Special Stochastic Systems Seminar
Abstract: Stochastic nonlinear filtering is one of the core areas of application of stochastic calculus. The basic object under consideration is a pair of stochastic processes $(X_t, Y_t)_{t \ge 0}$ where $(X_t)$ is called the signal process and $(Y_t)$ the observation process. The central problem in nonlinear filtering is the study of the measure valued process $(\Pi_t)$ which is the conditional distribution of $X_t$ given $\sigma\{Y_s: 0 \le s \le t\}$. This measure valued process is called the nonlinear filter. In the classical setting of nonlinear filtering, which is considered in this talk, the signal is taken to be a Markov process with values in some Polish space $E$ and the observations are given via the relation: \[ Y_t = \int_0^t h(X_s) ds + W_t, \] where $(W_t)$ is a standard $d$ dimensional Brownian motion independent of $(X_t)$ and $h$ is a map from $E \to \R^d$.
In this talk we will present various long time qualitative properties of the nonlinear filter. It can be shown that $(\Pi_t)$ is a $\clp(E)$ (the space of probability measures on $E$) valued Markov process and if $(X_t)$ is Feller and $h(\cc)$ is continuous then $(\Pi_t)$ is also Feller. Finally if $(X_t)$ has a unique invariant measure then under a mild technical condition $(\Pi_t)$ also admits a unique invariant measure. One of the applications of these results is the study of asymptotic errors for the nonlinear filter.
The nonlinear filter is computed using three pieces of information: the initial law of the signal, the transition probability function and the observation trajectory. In most practical problems one does not have access to the exact initial law or the transition probability function. Even in the ideal situation, in order to do explicit computations various approximations need to be made. Thus it is of central importance to study the sensitivity of the filter to such approximations. A crucial obstacle in the study of long term properties of approximate filters is that an incorrectly initialized filter, denoted hereafter as $(\ti \Pi_t)$, is in general not Markov. However, it can be shown that the pair process: $(X_t, \ti \Pi_t)$ is indeed a Markov process. In this talk we will consider the longstanding problem of existence and uniqueness of invariant measures for this pair process. The uniqueness of the invariant measure leads to various important results on the asymptotic behavior of approximate filters. Using recent results on asymptotic stability of filters we will present sufficient conditions for the uniqueness of the invariant measure for the above pair process to hold.
Brown University Center for Statistical Sciences Seminar
Monday, January 31, 2000... Continued
Brown University Center for Statistical Sciences Seminar
Abstract: We obtain close-form expressions for the prices and optimal hedging strategies of American put options in the presence of an ``up-and-out" barrier, both with and without constraints on the short-selling of stock. The constrained case leads to an interesting stochastic optimization problem of mixed optimal stopping/singular control type; this is reduced to a variational inequality, which is then solved explicitly.
Center for Fluid Mechanics Seminar
Abstract: Energy variational principles for various steady flows are considered. All those principles state that on the set of all `isovortical flows' of an ideal fluid the kinetic energy attains its stationary values at steady flows. We start with formulation of the Arnold principle in the simplest possible way. Then we give its generalizations for more sophisticated systems, such as stratified fluid, magneto-hydrodynamics (MHD) and dynamical system `solid+fluid'. All considerations are based on new forms of the `generalized isovorticity conditions' which are formulated in the spirit of the early Arnold's papers. Short discussion as well as several applications of stability criteria are given. The lecture is based on a series of joint papers with Prof. H.K.Moffatt.
Applied Mathematics Colloquium
Scientific Computing Seminar
Abstract: Adaptive Mesh Refinement (AMR) calculations carried out on structured meshes play an exceedingly important role in several areas of science and engineering. This is so not just because AMR techniques allow us to carry out calculations very efficiently but also because they model very precisely the multi-scale fashion in which nature itself works. Many AMR applications are also amongst the most computationally intensive calculations undertaken making it necessary to use parallel supercomputers for their solution. While class library-based approaches are being attempted for parallel AMR we point out in this talk that recent advances in the Fortran 90/95 standard and the OpenMP standard now make it possible to carry out highly parallel AMR calculations using language-based approaches. The language-based approaches offer several advantages over library-based approaches, the two principal ones being portability across parallel platforms and the best possible utilization of Distributed Shared Memory (DSM) hardware on machines that have such hardware. They also free up the applications scientist from being constrained by the static features of a class library. The choice of Fortran also ensures maximal reuse of pre-existing Fortran 77 applications and full Fortran 77-based processing efficiency on each computational node. Our implementation of the ideas presented here in the author's RIEMANN framework essentially permits any serial, uniform grid, stencil-based Fortran code to be turned into a parallel AMR code!
In this talk we first describe our strategy for using Fortran 90 in an object-oriented fashion. This permits AMR applications to be expressed in terms of familiar abstractions that are natural to the process of solving AMR hierarchies. We then describe the OpenMP features that are useful for parallel processing of AMR hierarchies in a load balanced fashion on multiprocessors. The automatic, parallel regridding of AMR hierarchies is also described. We then present a very efficient load balancer and show how it is to be used for load balanced solution of AMR hierarchies. Our load balancer is extremely general and should also see use in other disciplines. We follow this up with the application of the parallel AMR techniques developed here to the solution of elliptic and hyperbolic problems. For our elliptic problem we choose parallel, self-adaptive multigrid as an example. For our hyperbolic problem we choose time-dependent MHD as an example. In either case illustrative information is given about the adaptive processing of these systems. We also provide detailed scalability studies for both the above-mentioned problems which show that our methods scale extremely well up to several hundreds of processors!
PDE Seminar
Abstract: We consider astrophysical jet flow associated with star formation. This is modeled by a system of conservation laws. The non-linear nature of these differential equations requires numerical discretization in so-called conservation form. For our astrophysical problem this puts us into a quandary: internal variables like pressure and temperature can no longer be computed accurately. We propose the following way out: Embed the astrophysical jet model into a more complete model. There it is readily possible to compute the internal variables. Then we project these variables back to the original model. We can prove that the translation of this into a numerical procedure leads to reliable solutions. I will make this talk accessible to graduate students. It will be illustrated with pictures showing astronomical observations and numerical simulations.
Department of Mathematics Colloquium
<--- 2000 Index