Splitting Algorithms, Modern Operator Theory, and Applications (17w5030)

Arriving in Oaxaca, Mexico Sunday, September 17 and departing Friday September 22, 2017


(University of British Columbia)

(University of Goettingen)

(University of South Australia)


Over the past decade, many variants of operator splitting methods have been discovered and proposed, and some of these have found unexpected applications [ERT]. We will bring the leading researchers in the development of optimization algorithms based on Monotone Operator Theory splitting together with leading researchers in the field of partial differential equations. This will allow to survey the state of the art, to identify the most pressing open questions, to offer the opportunity to strike new collaborations, and thus to aid the understanding of these algorithms and their applications. Some of the relevant open questions concern the quantification of convergence rates, the understanding of the behaviour in infeasible cases or, and the lack of satisfying explanations of the behaviour especially in nonconvex settings. We will spend about 40\% of the workshop on algorithms and applications.

The theory of monotone operators [BaC,Sim] is relevant to this endeavour as it is the principal tool in understanding and analyzing the algorithms. Consequently, we will spend 15\% of the workshop on theoretical advances in monotone operator theory, especially as they pertain to algorithms and concrete, implementable methods.

Variational problems in PDEs model phenomena throughout the natural and social sciences, and their mathematical analysis is intimately connected with monotone operators, functional analysis and PDE theory. Some of the most exciting recent developments concern applications to modern materials science. 30\% of the workshop will be reserved for current developments in domain decomposition and operator splitting in PDEs. In view of the considerable divergence between monotone operator theory and its historical root in PDEs (the recent book [G] linking these areas is a welcome but rare exception), we will reserve 15\% of the time trying to bridge the communities through tutorials and talks from specialists in one area aiming to reach out to the other area.

This cross-fertilization has real potential to lead to unexpected advancements in either area. Throughout the workshop we will be building connections to the world outside of mathematics, namely to industry and physics (building on [BK] and [ERT]). We also aim to have an "open problem" discussion session on the last day.

~~~ Importance ~~~ The splitting algorithms that are the main topic of this workshop have found significant real world applications ranging from e.g., wavefront sensing [LBL] to road design [BK]. The open questions surrounding these algorithms are not only mathematically intrinsically beautiful but their resolution promises a real impact to the industrial world. The importance of this workshop lies in its potential to generate new knowledge that will make existing algorithms more efficient and expand their areas of applications.

~~~ Timeliness ~~ The usage of splitting methods and the corresponding research activities have increased significantly especially in the past five years; see, e.g., [BaC], [Com] and [BoCH], and the references therein. The idea to offer this workshop and to specifically aim at building a new bridge between Optimization and PDEs has found significant support in their communities. More than half of the proposed speakers below have been contacted, and all of them have expressed a strong interest in participating in this workshop. We hope that recognizing new ideas in one area will lead to advancements in the other as well.

~~~ Additional benefits ~~ As much as possible, we will aim to reach out and include younger researchers (graduate students, postdoctoral fellows, and assistant professors) as well as women. In order to make the workshop most productive to junior experts, we will ask some of speakers specifically to give survey talks/tutorials. The networking opportunities at this workshop will be import to younger researchers and researcher at smaller institutions to plan their careers and form collaborative research programs.

~~~ Other comments ~~ We note that Bauschke, Burachik and Luke have been involved in organizing a BIRS workshop in November 2009. That workshop has an accompanying Conference Proceedings volume [Banff], published by Springer, featuring 18 articles; this collection ranks in the upper tiers of the relevant Springer eBook Collection in 2012 with over 2500 Chapter downloads in 2011-2012.

~~~ References ~~~ [Banff] H.H. Bauschke, R.S. Burachik, P.L. Combettes, V. Elser, D.R. Luke and H. Wolkowicz (editors), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer 2011. [BaC] H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, 2011. [BCL] H.H. Bauschke, P.L. Combettes, and D.R. Luke: "Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization", Journal of the Optical Society of America 19(7) (2002), 1334-1345. [BK] H.H. Bauschke and V.R. Koch: "Projection methods: Swiss Army knives for solving feasibility and best approximation problems with halfspaces", in Infinite Products and Their Applications, in press. [BoCH] R.I. Bot, E.R. Csetnek and A. Heinrich: "A primal-dual splitting algorithm for finding zeros of sums of maximal monotone operators", SIAM Journal on Optimization, to appear. [Bre] H. Brezis: Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland/Elsevier, 1973. [Com] P.L. Combettes: "Systems of structured monotone inclusions: duality, algorithms and applicatons", SIAM Journal on Optimization, to appear. [DR] J. Douglas and H.H. Rachford: "On the numerical solution of heat conduction problems in two and three space variables", Transactions of the AMS 82 (1956), 421–439. [EB] J. Eckstein and D. Bertsekas: "On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators", Mathematical Programming Series A 55 (1992), 293–318. [ERT] V. Elser, I. Rankenburg, and P. Thibault: "Searching with iterated maps", Proceedings of the National Academy of Sciences 104(2) (2007), 418-423. [G] N. Ghoussoub: Self-dual Partial Differential Systems and Their Variational Principles, Springer, 2009. [HLN] R. Hesse, D.R. Luke, and P. Neumann: "Projection methods for sparse affine feasibility: results and counterexamples", http://arxiv.org/abs/1307.2009, July 2013 [LBL] D.R. Luke, J.V. Burke, and R.G. Lyon: "Optical wavefront econstruction: theory and numerical methods", SIAM Review 44 (2002), 169-224. [LM] P.-L. Lions and B. Mercier: "Splitting algorithms for the sum of two nonlinear operators", SIAM Journal on Numerical Analysis 16 (1979), 964–979. [MM] A. Mielke and S. Muller: Lower semicontinuity and existence of minimizers in incremental finite-strain elastoplasticity. ZAMM Z. Angew. Math. Mech., 86 (2006) 3:233-250. [M] A. Moameni: Non-convex self-dual Lagrangians: new variational principles of symmetric boundary value problems. J. Funct. Anal. 260 (2011), 2674–2715 [RS] R. Rossi, G. Savare, A. Segatti, U. Stefanelli: A variational principle for gradient flows in metric spaces, C.R. Math. Acad. Sci. Paris 349 (2011), 1224-1228. [Sim] S. Simons: From Hahn-Banach to Monotonicity, Springer, 2008.