Variational PDEs for Acceleration on Manifolds and Application to Diffeomorphisms

We consider the optimization of cost functionals on infinite dimensional manifolds and derive a variational approach to accelerated methods on manifolds. We demonstrate the methodology on the infinite-dimensional manifold of diffeomorphisms, motivated by optical flow problems in computer vision. We build on a variational approach to accelerated optimization in finite dimensions, and generalize that approach to infinite dimensional manifolds. We derive the continuum evolution equations, which are partial differential equations (PDE), and relate them to mechanical principles.