Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows
It provides a new numerical framework for gradient flows on manifolds, relevant for optimization and imaging applications, but the results are preliminary and tested only on specific problems.
The paper extends discrete gradient methods to Riemannian manifolds, introducing discrete Riemannian gradients for dissipative ODEs, and tests the resulting derivative-free optimization algorithm on eigenvalue problems and manifold-valued imaging tasks (InSAR and DTI denoising).
This paper concerns an extension of discrete gradient methods to finite-dimensional Riemannian manifolds termed discrete Riemannian gradients, and their application to dissipative ordinary differential equations. This includes Riemannian gradient flow systems which occur naturally in optimization problems. The Itoh--Abe discrete gradient is formulated and applied to gradient systems, yielding a derivative-free optimization algorithm. The algorithm is tested on two eigenvalue problems and two problems from manifold valued imaging: InSAR denoising and DTI denoising.