Optimization and Inference for Physical Flows on Networks

The Banff International Research Station will host the "Optimization and Inference for Physical Flows on Networks" workshop from March 5th to March 10th, 2017.

Mathematical models that describe the flow of fluids, movement of particles, or transfer of information over a network of channels appear in a wide range of fields of theoretical and scientific study, as well as important engineering applications. Particular interest is devoted to models for which the network structure is represented using graphs where flows on edges and conditions at nodes are governed by sets of physical laws, which may be steady or dynamic, and deterministic or random. Important examples of include the electric power grid, natural gas transmission systems, and traffic flows of air or land vehicles. Recent developments in applied mathematics and computation promise a breakthrough in our ability to learn about and optimally control them. These tasks can be mathematically posed as optimization/control and inference/learning problems. For example, the Optimal Power Flow (OPF) problem in electric transmission networks is an optimization problem that seeks to minimize the cost of power generation among distributed sources subject to the physical constraints of transmission networks. Alternatively, given a continuous stream of noisy/uncertain phase measurements throughout the power grid, we may infer congestion in the network. These problems can be solved together to actively control reactive power generation at different locations or change the network topology. As in the case of the power grid, the networks of interest may be very large, and may consist of sub-networks or components with physics and dynamics on multiple space and time scales. Therefore, the scaling properties of the mathematics and algorithms used to address these problems require careful investigation. Moreover, these problems tend to have intrinsic features such as non-linearities and non-convexities that are dictated by physical laws, which make the design of scalable algorithms a challenge.

The theoretical sides of the relevant mathematical fields, which include optimization, graph theory, statistical inference, dynamical systems, partial differential equations (PDEs), among others, have each given rise to sophisticated computational techniques. However, problems of optimization and inference for physical flow networks require methods from each of these disciplines to be further developed in coordination with one another. For instance, a very accurate and efficient method for simulation of flow dynamics governed by PDEs may be unsuitable for use within an optimization algorithm. Instead, appropriate models that are tractable for optimization, yet accurately represent the physics involved, may be used to develop optimal control algorithms with significantly improved scalability and stability. Similarly, inference algorithms for stochastic state estimation on networks, and the related marginal probabilities, need to be designed in a way that they provide accurate input for stochastic optimization problems. This workshop will bridge the mathematical gaps that challenge the solution of the motivating real-world problems.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disc iplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineeri ng Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).

BIRS Scientific Director, Nassif Ghoussoub
E-mail: birs-director[@]birs.ca
http://www.birs.ca/~nassif