Sparsity Regularization for Inverse Problems -- Theory, Algorithm and Application

反问题的稀疏正则化——理论、算法与应用

• 批准号: EP/M025160/1
• 负责人: Bangti Jin
• 金额: $ 12.52万
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2015
• 资助国家: 英国
• 起止时间: 2015 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FM025160%2F1
• 关键词: Sparsity; Regularization; Inverse; Problems; Theory

One fundamental task in many scientific and engineering disciplines is to probe the world around us, often in the form of deducing physical laws or determining the parameters therein from experimental data. This gives rise to a wide variety of inverse problems of inferring the cause of observed physical phenomena or determining medium properties of the concerned object from the measured responses. A model based approach to inverse problems assumes a known relation between the unknown physical quantity and the measured responses, in the form of linear or nonlinear equations. Inverse problems are numerically challenging to solve since they are unstable with respect to data noise. One of the most successful and powerful techniques is to incorporate a priori knowledge by means of regularization. This project focuses on one specific regularization technique based on sparsity constraints.In sparsity regularization, one looks for an approximate solution that has as few nonzero entries (or a small support) as possible. That is, we aim at explaining the physical phenomenon by simply using a small number of parameters. In this project, we study the mathematical theory, computational techniques and practical applications of the approach, especially by using a nonconvex penalty on the sought-for solution. There are several nonconvex penalties proposed in the literature, especially in statistics and machine learning, and we shall consider, for example, the popular l0 penalty, bridge penalty, smoothly clipped absolute deviation, minmax concavity penalty and clipped l1 penalties. Despite their popularity, their use and study in the context of inverse problems remain very limited. The proposed project examines these techniques in the framework of inverse problems theory. Specifically, we focus on the following objectives during the project period: (a) to develop a computational method of primal-dual active set type for efficiently solving the regularized model (with nonconvex penalties) and rigorously establish the convergence of the algorithm; (b) to develop applicable choice rules for the regularization parameter, and to analyze the structural properties of "local" minimizers; (3) to apply the nonconvex approach to tomography imaging, by combining it with an adaptive finite element method. The computational technique to be developed, primal dual active set algorithm, is widely applicable to many other areas, especially machine learning and statistics, since these nonconvex models have their origins and motivations there. Mathematically, the theory of sparsity regularization will shed valuable lights into the analysis of nonsmooth regularization, which shares common structures with many important mathematical models arising in imaging and signal processing. The specific choice rule for the regularization parameter will enable automatic parameter selection yet with rigorous theoretical justification, and improve the efficiency of the current practice based on tedious trial and error. The research in tomography imaging, i.e., the application of nonconvex penalty and adaptive algorithm, will directly impact the medical imaging community. There are a large group of researchers working on tomography imaging that will benefit directly from the research, and the obtained research results will be presented to them continuously. More generally, it provides guidance for developing efficient algorithms for nonlinear inverse problems for differential equations. In summary, the project will take sparsity regularization for inverse problems to the next level.

许多科学和工程学科的一项基本任务是探索我们周围的世界，通常以推导物理定律或从实验数据中确定其中的参数的形式。这就产生了各种各样的逆问题，推断所观察到的物理现象的原因，或确定介质的有关对象从测量的响应的属性。基于模型的反问题方法假设未知物理量和测量响应之间的已知关系，以线性或非线性方程的形式。反问题是数值求解的挑战，因为它们相对于数据噪声是不稳定的。最成功和最强大的技术之一是通过正则化的方式纳入先验知识。这个项目的重点是一个特定的正则化技术的基础上稀疏约束。在稀疏正则化，寻找一个近似的解决方案，有尽可能少的非零元素（或一个小的支持）。也就是说，我们的目标是通过简单地使用少量参数来解释物理现象。在这个项目中，我们研究的数学理论，计算技术和实际应用的方法，特别是通过使用一个非凸的惩罚所寻求的解决方案。在文献中，特别是在统计学和机器学习中，提出了几种非凸罚分，我们将考虑流行的l0罚分、桥罚分、平滑剪切绝对偏差、最小最大最小罚分和剪切l1罚分。尽管他们的普及，他们的使用和研究的背景下，反问题仍然非常有限。拟议的项目审查这些技术的框架内的反问题理论。具体而言，我们在项目期间的主要目标是：（a）开发一种有效求解正则化模型的原始-对偶活动集类型的计算方法（B）开发正则化参数的适用选择规则，并分析“局部”极小值的结构性质;（3）将非凸方法与自适应有限元方法相结合，应用于层析成像。待开发的计算技术，原始对偶活动集算法，广泛适用于许多其他领域，特别是机器学习和统计，因为这些非凸模型有其起源和动机。在数学上，稀疏正则化理论将为非光滑正则化分析提供有价值的启示，它与成像和信号处理中出现的许多重要数学模型具有共同的结构。正则化参数的特定选择规则将使参数的自动选择具有严格的理论依据，并提高目前基于繁琐的试错的实践效率。层析成像的研究，即，非凸惩罚和自适应算法的应用，将直接冲击医学影像界。有一大群研究人员致力于断层成像，将直接受益于研究，所获得的研究成果将不断呈现给他们。更一般地说，它为开发微分方程非线性反问题的有效算法提供了指导。总之，该项目将把反问题的稀疏正则化带到一个新的水平。

项目成果

Linearized reconstruction for diffuse optical spectroscopic imaging

漫射光学光谱成像的线性化重建

• DOI: 10.1098/rspa.2018.0592
• 发表时间: 2019
• 期刊: Mathematical, Physical and Engineering Sciences
• 作者: Ammari H

A variational Bayesian approach for inverse problems with skew-t error distributions

• DOI: 10.1016/j.jcp.2015.07.062
• 发表时间: 2015-11
• 期刊: J. Comput. Phys.
• 作者: Nilabja Guha;Xiaoqing Wu;Y. Efendiev;Bangti Jin;B. Mallick

Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data

• DOI: 10.1515/fca-2016-0005
• 发表时间: 2015-04
• 期刊: Fractional Calculus and Applied Analysis
• 影响因子: 3
• 作者: Bangti Jin;R. Lazarov;D. Sheen;Zhi Zhou

Acousto-electric tomography with total variation regularization

• DOI: 10.1088/1361-6420/aaece5
• 发表时间: 2018-08
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: Bolaji James Adesokan;Bjørn Jensen;Bangti Jin;K. Knudsen

The Linearized Inverse Problem in Multifrequency Electrical Impedance Tomography

• DOI: 10.1137/16m1061564
• 发表时间: 2016-01-01
• 期刊: SIAM JOURNAL ON IMAGING SCIENCES
• 影响因子: 2.1
• 作者: Alberti, Giovanni S.;Ammari, Habib;Zhang, Wenlong
• 通讯作者: Zhang, Wenlong