Derivatives with Respect to Metrics and Applications: Subgradient Marching Algorithm

Fethallah Benmansour (1) , Guillaume Carlier (1) , Gabriel Peyré (1) , Filippo Santambrogio (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper describes the Subgradient Marching algorithm to compute the derivative of the geodesic distance with respect to the metric. The geodesic distance being a concave function of the metric, this algorithm computes an element of the subgradient in $O(N^2 \log(N))$ operations on a discrete grid of $N$ points. It performs a front propagation that computes the subgradient of a discrete geodesic distance. Equipped with this Subgradient Marching, a Riemannian metric can be designed through an optimization process. We show applications to landscape modeling and to traffic congestion. Both applications require the maximization of geodesic distances under convex constraints, and are solved by subgradient descent computed with our Subgradient Marching. We also show application to the inversion of travel time tomography, where the recovered metric is the local minimum of a non-convex variational problem involving geodesic distances.

Mots clés: Geodesics; Eikonal equation; Subgradient descent; Fast Marching Method; Traffic congestion; Travel time tomography.; Travel time tomography

Domaines: Analyse numérique [math.NA]

Fethallah Benmansour, Guillaume Carlier, Gabriel Peyré, Filippo Santambrogio. Derivatives with Respect to Metrics and Applications: Subgradient Marching Algorithm. Numerische Mathematik, 2010, 116 (3), pp.357-381. ⟨10.1007/s00211-010-0305-8⟩. ⟨hal-00360794v2⟩