Organizers: Minh Bui and Sarah Strikwerda

Wednesday, November 11, 13:00 – 14:00
Speaker: Steven Gilmore
Title: Optimal Feedback Strategies of a Debt Management Problem and Fine Regularity for Nonlocal Balance Laws

Abstract: In this talk, we discuss some nonlinear and nonlocal PDEs and their applications. We focus on two areas: 1. Providing a detailed analysis of systems of Hamilton-Jacobi equations arising from models of optimal debt management with bankruptcy risk. In particular, we establish the existence results for the system and recover the optimal feedback strategy of repayment for the borrower. 2. Study deeper regularity of the Burgers-Poisson equation, a nonlinear dispersive model derived as a simplified model of shallow water waves.

Wednesday, November 4, 13:00 – 14:00
Problem Session

Wednesday, October 28, 13:00 – 14:00
Speaker: Sarah Strikwerda
Title: Analysis of Equations of Nonlinear Poroelasticity

Abstract: In this talk, I will present the analysis in the paper “Analysis and Numerical Approximations of Equations of Nonlinear Poroelasticity” by Yahzhao Can, Song Chen, and A.J. Meir. I will focus on how Rothe’s method was used to show the existence of a weak solution.

Wednesday, October 21, 13:00 – 14:00
Problem Session

Wednesday, October 14, 13:00 – 14:00
Speaker: Steven Gilmore
Title: Debt Management 101

Abstract: We discuss a system of Hamilton-Jacobi equations resulting from a stochastic optimal debt management model. In particular, the model relates monetary policy to sovereign debt vulnerability. We will give an overview of past results and present future questions for study. 

Wednesday, October 7, 13:00 – 14:00
Speaker: Minh Bui
Title: Warped Proximal Iterations for Monotone Inclusions

Abstract: Resolvents of set-valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclu- sions. We propose a generalization of this notion, called warped resolvent, which is constructed with the help of an auxiliary operator. The properties of warped resolvents are investigated and connections are made with existing notions. Abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems.

Wednesday, September 30, 13:00 – 14:00
Problem Session

Wednesday, September 16, 13:00 – 14:00
Speaker: Zev Woodstock
Title: Signal Recovery from Nonlinear Transformations

Abstract: We consider the problem of recovering a signal from nonlinear transformations, under convex constraints modeling a priori information. Standard feasibility and opti- mization methods are ill-suited to tackle this problem due to the nonlinearities. We show that, in many common applica- tions, the transformation model can be associated with fixed point equations involving firmly nonexpansive operators. In turn, the recovery problem is reduced to a tractable common fixed point formulation, which is solved efficiently by a provably convergent, block-iterative algorithm. Applications to signal and image recovery are demonstrated. Inconsistent problems are also addressed.

Wednesday, September 9, 13:30 – 14:30
Speaker: Steven Gilmore
Title: SBV Regularity for Hyperbolic Conservation Laws

Abstract: In this talk, we discuss a regularity theorem for admissible solutions of a class of hyperbolic conservation laws. We will consider global solutions to the Cauchy problem, which are known to (generally) lose their regularity in finite time. It has been shown that these solutions are functions of bounded variation i.e. L^1 functions with distributional derivatives which are Radon measures. The singular part of this measure can be decomposed into the jump part which is of dimension n-1 and the Cantor part which is of “fractal dimension” between n and n-1. We will explore under what assumptions the derivatives of solutions do not contain a Cantor part and outline the proof of the positive result. 

Wednesday, September 2, 13:00 – 14:00
Speaker: Sarah Strikwerda
Title: Necessary Optimality Conditions for Poro-Viscoelastic Model

Abstract: We consider an optimal control problem subject to a nonlinear poro-visco-elastic model with applications to fluid flows through biological tissues. In particular, our goal is to optimize the fluid pressure using either distributed or boundary control. I will present an explanation on why traditional methods of deriving necessary optimality conditions fail without increasing regularity of the solution to the model as well as a method to increase the regularity.

Wednesday, August 26, 13:00 – 14:00
Problem Session

Wednesday, August 19, 13:00 – 14:00
Speaker: Minh Bui
Title: Bregman Forward-Backward Operator Splitting

Abstract: We propose an iterative method for finding a zero of the sum of two maximally monotone operators in reflexive Banach spaces. One of the operators is single-valued, and the method alternates an explicit step on this operator and an implicit step on the other one. Both steps involve the gradient of a convex function that is free to vary over the iterations. The convergence of the resulting forward-backward splitting method is analyzed using the theory of Legendre functions, under a novel assumption on the single-valued operator that captures various existing properties. When applied to minimization problems, rates are obtained for the objective values. The proposed framework unifies and extends several iterative methods which have thus far not been brought together, and it is also new in Euclidean spaces.