The lecture will consider the case of evolution equations, endowed with a "free energy," for which the initial value problem possesses multiple solutions, with an eye to examining whether the particular solution that maximizes the rate of energy decay enjoys a special status.
The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. Such phenomena, also known as the Talbot effect, have been observed in dispersive waves, optics, and quantum mechanics, and lead to intriguing connections with exponential sums arising in number…
In this talk we discuss an algorithm to overcome the curse of dimensionality, in possibly non-convex/time/state-dependent Hamilton-Jacobi partial differential equations. They may arise from optimal control and differential game problems, and are generally difficult to solve numerically in high dimensions. A major contribution of our works is to consider an optimization problem over a single…
We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. We have to develop for this new laws of large numbers at the exponential scale. But our…
I will discuss a nonparametric modeling approach for forecasting stochastic dynamical systems on smooth manifolds embedded in Euclidean space. This approach allows one to evolve the probability distribution of non-trivial dynamical systems with an equation-free modeling. In the second part of this talk, I will discuss a nonparametric estimation of likelihood functions using data-driven basis functions and the…
From crawling cells to orca whales, swimming in nature occurs at different scales. The study of swimming across length scales can shed light onto the biological functions of natural swimmers or inspire the design of artificial swimmers with applications ranging from targeted drug delivery to deep-water explorations. In this talk, I will present experiments and simulations…
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…