Events
Differential Equations Seminar
- Events
- Differential Equations Seminar
Michael Shearer, North Carolina State University, Riemann Problems for the BBM Equation
ZoomThe BBM equation is a nonlinear dispersive scalar PDE related to the KdV equation. However, it has a non-convex dispersion relation that introduces a variety of novel wave structures. These waves are highlighted by considering numerical solutions of Riemann problems, in which a smoothed step function initial condition u(x,0) exhibits long-time behavior that is a…
Luis Briceno, Universidad Técnica Federico Santa María, Chile, Splitting algorithms for non-smooth convex optimization: Review, projections, and applications
ZoomIn this talk we review some classical algorithms for solving structured convex optimization problems, passing from gradient descent to proximal iterations and going further to modern proximal primal-dual splitting algorithms in the case of more complicated objective functions. We put special attention to constrained convex optimization, in which we accelerate the performance of the algorithms…
Boris Muha, University of Zagreb, Croatia, Analysis of Moving Boundary Fluid-Structure Interaction Problems Arising in Hemodynamics
ZoomFluid-structure interaction (FSI) problems describe the dynamics of multi-physics systems that involve fluid and solid components. These are everyday phenomena in nature, and arise in various applications ranging from biomedicine to engineering. Mathematically, FSI problems are typically non-linear systems of partial differential equations (PDEs) of mixed hyperbolic-parabolic type, defined on time-changing domains. In this lecture…
Konstantin Tikhomirov, Georgia Institute of Technology, USA, Littlewood-Offord inequalities, and random matrices
ZoomI will discuss some aspects of two recent results on singularity of Bernoulli matrices. I will emphasize the use of new Littlewood-Offord-type inequalities in the proofs of the results. Partially based on a joint work with A.Litvak. Zoom meeting Link
Rupert L. Frank, California Institute of Technology, A ‘liquid-solid’ phase transition in a simple model for swarming
ZoomWe consider a non-local optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. In particular, we show that in the large mass regime the ground state density profile is the characteristic function of a round ball. An essential ingredient in our proof…
Petronela Radu, University of Nebraska-Lincoln, USA, Nonlocal models: theoretical and applied aspects
ZoomThe emergence of nonlocal theories as promising models in different areas of science (continuum mechanics, biology, image processing) has led the mathematical community to conduct varied investigations of systems of integro-differential equations. In this talk I will present some recent results on systems of integral equations with weakly singular kernels, flux-type boundary conditions, as well…
Teemu Saksala, North Carolina State University, Generic uniqueness and stability for the mixed ray transform
ZoomWe consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1+1 and 2+2 tensors fields. We show how the…
Oliver Tse, Eindhoven University of Technology, Jump processes as generalized gradient flows
ZoomThe study of evolution equations in spaces of measures has seen tremendous growth in the last decades, of which resulted in general metric space theories for analyzing variational evolutions—evolutions driven by one or more energies/entropies. On the other hand, physics and large-deviation theory suggest the study of generalized gradient flows—gradient flows with non-homogeneous dissipation potentials—which…
Francisco J. Silva, Université de Limoges, Analytical and numerical aspects of variational mean field games
ZoomMean Field Games (MFGs) have been introduced independently by Lasry-Lions and Huang, Malhamé and Caines in 2006. The main purpose of this theory is to simplify the analysis of stochastic differential games with a large number of small and indistinguishable players. Applications of MFGs include models in Economics, Mathematical Finance, Social Sciences and Engineering. In…
Braxton Osting, University of Utah, Consistency of archetypal analysis
ZoomArchetypal analysis is an unsupervised learning method that uses a convex polytope to summarize multivariate data. For fixed k, the method finds a convex polytope with k vertices, called archetype points, such that the polytope is contained in the convex hull of the data and the mean squared distance between the data and the polytope…
Nick Barron, University of Loyola, Applications of Quasiconvex functions to HJ Equations and Optimal Control
ZoomQuasiconvex functions, a major generalization of convex functions, naturally arise in calculus of variations, optimal control and differential games in L-infinity. This connection with HJ equations, representation formulas, obstacle problems, and reach-avoid problems will be discussed. Zoom meeting ID: 802 764 2791
Geng Chen, University of Kansas, Poiseuille flow of nematic liquid crystals via Ericksen-Leslie model
ZoomIn this talk, we will discuss a recent global existence result on the Poiseuille flow of nematic liquid crystals via full Ericksen-Leslie model. The existing results on the Ericksen-Leslie model for the liquid crystals mainly focused on the parabolic and elliptic type models by omitting the kinetic energy term. In this recent progress, we established…
Paata Ivanisvili, North Carolina State University, Enflo’s problem
ZoomA nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. I will speak about the joint work with…
Andrew Papanicolaou, NC State, Consistent Inter-Model Specification for Time-Homogeneous SPX Stochastic Volatility and VIX Market Models
ZoomThis work explores the recovery stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is…
Marta Lewicka, University of Pittsburgh, Expansions of averaging operators and applications
ZoomIn my talk, I will explain the approach of finding solutions to nonlinear PDEs via tug-of-war games. I will focus on the context of p-Laplacian and the non-local geometric p-Laplacian. Zoom meeting: Link
Hédy Attouch, Université Montpellier II, France, Acceleration of first-order optimization algorithms via inertial dynamics with Hessian driven damping
ZoomIn a Hilbert space, for convex optimization, we report on recent advances regarding the acceleration of first-order algorithms. We rely on inertial dynamics with damping driven by the Hessian, and the link between continuous dynamic systems and algorithms obtained by temporal discretization. We first review the classical results, from Polyak's heavy ball with friction method…
Alexander Reznikov, Florida State University, Discrete minimization problems on non-rectifiable sets
ZoomWe survey the known results concerning the minimal Riesz energy on sufficiently smooth sets, and present some new results on fractal sets, which are the key examples of non-rectifiable sets. In particular, we will talk about the connection of these problems to ergodic theory and probability theory: the key step in our proofs is applying…
Roberto Cominetti, Universidad Adolfo Ibáñez, Chile, Convergence rates for Krasnoselskii-Mann fixed-point iterations
ZoomA popular method to approximate a fixed point of a non-expansive map is C is the Krasnoselskii-Mann iteration. This covers a wide range of iterative methods in convex minimization, equilibria, and beyond. In the Euclidean setting, a flexible method to obtain convergence rates for this iteration is the PEP methodology introduced by Drori and Teboulle…