Calculus of Variations and Geometric Measure Theory

F. Benmansour - G. Carlier - G. Peyré - F. Santambrogio

Derivatives with respect to metrics and applications: Subgradient Marching Algorithm

created by santambro on 16 Jan 2009
modified on 07 Apr 2010


Accepted Paper

Inserted: 16 jan 2009
Last Updated: 7 apr 2010

Journal: Numerische Mathematik
Year: 2010


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.

Keywords: Traffic congestion, Fast Marching Method, subgradient descent, Geodesics., Eikonal equation, travel time tomography