Primal-dual formulation of the Dynamic Optimal Transport using Helmholtz-Hodge decomposition

This work deals with the resolution of the dynamic optimal transport (OT) problem between 1D or 2D images in the fluid mechanics framework of Benamou-Brenier [6]. The numerical resolution of this dynamic formulation of OT, despite the successful application of proximal methods [36] is still computationally demanding. This is partly due to a space-time Laplace operator to be solved at each iteration, to project back to a divergence free space. In this paper, we develop a method using the Helmholtz-Hodge decomposition [23] in order to enforce the divergence-free constraint throughout the iterations. We prove that the functional we consider has better convexity properties on the set of constraints. In particular we explain that in 1D+time, this formulation is equivalent to the resolution of a minimal surface equation. We then adapt the first order primal-dual algorithm for convex problems of Chambolle and Pock [12] to solve this new problem, leading to an algorithm easy to implement. Besides, numerical experiments demonstrate that this algorithm is faster than state of the art methods for dynamic optimal transport [36] and efficient with real-sized images

Noncommutative de Sitter and FRW spaces

Several versions of fuzzy four-dimensional de Sitter space are constructed using the noncommutative frame formalism. Although all noncommutative spacetimes which are found have commutative de Sitter metric as a classical limit, the algebras and the differential calculi which define them have many differences which we derive and discuss.Comment: 20 page

Log-periodic route to fractal functions

Log-periodic oscillations have been found to decorate the usual power law behavior found to describe the approach to a critical point, when the continuous scale-invariance symmetry is partially broken into a discrete-scale invariance (DSI) symmetry. We classify the `Weierstrass-type'' solutions of the renormalization group equation F(x)= g(x)+(1/m)F(g x) into two classes characterized by the amplitudes A(n) of the power law series expansion. These two classes are separated by a novel ``critical'' point. Growth processes (DLA), rupture, earthquake and financial crashes seem to be characterized by oscillatory or bounded regular microscopic functions g(x) that lead to a slow power law decay of A(n), giving strong log-periodic amplitudes. In contrast, the regular function g(x) of statistical physics models with ``ferromagnetic''-type interactions at equibrium involves unbound logarithms of polynomials of the control variable that lead to a fast exponential decay of A(n) giving weak log-periodic amplitudes and smoothed observables. These two classes of behavior can be traced back to the existence or abscence of ``antiferromagnetic'' or ``dipolar''-type interactions which, when present, make the Green functions non-monotonous oscillatory and favor spatial modulated patterns.Comment: Latex document of 29 pages + 20 ps figures, addition of a new demonstration of the source of strong log-periodicity and of a justification of the general offered classification, update of reference lis