Lefschetz Center for Dynamical Systems Seminar
MS-B284, Los Alamos, New Mexico, 87545 | |
**PLEASE NOTE CHANGE OF LOCATION FOR THIS WEEK ONLY** |
Abstract: A dispersion-managed (DM) system, which is a system with periodical variation of dispersion along an optical fiber, is one of the key components of current development of ultrafast high-bit-rate optical communication lines. DM system is governed by a nonlinear Schroedinger equation with variable dispersion. Provided nonlinearity is small enough this equation can be averaged over period of dispersion variation resulting in path-averaged integro-differential Gabitov-Turitsyn equation. The soliton solution is obtained by iterating the path-averaged equation analytically and numerically. An efficient numerical algorithm for obtaining of DM soliton shape is developed. The envelope of soliton oscillating tails is found to decay exponentially in time while the oscillations are described by a quadratic law.
Brown University Center for Statistical Sciences Seminar
Abstract: Recently, it has been of great interest to estimate causal effects of treatment regimens in development of HIV therapies. The HIV Epidemiology Research Study (HERS) enrolled HIV infected women and they received antiretroviral therapy regimens, such as single antiretroviral therapies or combinations of two or more of them, known as highly active antiretroviral therapy (HAART) for over six years. In this paper, we estimate the long term causal effects of the therapies on CD4 count, isolating it in the presence of time varying confounding by fitting the marginal structural models (MSMs) introduced by Robins (1998). We address several potential complications, such as high rate of attrition and lack of availability of HAART for the first two years of follow up. We fit MSMs under assumption that all confounders are measured, and conduct sensitivity analysis to quantify potential bias due to unmeasured confounders.
Stochastic Systems Seminar
with I. Kontoyiannis (Brown University); Christof Schuette and Wilhelm Huisinga (Freie Universitt Berlin) | |
**PLEASE NOTE CHANGE OF LOCATION FOR THIS WEEK ONLY** |
Abstract: The classical description of molecular processes deals with the microscopic configuration of the molecule, i.e., all atomic positions and momenta, and leads to a mathematical model in terms of Hamiltonian equations of motions for all atoms in the system. Markovian models, such as the stochastic differential equation (SDE) Langevin models, are a common alternative. For models arising in biological systems with hundreds or thousands of atoms, a Markovian model is far too complex for exact solution. How can we devise alternative models which capture essential features?
Given a Markov process on a general state space, in continuous time, we consider the following questions:
(i) When does the semigroup have a discrete spectrum? When is there a `spectral gap'?
(ii) When can the generator be approximated by a finite-rank operator, thereby approximating the complex process by a finite state-space Markov chain?
(iii) What is the physical significance of high-order eigenfunctions?
We find that a spectral gap is equivalent to V-uniform ergodicity, which is characterized by a simple Lyapunov function criterion. A strengthening of this assumption implies that the spectrum is discrete. The stronger Lyapunov criterion is closely related to the standard test for exponential tightness in the Donsker-Varadhan theory of large deviations.
Real eigenfunctions provide a decomposition of the state space into `almost' -absorbing subsets. It is shown that the process mixes rapidly in each of these subsets prior to exiting.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: In this talk we will present an overview of two statistical problems, related to dynamical systems and long range dependence, for which both positive and negative results have been established.
The first problem is how to remove observation noise from measurements of the successive states of a deterministic dynamical system. We will describe a simple filtering scheme proposed by S. Lalley and establish its consistency with suitably bounded noise. We will also present a nice result of Lalley's showing that, for many dynamical systems, the removal of Gaussian observation noise is not possible.
The second problem is hypothesis testing for families of dependent processes: given two known, disjoint families of ergodic processes, when is it possible to establish the membership of a process in their union solely from increasing obsevations of that process? We will briefly discuss some sufficient conditions for distinguishability, and give an example of two families that cannot be distinguished by a procedure. If time permits we will discuss an application of the negative result to the problem of density estimation from ergodic processes.
Scientific Computing Seminar
Abstract: I'll present a fast modular numerical method for solving general moving interface problems. It simplifies code development by providing a black-box solver which moves a given interface one step with given normal velocity. The method evaluates an explicit second-order semi-Lagrangian advection formula with efficient geometric algorithms, and extracts the interface by fast adaptive contouring. Spatial and temporal resolutions are decoupled, permitting grid-free adaptive refinement of the interface. A modular implementation couples to fast geometry and PDE solvers to solve complex moving interface problems.
<--- 2001 Index