Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows

TL;DR

This paper extends discrete gradient methods to Riemannian manifolds, introducing discrete Riemannian gradients and applying them to optimize dissipative systems, including eigenvalue and imaging problems.

Contribution

It introduces discrete Riemannian gradients and demonstrates their effectiveness in optimization and imaging applications on manifolds.

Findings

Successful application to eigenvalue problems

Effective in InSAR and DTI denoising tasks

Provides a derivative-free optimization algorithm

Abstract

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.