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

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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.