Variational principles are at the core of the formulation of mechanical problems. What happens in the presence of symmetry when variables can be eliminated? I will discuss the geometry underlying this reduction process and present the induced constrained variational principle and the associated Euler-Lagrange equations. The rigid body and the Euler equations for ideal fluids are examples of such reduced Euler-Lagrange equations in convective and spatial representations, respectively. This geometric structure permits the introduction of a new class of optimal control problems that have the remarkable property that the control satisfies precisely these reduced Euler-Lagrange equations. As an example, it is shown that geodesic motion for the normal metric can be controlled by geodesics on the symmetry group. In the case of fluids, these optimal control problems yield the classical Clebsch variables and singular solutions for the Camassa-Holm equation. Relaxing the constraint to a quadratic penalty yields associated optimization problems. Time permitting, the equations of metamorphosis dynamics in imaging will be deduced from this optimization problem.

The Principal Component Decomposition (POD) technique has been used as a model reduction tool for many applications in engineering and science. In principle, one begins with an ensemble of data, called snapshots, collected from an experiment or laboratory results. The beauty of the POD technique is, when it is applied, the entire data set can be represented by the smallest number orthogonal basis elements. It is such capability that allows us to reduce the complexity and dimensions of many physical applications. Mathematical formulations and numerical schemes for the POD method will be discussed along with three applications, satellite photo image reconstruction, cancer detection with DNA microarrays, and stock allocation optimization.

This talk will introduce and solidify some concepts in TV Minimization via Chambolle Duality. I will also give an intro into image segmentation and talk about some recent work in this area.