Past

Spectral methods are a class of methods for solving PDEs numerically.

If the solution is analytic, it is known that these methods converge

exponentially quickly as a function of the number of terms used.

The basic spectral method only works in regular geometry (rectangles/disks).

A huge amount of effort has gone into extending it to

domains with a complicated geometry. Domain decomposition/spectral

element methods partition the domain into subdomains on which the PDE

can be solved (after transforming each subdomain into a

regular one). We take the dual approach - embedding the domain into

a larger regular domain - known as the fictitious domain method or

domain embedding. This method is extremely simple to implement and

the time complexity is almost the same as that for solving the PDE

on the larger regular domain. We demonstrate exponential convergence

for Dirichlet, Neumann and nonlinear problems. Time permitting, we

shall discuss extension of this technique to PDEs with discontinuous

coefficients.

A singular perturbation analysis is presented to analyze various PDE models in a
two-dimensional domain that contain localized regions of non-uniform behavior. A
key theme of this talk is to present a unified mathematical approach, based on
an asymptotic analysis involving logarithmic series and certain Green's function
techniques, that can be used to treat a variety of PDE models such as diffusion
or eigenvalue problems in perforated domains or reaction-diffusion models with
spot-type behavior.

    I sketch, using Kichenassamy - Rendall ideas, a simplified and generalized proof of the Fuchs theorem for differential equations with a singularity. I use the theorem to construct solutions of polarized and half polarized U(1) symmetric vacuum spacetimes with "Asymptotically Velocity Terms Dominated" (AVTD) behaviour near the singularity. I show that the definition I give of half polarization for U(1) symmetric vacuum space-times is a necessary and sufficient condition for non polarized such spacetimes to have this AVTD behaviour.  

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.

  The talk is based on my recent joint work with Zhechun Hu and Wei Sun. We consider a nonlinear filtering problem for general right continuous Markov processes associated with semi-Dirichlet forms. We show that in our general setting the filtering processes satisfy also DMZ (Duncan-Mortensen-Zakai) equation. The uniqueness of the solutions of the filtering equations are verified through their Wiener chaos expansions. Our results on the Wiener chaos expansions for nonlinear filters with possibly unbounded observation functions are novel and have their own interests. We investigate further the absolute continuity of the filtering processes with respect to the reference measures and derive the density equations for the filtering processes.