NCSU Differential Equations/Nonlinear Analysis Seminar Schedule Spring 2022

Wednesday, Feb 16, 15:00-16:00

Zoom meeting: Link
Speaker: Barbara Keyfitz, The Ohio State University, USA
Title: Hyperbolic Conservation Laws and Stability in L^2
Abstract: Recently there has been considerable research into the stability of shocks in systems of conservation laws, with stability understood in some square-integrable sense. In this talk I will give some background on systems of nonlinear hyperbolic partial differential equations (known as conservation laws), and on the issues concerning well-posedness. There are reasons that the still-unsolved problem of existence of solutions in more than a single space dimension is expected to involve L-2. We present a simple, rather discouraging, example that shows why that goal seems elusive. This is joint work with Hao Ying.

Wednesday, Feb 23, 15:00-16:00

Zoom meeting: Link
Speaker: Terry Rockafellar, University of Washington, USA
Title: Augmented Lagrangian Methods and Local Duality in Nonconvex Optimization
Abstract: Augmented Lagrangians were first employed in an algorithm for solving nonlinear programming problems with equality constraints. However, the approach was soon extended to inequality constraints and shown in the case of convex programming to correspond to applying the proximal point algorithm to solve a dual problem. Recent developments make it possible now to articulate that ALM approach in extensions far beyond classical nonlinear programming. This is tied to revelations of a kind local dual problem, based on advances in understanding second-order sufficient conditions for local optimality. Surprising insights about stepsizes are obtained even for the classical NLP implementation.

Wednesday, Mar 02, 15:00-16:00

Zoom meeting: Link
Speaker: Teemu Pennanen, King’s College London
Title: Convex duality in nonlinear optimal transport
Abstract: We study problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of alarge class of related problems in probability theory and allows for generalizations of the classical problem formulations. General results on convex duality yield dual problems and optimality conditions for these problems. When the objective takes the form of a convex integral functional, we obtain more explicit optimality conditions and establish the existence of solutions for a relaxed formulation of the problem. This covers, in particular, the classical mass transportation problem and its nonlinear generalizations.

Wednesday, Mar 09, 15:00-16:00

Zoom meeting: Link
Speaker: Ivan Yotov, University of Pittsburgh
Title: A nonlinear Stokes-Biot model for the interaction of a non-Newtonian fluid with poroelastic media
Abstract: A nonlinear model is developed for fluid-poroelastic structure interaction with quasi-Newtonian fluids that exhibit a shear-thinning property. The flow in the fluid region is described by the Stokes equations and in the poroelastic medium by the quasi-static Biot model. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type, which is weakly enforced through a Lagrange multiplier. We establish existence and uniqueness of the solution of the weak formulation using non-Hilbert
spaces. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. Applications to hydraulic fracturing and arterial flows are presented.

Wednesday, Mar 23, 15:00-16:00

Zoom meeting: Link
Speaker: Juan Carlos, Centro de Modelización Matemática, Ecuador
Title: Bilevel learning for inverse problems
Abstract: In recent years, novel optimization ideas have been applied to several inverse problems in combination with machine learning approaches, to improve the inversion by optimally choosing different quantities/functions of interest. A fruitful approach in this sense is bilevel optimization, where the inverse problems are considered as lower-level constraints, while on the upper-level a loss function based on a training set is used. When confronted with inverse problems with nonsmooth regularizers or nonlinear operators, however, the bilevel optimization problem structure becomes quite involved to be analyzed, as classical nonlinear or bilevel programming results cannot be directly utilized. In this talk, I will discuss on the different challenges that these problems pose, and provide some analytical results as well as a numerical solution strategy.

Wednesday, Mar 30, 15:00-16:00

Zoom meeting: Link
Speaker: Stéphane Gaubert, École Polytechnique, France
Title: What tropical geometry tells us about linear programming and zero-sum games
Abstract: Tropical convex sets arise as “log-limits” of parametric families of classical convex sets. The tropicalizations of polyhedra and spectrahedra are of special interest, since they can be described in terms of deterministic and stochastic games with mean payoff. In that way, one gets a correspondence between classes of zero-sum games, with an unsettled complexity, and classes of semi-algebraic convex optimization problems. We shall discuss applications of this correspondence, including a counter example, showing that interior point methods are not strongly polynomial.

Wednesday, April 6, 15:00-16:00

Zoom meeting: Link
Speaker: Bianchini Stefano, SISSA, ITALY
Title: TBA

Wednesday, April 13, 15:00-16:00

Zoom meeting: Link
Speaker: Robin Neumayer, CMU, USA
Title: Quantitative Faber-Krahn Inequalities and Applications
Abstract: Among all drum heads of a fixed area, a circular drum head produces the vibration of lowest frequency. The general dimensional analogue of this fact is the Faber-Krahn inequality: balls have the smallest principal Dirichlet eigenvalue among subsets of Euclidean space with a fixed volume. I will discuss new quantitative stability results for the Faber-Krahn inequality on Euclidean space, the round sphere, and hyperbolic space, as well as an application to the Alt-Caffarelli-Friedman monotonicity formula used in free boundary problems. This is based on joint work with Mark Allen and Dennis Kriventsov.

Wednesday, April 20, 15:00-16:00

Zoom meeting: Link
Speaker: Francesca Bucci, Università degli Studi di Firenze, Italy
Title: Riccati theory in the realm of PDE’s: state of the art and recent advances in the optimal control of evolution equations with memory
Abstract: The well-posedness of Riccati equations plays a central role in the study of the optimal control problem with quadratic functionals for linear partial differential equations (PDEs). Indeed, it allows the synthesis of the optimal control by solving the Riccati equation corresponding to the minimization problem, and then of the closed-loop equation. In this lecture I will first recall the principal steps of an established approach to the linear-quadratic problem for infinite-dimensional systems representing PDEs with distributed or boundary control. The failures or challenges stemming from the presence of unbounded control operators, combined with a hyperbolic or composite dynamics, will be highlighted. Finally, I will outline how the said approach proves successful even in the case of certain evolution equations with finite memory, thereby providing a first extension (to the realm of PDE’s) of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context.

(Talk based on past work with P. Acquistapace (Pisa) and I. Lasiecka (Memphis), as well as on ongoing joint work with P. Acquistapace.)