反问题多尺度方法

• 批准号: 1913129
• 负责人: Yunan Yang
• 金额: $ 17.63万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1913129&HistoricalAwards=false
• 关键词: Optimal; Transport; Based; Multiscale; Method

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.

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，包括地震源反演和碳氢化合物勘探在内的几个地球物理学领域以及数据科学中的许多机器学习应用中找到。拟议的研究将连接最优运输，一个经典的分析主题，与许多广泛使用的方法在数据驱动的问题。本文的研究成果将有助于更好地理解现有的数值方法，促进高精度、快速收敛的反问题求解新技术的发展。广泛的应用还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，将最优输运的吸引人的理论特性转化为现代科学和工程中的各种应用。拟议的研究分析了基于最优输运的地震反演中固有的多尺度特征，以建立求解一般非线性大规模反问题的鲁棒算法。重点是设计约束局部优化中的目标函数。测量模型预测和数据之间的最小二乘失配的标准方法是使用频率步进和加权方法，其中不同的频率被单独处理;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定排序通过缓解基于梯度的优化中的局部极小值问题和加速收敛来解决反演中的两个最大挑战。PI最近的工作介绍了一个框架，地震反演问题使用Wasserstein距离作为目标函数。使用最优传输理论，PI证明了该度量提供了一个凸优化景观，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值。拟议的研究将调查最佳运输为基础的反演与现有的频率步进和加权方法之间的联系，以扩展最佳运输技术的非线性反问题超越地震学。特别是，PI将为定量光声层析成像（QPAT）和低温电子显微镜（cryo-EM）制定最佳的基于传输的反演。在这项工作中的方法将开发使用现有的框架迭代方法和动力系统的收敛性分析。从这项研究的理论结果将阐明数据拟合（残差减少）和模型拟合（解决方案误差）之间的关系，在各种数据驱动的反问题和迭代方法。该奖项反映了NSF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。

项目成果

Anderson Acceleration for Seismic Inversion

• DOI: 10.1190/geo2020-0462.1
• 发表时间: 2020-08
• 期刊: ArXiv
• 作者: Yunan Yang

Adjoint DSMC for nonlinear Boltzmann equation constrained optimization

• DOI: 10.1016/j.jcp.2021.110404
• 发表时间: 2020-09
• 期刊: J. Comput. Phys.
• 作者: R. Caflisch;Denis A. Silantyev;Yunan Yang

New likelihood functions and level-set prior for Bayesian full-waveform inversion

用于贝叶斯全波形反演的新似然函数和水平集先验

• DOI: 10.1190/segam2020-3428223.1
• 发表时间: 2020
• 期刊: SEG Technical Program Expanded Abstracts 2020
• 作者: Dunlop, Matt;Yang, Yunan
• 通讯作者: Yang, Yunan

The quadratic Wasserstein metric for inverse data matching

• DOI: 10.1088/1361-6420/ab7e04
• 发表时间: 2019-11
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: Bjorn Engquist;Kui Ren;Yunan Yang

The convexity of optimal transport-based waveform inversion for certain structured velocity models

某些结构化速度模型的最优输运波形反演的凸性

• DOI: 10.1137/20s1361870
• 发表时间: 2021
• 期刊: SIAM Undergraduate Research Online
• 作者: Mahankali, Srinath