Spring 2017
Time & Location: All talks are on Wednesdays in Gibson Hall 414 at 3:00 PM unless otherwise noted.
Organizer: Gustavo Didier
February 8
Scalings and saturation in infinite-dimensional control problems with applications to stochastic partial differential equations.
David HerzogTulane University
Abstract:
We discuss scaling methods which can be used to solve low mode control problems for nonlinear partial differential equations. These methods lead naturally to a infinite-dimensional generalization of the notion of saturation, originally due to Jurdjevic and Kupka in the finite-dimensional setting of ODEs. The methods will be highlighted by applying them to specific equations, including reaction-diffusion equations, the 2d/3d Euler/Navier-Stokes equations and the 2d Boussinesq equations. Applications to support properties of the laws solving randomly-forced versions of each of these equations will be noted.
February 15
Continuum approximation of invasion probabilities for stochastic population models
Rebecca BorcheringUniversity of Florida, Mathematics
Abstract:
When an individual with a novel trait is introduced in a new environment, we would like to understand what drives the likelihood that its lineage will persist. In deterministic population models, whether the invasive population “succeeds” often depends on whether the parameters of the system fall in a super- or sub-critical regime. In stochastic population models, the parameters must be super-critical for there to be a substantial of invasion, but even in the super-critical regime, chance alone allows for many invasive lineages to quickly go extinct.
In this talk, we compare popular continuum approximations for the invasion probability to its exact solution. In particular, methods known as "Diffusion (or Stochastic Differential Equation) Approximation" and "Exponential Approximation" are derived. We find analytical expressions for these approximations in the large population limit and then use numerical methods to evaluate the performance of the approximation methods for finite populations. Interestingly we find that the diffusion approximation fails to obtain the correct large population limit, but can perform well for small populations that experience near critical dynamics. The exponential approximation obtains the right large population limit in the supercritical regime, but fails to capture nonmontonic characteristics of the invasion probability for small to intermediate sized populations.
March 8
Simulating the Mean Efficiently and to a Given Tolerance
Fred HickernellIllinois Institute of Technology (HOST James Hyman)
Abstract:
Various application problems are formulated as means of random variables, e.g., option pricing, multivariate probabilities, and Sobol' indices. When the distribution of the random variable is complicated, computer simulation can be used to approximate the population mean by a sample mean. Two big questions are how to draw the sample to minimize error, and what sample size is needed to guarantee the desired accuracy. This talk describes our best answers so far. The sampling error can be decomposed into a product of three factors.
One of these factors, the discrepancy, describes the efficiency of simulation. By assuming that the distribution of the random variable is nice, in some mathematically precise way, we can rigorously bound the error of the simulation and thus determined the sample size required. Host: Mac Hyman
Two recent papers that are related to this talk: https://arxiv.org/abs/1702.01491 and https://arxiv.org/abs/1702.01487
March 13
(Special Day)
Transition probabilities for degenerate diffusions arinsing in population genetics
Camelia PopUniversity of Minnesota
Abstract:
We provide a detailed description of the structure of the transition probabilities and of the hitting distributions of boundary components of a manifold with corners for a degenerate strong Markov process arising in population genetics. The Markov processes that we study are a generalization of the classical Wright-Fisher process. The main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward Kolmogorov equation defined on a compact manifold with corners, which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities. This is joint work with Charles Epstein.
March 15
(Joint with Applied Math)
Asymptotic analysis for randomly forced magnetohydrodynamics
Susan FriedlanderUniversity of Southern California
Abstract:
We consider the three dimensional magnetohydrodynamics (MHD) equations in the presence of stochastic forcing as a model for magnetostrophic turbulence. For scales relevant to the Earth's fluid core this MHD system is very rich in small parameters. We discuss results concerning the asymptotics of the stochastically forced PDEs in the limit of vanishing parameters. In particular we establish that the system sustains ergodic statistically steady states thus providing a rigorous foundation for magnetostrophic turbulence.
March 22
Long and medium time behavior of stochastically modeled biochemical reaction networks
David AndersonUniversity of Wisconsin
Abstract:
If the abundances of the constituent molecules of a biochemical reaction system are sufficiently high then their concentrations are typically modeled by a coupled set of ordinary differential equations (ODEs). If, however, the abundances are low then the standard deterministic models do not provide a good representation of the behavior of the system and stochastic models are used.
In this talk, I will focus on stochastic models of biochemical reaction systems and discuss two topics. In the first, I will provide a large class of interesting nonlinear models for which the stationary distribution can be solved for explicitly. Such results are particularly useful for averaging purposes in multi-scale settings. The second topic will pertain to an interesting class of models that exhibits fundamentally different long-term behavior when modeled deterministically or stochastically (the ODEs predict stability, the stochastic model goes extinct with probability one). Recent work pertaining to the distribution of the stochastic model on compact time intervals (similar to the study of the quasi-stationary distribution) resolves the apparent discrepancy between the modeling choices.
April 12
The Causes of Metastability and Their Effects on Transition Times
Katie NewhallUniversity of North Carolina - Chapel Hill
Abstract:
Many experimental systems can spend extended periods of time relative to their natural time scale in localized regions of phase space, transiting infrequently between them. This display of metastability can arise in stochastically driven systems due to the presence of large energy barriers, or in deterministic systems due to the presence of narrow passages in phase space. To investigate metastability in these different cases, I take a Langevin equation and determine the effects of small damping, small noise, and dimensionality on the dynamics and mean transition time. Of particular interest is what happens in the infinite dimensional limit, a stochastic partial differential equation, and the question of what ensemble this system appears to sample over time. Both analytical and numerical results will be presented.
April 19
Bayesian models for overdispersed binomial responses using Stan
Aleksandra GorzyckaTulane University
Abstract:
In applying probability models it is often found that the data exhibit greater variability than is predicted by the implicit mean-variance relationship for the assumed distribution. This phenomenon of overdispersion has been widely considered in the literature, particularly in relation to the binomial distribution. I will present a method for fitting probability models to data sets in a Bayesian framework using the probabilistic programming language Stan, with application to food consumption survey data collected by the World Food Programme in the Democratic Republic of the Congo.
