Douglas-Rachford splitting: static properties, asymptotic behaviour, variants, extensions and applications

Douglas-Rachford 分裂：静态属性、渐近行为、变体、扩展和应用

• 批准号: RGPIN-2018-03703
• 负责人: Bauschke, Heinz
• 金额: $ 3.28万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=761807
• 关键词: Douglas; Rachford; splitting; static; properties

Numerous problems in mathematics, engineering and the physical sciences may be modelled as finding the minimizer of the sum of two functions f and g, or as finding a zero of the sum of two operators A and B. In many cases, the operators A and B are simple enough so that it is easy to compute their resolvents. Splitting methods combine these individual resolvents into a new operator to iteratively solve the original sum problem. A famous splitting method is the Douglas-Rachford algorithm (DRA) which was originally used to solve certain heat equations numerically. In 1979, P.-L. Lions (a Fields medalist) and B. Mercier established the seminal result that DRA works and can find minimizers of f+g when both functions are convex and possibly even nonsmooth (or, more generally, when A and B are monotone and possibly set-valued). The literature on DRA has exploded ever since and nowadays DRA is successfully applied even when there are no rigorous convergence results in many areas including signal processing, machine learning, image reconstruction, and phase retrieval. Various tantalizing and important questions concerning DRA and its variants remain open including: 1) What happens when the functions f and g are not convex (or A and B are not monotone)? 2) What happens if the underlying problem has no solution (e.g., because of noisy data)? 3) How fast is the convergence to a solution? 4) What happens if we work in infinite-dimensional (function or sequence) spaces?To make progress on these problems, I shall use modern tools from Convex Analysis, Monotone Operator Theory and Variational Analysis to study static properties and asymptotic behaviour of DRA and its variants. The results obtained will not only advance current scientific and mathematical knowledge but also potentially improve manufacturing and industrial processes such as floor planning and the design of roads. Last, but not least, the skills my students and postdoctoral fellows acquire will prepare them for successful careers in industry and academia thereby fostering Canada's leadership in Science and Engineering.

数学，工程和物理科学方面的许多问题可能被建模为找到两个功能F和G的总和的最小化器，或者找到两个操作员A和B的总和中的零。在许多情况下，操作员A和B非常简单，以便易于计算分辨率。分裂方法将这些单独的分解结合到新的操作员中，以迭代解决原始总和问题。 一种著名的分裂方法是道格拉斯 - 拉赫福德算法（DRA），该算法最初用于数值求解某些热方程。 1979年，P.-L。狮子（A Fields奖章）和B. Mercier确立了开创性的结果，即DRA起作用，并且在两个函数都是凸面，甚至可能不平滑时（或更一般而言，当A和B是单调且可能值得值值时）时可以找到F+G的最小化。从那以后，关于DRA的文献爆炸了，如今，即使在许多领域都没有严格的合并结果，包括信号处理，机器学习，图像重建和相位检索。 有关DRA及其变体的各种诱人和重要问题仍然开放，包括：1）当功能F和G不是凸（或A和B不是单调）时会发生什么？ 2）如果潜在问题没有解决方案（例如，由于嘈杂的数据）会发生什么？ 3）收敛到解决方案的速度多快？ 4）如果我们在无限维（功能或序列）空间中工作会发生什么？为了在这些问题上取得进展，我将使用凸分析，单调操作者理论和变分分析中的现代工具来研究DRA及其变体的静态性质和渐近行为。获得的结果不仅将推进当前的科学和数学知识，而且还可以改善制造业和工业过程，例如平面图和道路设计。最后但并非最不重要的一点是，我的学生和博士后研究员获得的技能将为他们在工业和学术界的成功职业做好准备，从而促进了加拿大在科学和工程领域的领导。