Invited speakers include:
Thursday, Oct 5
Location: Building Room Name
3:30 - 4:30, Norman Mayer Room 101, Mark Freidlin
Title: "On The Mathematical Works of Alexander Wentzell"
Friday, October 6
Location: Building Room Name
10:00-11:00, Stanley Thomas 316, Nicolai Krylov
Title: "Poisson stochastic process and basic Shauder and Sobolev estimates in the theory of parabolic equations"
11:00-12:00, Stanley Thomas 316, Sri Namachchivaya
Title: "Random and Data Driven Dynamical Systems"
2:30-3:30, Gibson Hall 126, Katie Newhall
Title: "Metastability of the Nonlinear Wave Equation"
3:30-4:30, Gibson Hall 126, Eric Vanden-Eijnden
Title: "A tour on the computational side of large deviation theory"
Saturday, October 7
Location: Building Room Name
9:00-10:00, Gibson Hall 126, Sandra Cerrai
Title: "Fast Flow Asymptotics for Stochastic Incompressible Viscous Fluids in the Plane and SPDEs on Graphs"
10:30-11:30, Gibson Hall 126, Leonid Koralov
Title: "Large Time Behavior of Randomly Perturbed Dynamical Systems"
11:30-12:30, Gibson Hall 126, Jay Newby
Title: "Spontaneous excitability in the Morris-Lecar model driven by ion channel noise"
Title: Fast Flow Asymptotics for Stochastic Incompressible Viscous Fluids in the Plane and SPDEs on Graphs
Fast advection asymptotics for a stochastic reaction-diffusion-advection equation is studied in this paper. To describe the asymptotics, one should consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow."
Title: On The Mathematical Work of Alexander Wentzell
I will review the mathematical achievements of Alex Wentzell: Boundary conditions for diffusion processes and PDEs, Large deviations and long-time influence of random perturbations of dynamical systems, Eigenvalues of stochastic matrices with exponentially small transition probabilities and related elliptic operators, Asymptotic expansions in limit theorems for stochastic processes, Modified averaging principle for stochastic perturbations, and Various problems for PDEs with a small parameter will be considered.
We study several asymptotic problems: averaging of incompressible flows with ergodic components, transition from homogenization to averaging regimes in periodic flows, behavior of randomly perturbed flows with regions where a strong flow creates a trapping mechanism. The problems are related by a common set of probabilistic techniques that are used to solve them.
We show how knowing Shauder and Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the SAME constants as in the case of one-dimensional heat equation.
The method is based on using the Poisson stochastic process. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results. Joint work with E. Priola.
I will present a general overview of several engineered and natural systems with uncertain mathematical models, the multidisciplinary methods required for their analysis, and relevant results. The collection of new mathematical techniques that I will describe lies at the confluence of three important areas: dynamical systems; control and estimation (data assimilation); and information theory. The first part of the talk focuses on the challenges in data assimilation that arise from the interactions between uncertainties, nonlinearities, and observations. I will present rigorous reduced-order data assimilation techniques for high dimensional multi-scale problems. In particular, I will outline how scaling interacts with filtering via stochastic averaging. Optimal sensor placement based on information theoretic concepts will also be discussed. The second part of the talk brings together three interesting themes in dynamical systems — resonances, domains of attraction, and large deviations. The subtleties of their interactions are explored in a canonical way by combining the ideas from dynamical systems, homogenization methods, and large deviations.
Intrinsic noise from molecular fluctuations of voltage-gated ion channels cause spontaneous activity that propagates into and affects local neural network function. A spontaneous action potential is a physical example of a new type of first-exit-time problem: the random time to initiate an excitable event in an excitable system with a single fixed point. I will show how noise induced excitable events in the stochastic Morris-Leccar model are initiated through a predictable sequence of events. In other words, a single mechanism explains how spontaneous activity is generated. Moreover, the generating mechanism contradicts the current understanding of this phenomena. It is widely believed that spontaneous activity in most neurons is driven primarily by fast sodium channels, because these channels govern the fast initiation stage of an action potential. Potassium channels respond much more slowly and are responsible for reseting the membrane voltage at the final stage of the action potential. Contrary to the standard paradigm, metastable dynamics predicts that the primary driving force behind spontaneous initiation of an action potential is the random opening and closing potassium channels.
I will discuss the long-time dynamics of infinite energy solutions to a wave equation with nonlinear forcing. Of particular interest is when these solutions display metastability in the sense that they spend long periods of time in disjoint regions of phase-space and only rarely transition between them. This phenomenon is quantified by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. Numerical results suggest a regime for which the dynamics are not fundamentally different from that observed in the stochastic counterpart in which random noise and damping terms are added to the equation, as well as a regime for which successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.
Unlikely events matter. Massive earthquakes, giant hurricanes, global ﬁnancial crises, and pandemics are just a few examples of events that are rare but have catastrophic consequences. There are also many other situations in which the occurrence of a rare event is less dramatic but important nonetheless: for instance, typical electronic components are required to be extremely reliable, with a very low probability of failure, as are many other engineering devices used in the automobile, aerospace, and medical industries. In all of these examples, it is desirable to accurately estimate the probability and rate of occurrence of rare events. Fredlin-Wentzell large deviation theory (LDT) is the right framework to address these questions in many situations: LDT gives not only the desired probability of the event, but also its most likely pathway (MLP) of occurrence, which is often predictable. In this talk I will describe numerical tools that can be build upon LDT, permit to compute the MLP and can be integrated in importance sampling Monte Carlo sampling techniques and/or data assimilation strategies to dramatically improve their efficiency. I will illustrate these tools on examples arising in various applications, including molecular dynamics, material sciences, and geophysical flows.
Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu