Organizers: Sarah Strikwerda and Adam Pickarski

Tuesday, March 23, 15:00-16:00
Speaker: Evangelia Ftaka
Title: Calculus of Variations
Abstract: We will discuss the Euler-Lagrange equation (multidimensional case) as necessary condition to the problem of minimizing variational integrals over C^2 functions with boundary constraints, as well as with constraints on the feasible minimizers (Lagrange Multiplier). We will also discuss why the E-L equation can be interpreted as the gradient of the variational integral, and why the gradient descent method can be used to approximate solutions of the E-L equation.

Wednesday, March 9, 15:00-16:00
Speaker: Adam Pickarski
Title: Multidimensional Scaling Properties
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 multi-dimensional 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 and quantify rates of convergence of the algorithm.

Wednesday, March 2 15:00-16:00
Speaker: Andrew Murdza
Title: Functions of Bounded Variation
Abstract: Properties of functions of bounded variation

Wednesday, February 23 15:00-16:00
Speaker: Matthew Broussard
Title:  Introduction to the Calculus of Variations
Abstract: An introduction to the problem of minimizing variational integrals over twice and single continuously differentiable functions with boundary constraints, as well as statements of the strong and weak Euler Lagrange equations. Some special minimizing problems will be introduced, and finally a discussion of minimizing without boundary constraints. 

Wednesday, February 16 15:00-16:00
Speaker: Sarah Strikwerda
Title:  Analysis and Numerics of Optimal Control in Fluid Flows
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 reference an application of this problem to the human eye. We discuss the theoretical results for the problem as well as the methods to numerically approximate an optimal control. 

Wednesday, February 9, 14:00-15:00
Speaker: Andrew Shedlock
Title: The Calderon Problem: An Inverse Problem in Electrical Impedance Tomography
Abstract: Electrical Impedance Tomography is a classic inverse problem in which one attempts to study properties of the interior of an object based on its boundary values. In the Calderon Problem, we assume the voltage potential satisfies a conductivity equation on the interior with boundary values. Calderon Problem asks questions on the well-posedness of the Dirichlet to Neumann (DN) Boundary Map which takes in boundary values and outputs the outward directional flux on each point of the boundary. Further, with the DN Map one can ask if we can reconstruct the conductivity of the interior of our region. 

Wednesday, February 2, 15:00-16:00
Title: Problem Session
Abstract: We work through Analysis problems together.

Wednesday, January 26, 15:00-16:00
Title: Expectations
Abstract: We organize the schedule