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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688925
• 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、工程和物理科学中的许多问题可以建模为寻找两个函数 f 和 g 之和的最小值，或者寻找两个算子 A 和 B 之和的零。在许多情况下，算子 A 和 B 足够简单，因此很容易计算它们的解。分裂方法将这些单独的解决方案组合成一个新的算子，以迭代地解决原始求和问题。*** 著名的分裂方法是 Douglas-Rachford 算法 (DRA)，它最初用于数值求解某些热方程。 1979 年，P.-L。 Lions（菲尔兹奖获得者）和 B. Mercier 建立了 DRA 起作用的开创性结果，并且当两个函数都是凸函数甚至可能是非光滑函数时（或者更一般地，当 A 和 B 是单调的并且可能是集值函数时），可以找到 f+g 的最小化值。从那时起，有关 DRA 的文献呈爆炸式增长，如今，即使在信号处理、机器学习、图像重建和相位检索等许多领域没有严格的收敛结果的情况下，DRA 也得到了成功的应用。 *** 关于 DRA 及其变体的各种诱人且重要的问题仍然悬而未决，包括：*** 1) 当函数 f 和 g 不是凸的（或者 A 和 B 不是单调的）时会发生什么？*** 2) 如果根本问题无解（例如，由于噪声数据）会发生什么？*** 3) 收敛到解决方案的速度有多快？*** 4) 如果我们在无限维（函数或函数）中工作会发生什么？ 序列）空间？***为了在这些问题上取得进展，我将使用凸分析、单调算子理论和变分分析等现代工具来研究 DRA 及其变体的静态属性和渐近行为。获得的结果不仅将推进当前的科学和数学知识，而且有可能改善制造和工业流程，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界和学术界的成功职业生涯做好准备，从而培养加拿大在科学和工程领域的领导地位。