Applied Mathematics and Mathematical Physics
Imperial College London
Jet particle methods for ideal fluids and shape analysis
We will consider a regularized model of an ideal fluid, with a regularization parameter, sigma, for which the model converges to the Euler equations as sigma vanishes. We will then do an infinite hierarchy of infinite dimensional reductions by symmetry to obtain particle like solutions to the regularized Euler equations. The resulting hierarchy of particles will contain (finite dimensional) internal group symmetries, jet groups, which serve as a shadow of the particle relabeling symmetry of the full fluid system. The Noether theorem associated to the symmetry of these particle solutions is a shadow of the Kelvin circulation theorem. Finally, collisions between particles at one level of the hierarchy will result in infinite-time mergers into particles at the next level in the hierarchy. Numerical experiments will suggest that such collisions are not rare events and we will interpret such events as a cascade to finer scales. We will then show how this mathematics may be used for the purpose of large deformation shape analysis. The internal group variables of these particles are manifestations of spatial gradients of diffeomorphisms. This identification allow one to advect higher-order geometric quantities such as tensor fields and curvatures. Such an ability is important in medical imaging wherein one is concerned with objects such as diffusion tensors, and surface curvatures.
Sponsored by the Department of Mathematical Sciences
Refreshments will be served in the FO 2.604 30 minutes prior to the talk