In this talk we will discuss new classes of models that seek to describe evolution of a congregation of agents based on laws of self-organization. These models appear in a broad range of applications -- from biological sciences to social behavior. We focus on two long time phenomena: flocking and alignment. It has been a mathematical challenge…
This talk focuses on differential problems describing the flow of a viscous fluid in deformable domains. Such problems include flow in compliant tubes, often adopted for the modeling of arterial blood flow, and flow through deformable porous media, often adopted for the modeling of tissue perfusion. The mixed hyperbolic-parabolic-elliptic nature of these systems guides the…
This talk will be devoted to a fundamental question on the compactness of sets of solutions. The key concept in this study is the Kolmogorov epsilon-entropy which is the logarithm of the minimum number of elements in an epsilon-covering of a given (totally bounded) set. I will use this concept to provide a sharp estimate…
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical…
In nonlinear conservation laws the flux function f(u) is usually single valued, but in many important applications it is hysteretic, i.e., it assigns different values depending on whether the input u(t) is increasing or decreasing in t. We present our recent results on a hysteresis model built with a collection of auxiliary ODEs under constraints. The model shares some similarities…
We consider a controllability problem for the evolution of a set in V(t). This problem was originally motivated by a model where a dog controls a flock of sheep. Here, V(t) is the region occupied by the sheep and the position of the dog is regarded as a control function. We will discuss necessary conditions…
In this talk I’ll describe an overview of results on the Navier-Stokes equations for viscous compressible flows both in the deterministic and stochastic frame- work. A contrast will be drawn between the one-dimensional flows and the multidimensional case. In the case of 1d isentropic compressible flow, the existence of invariant mea- sures will be established…
This talk will address deterministic mean field games for which agents are restricted in a closed domain of R^n with smooth boundary. In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of solutions to the minimization…
In recent years, the study of evolution equations featuring a fractional Laplacian has received much attention due to the fact that they have been successfully applied into the modelling of a wide variety of phenomena, ranging from biology, physics to finance. The stochastic process behind fractional operators is linked, in the whole space, to an…
In the first part of this work, we consider the strategy of realizing the solution of the three-dimensional linear wave equation with radial Cauchy data as a limit of radial exterior solutions satisfying vanishing Neumann and Dirichlet conditions, on the exterior of vanishing balls centered at the origin. We insist on robust arguments based on energy methods and strong convergence. Our findings show that while one…
We consider nonparametric estimation of probability measures for parameters in problems where only aggregate (population level) data are available. We summarize an existing computational method for the estimation problem which has been developed over the past several decades. Theoretical results are presented which establish the existence and consistency of very general (ordinary, generalized and other)…
The incompressible Euler equation of fluid mechanics describes motion of ideal fluid, and was derived in 1755. In two dimensions, global regularity of solutions is known, and double exponential in time upper bound on growth of the derivatives of solution goes back to 1930s. I will describe a construction of example showing sharpness of this…
Conservation laws with discontinuous flux functions arise in various models. In this talk we consider solutions to a class of conservation laws with discontinuous flux, where the flux function is discontinuous in both time and space, but regulated in the two variables. Convergence and the uniqueness of the vanishing viscosity limit for the viscous equation…
In this talk, we shall review the Hamilton-Jacobi theory for A Fully Convex Bolza (FCB) problems when the data has no implicit state constraints and is coercive, in which case the minimizing class of arcs are Absolutely Continuous (AC).
His talk concerns the study of criticality of Lagrange multipliers in variational systems that have been recognized in both theoretical and numerical aspects of optimization and variational analysis. In contrast to the previous developments dealing with polyhedral KKT systems and the like, we now focus on general nonpolyhedral systems that are associated, in particular, with…
Crumples in a sheet of paper, wrinkles on curtains, cracks in metallic alloys, and defects in superconductors are examples of patterns in materials. A thorough understanding of the underlying phenomenon behind the pattern formation provides a different prospective on the properties of the existing materials and contributes to the development of new ones. In my talk…
We discuss mathematical modeling and analysis of the incompressible viscous flow at the interface of permeable media. Very recently, a simplified theory with asymptotic modeling and related approximations was extensively developed by to provide physically relevant jump interface conditions for the two- or three-dimensional non-inertial flow at the interface of a permeable medium. The results…
In this talk we will propose a new framework to model control systems in which a dynamic friction occurs. In particular, such a framework is motivated by the study of the movement of a robotic root tip in the soil. The model consists in a controlled differential inclusion with a dissipative, upper semi-continuous right hand…
In this talk we discuss a point-wise state constraint problem for a general class of PDEs optimal control problems and sparsity optimization. We use the penalty formulation and derive the necessary optimality condition based on the Lagrange multiplier theory.The existence of Lagrange multiplier associated with the point-wise state constraint as a measure is established. Also we…
A fundamental problem in Bayesian inference and statistical machine learning is to efficiently sample from probability distributions. Standard Markov chain Monte Carlo methods could be prohibitively expensive due to various complexities of the target distribution, such as multimodality, high dimensionality, large datesets, etc. To improve the sampling efficiency, several new interesting ideas/methods have recently been proposed in the community…