Organizer: Zev Woodstock

Wednesday, November 20, SAS 3282, 12:00 – 13:00

Speaker: Steven Gilmore
Title: Proof of Rademacher’s Theorem.

Wednesday, November 13, SAS 3282, 14:00 – 15:00

Speaker: Prerona Dutta
Title: Metric entropy for functions of bounded total generalized variation.

Abstract: In this talk, we establish sharp estimates for a minimal number of bits needed to present all bounded generalized variation functions taking values in a general totally bounded metric space up to an accuracy of epsilon >  0 with respect to L^1 -distance.

Wednesday, November 06, SAS 3282, 12:00 – 13:00

Speaker: Ethan King
Title: Gauge optimization and duality by Friedlander et al.

Abstract: The Moreau envelope is one of the key convexity-preserving functional operations in convex analysis, and it is central to the development and analysis of many approaches for convex optimization. This paper develops the theory for an analogous convolution operation, called the polar envelope, specialized to gauge functions. Many important properties of the Moreau envelope and the proximal map are mirrored by the polar envelope and its corresponding proximal map. These properties include smoothness of the envelope function, uniqueness, and continuity of the proximal map, which play important roles in duality and in the construction of algorithms for gauge optimization. A suite of tools with which to build algorithms for this family of optimization problems is thus established.

Wednesday, October 30, SAS 3282, 14:00 – 15:00

Speaker: Farid Benmouffok
Title: Exercises and Problems day.

Wednesday, October 23, SAS 3282, 12:30 – 13:00
Speaker:  Steven Gilmore
Title: : A model of debt with bankruptcy risk and currency devaluation

Abstract: We consider a system of Hamilton-Jacobi equations, arising from a stochastic optimal debt management problem in an infinite time horizon with exponential discount, modeled as a noncooperative interaction between a borrower and a pool of risk-neutral lenders. In this model, the borrower is a sovereign state that can decide how much to devaluate its currency and which fraction of its income should be used to repay the debt.  Moreover, the borrower has the possibility of going bankrupt at a random time and must declare bankruptcy if the debt reaches a threshold. When bankruptcy occurs, the lenders only recover a fraction of their capital. To offset the possible loss of part of their investment, the lenders buy bonds at a discounted price which is  not given  a priori. This leads to a nonstandard optimal control problem. We establish an existence result of solutions to this system and in turn recover optimal feedback  payment strategy and currency devaluation

Wednesday, October 23, SAS 3282, 12:00 – 12:30
Speaker: Ben Freedman
Title: Existence of Solutions to Nonlinear Continuous Boundary Value Problems on Infinite Intervals

Abstract: In this talk, we will consider boundary value problems on infinite intervals subject to weakly nonlinear boundary conditions. Results are obtained for such problems in the context of differential equations, and the framework we present allows us to establish sufficient conditions for the existence of solutions as well as a qualitative description of solutions according to a parameter. 

Wednesday, October 16, SAS 3282, 14:00 – 15:00

Speaker: Alex Smirnov, (UNCC)
Title: Convexification for an inverse problem with single measurement for a 1D hyperbolic equation.

Abstract: The questions of stability and global convergence for coefficient inverse problems are the most critical and challenging in numerical applications. To address these challenges, a novel numerical method, based on the concept of convexification was proposed. The globally convergent numerical method was developed to solve a coefficient inverse problem with a single measurement for a 1D hyperbolic equation. This approach constructs a weighted Tikhonov-like functional, which is strictly convex on an a priori chosen ball of an arbitrary radius in an appropriate Hilbert space. The strict convexity is ensured via the presence of the Carleman Weight Function in the functional. Then, the gradient projection method converges to the exact solution starting from an arbitrary point of that ball. The numerical simulations demonstrate the stability and accuracy of the proposed method.

Wednesday, October 9, SAS 3282, 12:00– 13:00

Speaker: Steven Gilmore
Title: A note on admissible solutions of 1D scalar conservation laws and 2D Hamilton-Jacobi equations.

Abstract: Let C be an open set and f, with f” > 0. In this note we prove that entropy solutions of Dtu+Dxf(u) = 0 belong to SBVloc. As a corollary we prove the same property for gradients of viscosity solutions of planar Hamilton?Jacobi PDEs with uniformly convex Hamiltonians.

Wednesday, October 2, SAS 3282, 14:00– 15:00

Speaker: Zev Woodstock
Title:  Recession Theory and a few existence results.

Wednesday, September 25, SAS 3282, 12:00 – 13:00

Speaker: Sarah Strikwerda
Title: Optimal Controls of PDE, Fredi Tröltzsch  [Chapters 2 and 8].

Wednesday, September 18, SAS 3282, 14:00 – 15:00

Speaker: Minh Bui.
Title:  Proximal Iterations.

Wednesday, September 11, SAS 3282, 12:00 – 13:00

Speaker: None (Discussion session)
Title: Exercises and Problems day.

Wednesday, September 4, SAS 3282, 14:00 – 15:00

Speaker: None (Discussion session)
Title: Exercises and Problems day.