演講訊息 TALKS

We study a class of evolution equations within the framework of the so-called Euler- Poincaré differential (EPDiff) equations. The EPDiff equations arise in studies related to the Riemannian geometry of groups of diffeomorphisms, as well as the fluid mechanics of nonlinear waves. The evolution of the dependent variable of an EPDiff system is coupled to an auxiliary field defined by the action of an invertible elliptic operator onto the base field. We study the finite-dimensional particle formulation of the EPDiff equations that is in a manner reminiscent of the streamfunction-vorticity formulation of the Euler equations in hydrodynamics. With the help of the Lagrangian formulation of the EPDiff evolution equations, we develop particle algorithms for the multidimensional EPDIFF system in both its dispersionless and dispersive forms. Our focus of the particle algorithms is on the class of non-smooth kernels, and in particular the “conon” case, for which the motion of particles are restricted in a one-dimensional channel. The one-channel reduction reduces the particle system to the “completely integrable” one-dimensional case, even though the “single channel” solution retains its dependence on two spatial dimensions. We study collision cases of these non-smooth kernels in a one-dimensional channel and bring forth the effects, due to the lack of Lipshitz continuity in the associated systems of ordinary differential equations. We also demonstrate that potentially the particle algorithms can be used as a numerical algorithm for solving the EPDIFF equations. We compare the numerical solutions between the particle algorithm and a splitting pseudospectral method developed in this paper. We find that the two results are virtually indistinguishable, provided a sufficiently large number of particles is used for the particle algorithm. Finally we link the N-particle algorithm to the application of template matching. Template matching is commonly used in problems of image registration and pattern recognition. Template matching can be formulated as finding the shortest or least expensive path of continuous deformation of one geometric object (reference) into another one (target). In this context, the time-dependent deformation process is the so-called geodesic evolution, and the derivation of the geodesic evolution equations falls into the Euler-Poincaré theory, which produces the EPDiff motion equations. Therefore, mathematically the EPDiff equations describe geodesic motion on the diffeomorphism group, and have the same form as the template matching equations. We propose a template matching algorithm based on the extended Kalman filter.