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都可以建模为找到两个函数f和g的和的最小值，或者找到两个算子a和B的和的零。在许多情况下，算子a和B足够简单，因此很容易计算出它们的解。拆分方法将这些单独的解合并成一个新的运算符，迭代地解决原始和问题。***一种著名的分裂方法是Douglas-Rachford算法（DRA），它最初用于数值求解某些热方程。1979年，p - l。Lions （Fields奖得主）和B. Mercier建立了一个开创性的结果，即DRA可以工作，并且可以在两个函数都是凸的甚至可能是非光滑的情况下找到f+g的最小值（或者更一般地说，当a和B是单调的并且可能是集值的）。从那时起，关于DRA的文献爆炸式增长，如今，即使没有严格的收敛结果，DRA也成功地应用于许多领域，包括信号处理、机器学习、图像重建和相位检索。***关于DRA及其变体的各种诱人而重要的问题仍然开放，包括：*** 1)当函数f和g不是凸的（或A和B不是单调的）会发生什么？*** 2)如果潜在的问题没有解决方案（例如，由于嘈杂的数据）会发生什么？*** 3)收敛到一个解的速度有多快？*** 4)如果我们在无限维（函数或序列）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及其变体的静态性质和渐近行为。所获得的结果不仅将推进当前的科学和数学知识，而且还可能改善制造和工业过程，如地板规划和道路设计。最后，但并非最不重要的是，我的学生和博士后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，从而促进加拿大在科学和工程领域的领导地位。