Stochastic Systems Seminar
Abstract:
I will discuss an importance sampling method for certain rare event problems
involving small noise diffusions. Standard Monte Carlo schemes for these
problems behave exponentially poorly in the small noise limit. Previous work
in rare event simulation has focused on developing, in specific situations,
estimators with optimal exponential variance decay rates. This criterion
still allows for exponential growth of the statistical relative error. I will
introduce an estimator related to a deterministic control problem that not only
has an optimal variance decay rate under certain conditions, but that can even
have vanishingly small statistical relative error in the small noise limit.
The method can be seen as the limit of a well known zero variance importance
sampling scheme for diffusions which requires the solution of a second order
partial differential equation.
I will also give several numerical illustrations of our results.
****CANCELLED****Stochastic Systems Seminar
Pattern Theory Seminar
Abstract: Arnold in the 60's showed that Euler's equation of fluid flow is a geodesic in the space of volume preserving diffeomorphisms. Ulf Grenander and Mike Miller pioneered the application of metrics on the full group of diffeomorphisms to geodesic morphing of body parts in medical images. It turns out that there is a rich flora of diverse Riemannian metrics on these shape spaces which many researchers are now beginning to uncover. I will sketch some of these and some of the key ideas which go into it.