Organizer: Sarah Strikwerda

Wednesday, September 8, 15:00-16:00
Speaker: Sarah Strikwerda
Title: Optimal Control  in Fluid Flows through Deformable Porous Media
Abstract: We consider an optimal control problem subject to a poro-visco-elastic model with applications to fluid flows through biological tissues. Our goal is to optimize the fluid pressure and solid displacement using distributed or boundary control. We discuss an application of this problem to a tissue in the human eye. Previous literature on well- posedness of the poro-visco-elastic model are reviewed. Results on the existence of an optimal control and the associated necessary optimality conditions are presented.

Wednesday, September 15, 15:00-16:00
Speaker: Andrew Murdza
Title: Discussion of Smooth Manifolds and Smooth Maps

Wednesday, September 22, 15:00-16:00
Speaker: Andrew Shedlock
Title: Extension of the Method of Characteristics for Viscous Burgers’ Equation via a Stochastic Differential Equation
Abstract: The inviscid Burgers’ Equation is a first-order nonlinear PDE which is derived from the Navier-Stokes equations and is traditionally solved using the method of characteristics until the formation of shockwaves emerges. The method of characteristics, while a powerful tool, cannot be applied to the viscous Burgers’ equation which is a nonlinear second-order PDE with a viscosity parameter. The viscosity parameter is directly associated with the Reynold’s number of a fluid and numerical simulations with high Reynolds number are notoriously difficult in computational fluid dynamics. To avoid the traditional difficulties with high Reynolds number flow we work with a stochastic differential equation and present results that show this stochastic representation has results which are a natural extension of the method of characteristics and some interesting results of a numerical we developed.

Wednesday, September 29, 15:00-16:00
Problem Session

Wednesday, October 13, 14:00-15:00
Speaker: Adam Pickarski
Abstract: In this work we consider theoretical properties for a specific type of dimension reduction problem, which seeks to accurately map features of high dimensional probability distributions into a lower dimensional space. Specifically, we study a variant of multidimensional scaling, which is posed as an optimization problem in which mappings seek to preserve pairwise distances, and which is known to generate non-linear mappings. We study a population (i.e. continuum) limit of this optimization problem, and prove fundamental properties of the associated mappings. In particular, we show that optimal mappings exist, by first studying a relaxed variational problem, and then by showing through algebraic methods that the solution to the relaxed problem is associated with a piecewise smooth embedding. Along the way we also identify a representation formula for the solution to the problem, and propose a new algorithm for solving this problem which has the potential to greatly improve computational efficiency.

Wednesday, October 20, 14:00-15:00
Problem Session

Wednesday, October 27, 14:00-15:00
Speaker: Diego Cornejo
Title: Johnson and Lindenstrauss Dimensional Reduction
Abstract: In this talk, we are going to prove a dimensional reduction result using elementary probabilistic techniques. More precisely, we will show that a set of n points in high dimensional Euclidean space can be mapped into an O(log n/ epsilon^2)-dimensional Euclidean space.

Wednesday, November 3, 16:00-17:00
Speaker: Minh Bui
Title: Title: A Warped Resolvent Algorithm to Construct Nash Equilibria
Abstract: We propose an asynchronous block-iterative decomposition algorithm to solve Nash equilibrium problems involving a mix of nonsmooth and smooth functions acting on linear mixtures of strategies. The methodology relies heavily on monotone operator theory and in particular on warped resolvents.

Wednesday, November 17, 14:00-15:00
Speaker: Prerona Dutta
Title: Metric entropy for generalized BV functions
Abstract: Inspired by a question posed by Lax in 2002, the study of metric entropy for nonlinear partial differential equations has received increasing attention in recent years. This talk demonstrates methods to obtain sharp upper and lower bounds on the metric entropy for a class of bounded total generalized variation functions taking values in a general totally bounded metric space. Thereafter we use this result to establish metric entropy estimates for the set of entropy admissible weak solutions to a scalar conservation law with weak genuinely nonlinear flux. Estimates of this type could provide a measure of the order of resolution of a numerical method required to solve the equation.