Olivier Besson

Olivier Besson, Julien Straubhaar

A finite element method for the numerical solution of the anisotropic Navier-Stokes equations in shallow domain is presented. This method take into account aspect ratio in the hydrostatic approximation of the Navier-Stokes equations \cite{beslay,azthese,azgui}. A projection method \cite{guermond,shen} is used for the time discretization. The linear systems are solved via a some preconditioned conjugate algorithm, well adapted to massively parallel computers \cite{julien_these,julien_precond,julien_precond_paral}. Some results are presented for the wind driven water circulation in the North Atlantic.

Olivier Besson, Martine Picq, Jérôme Pousin

This work originates from a heart's images tracking which is to generate an apparent continuous motion, observable through intensity variation from one starting image to an ending one both supposed segmented. Given two images $ρ_0$ and $ρ_1$, we calculate an evolution process $ρ(t,\cdot)$ which transports $ρ_0$ to $ρ_1$ by using the optical flow. In this paper we propose an algorithm based on a fixed point formulation and a space-time least squares formulation of the transport equation for computing a transport problem. Existence results are given for a transport problem with a minimum divergence for a dual norm or a weighted $H^1_0$-semi norm, for the velocity. The proposed transport is compare with the transport introduced by Dacorogna-Moser. The strategy is implemented in a 2D case and numerical results are presented with a first order Lagrange finite element, showing the efficiency of the proposed strategy.

Olivier Besson, Nicolas Dobigeon, Jean-Yves Tourneret

In this letter, we consider two sets of observations defined as subspace signals embedded in noise and we wish to analyze the distance between these two subspaces. The latter entails evaluating the angles between the subspaces, an issue reminiscent of the well-known Procrustes problem. A Bayesian approach is investigated where the subspaces of interest are considered as random with a joint prior distribution (namely a Bingham distribution), which allows the closeness of the two subspaces to be adjusted. Within this framework, the minimum mean-square distance estimator of both subspaces is formulated and implemented via a Gibbs sampler. A simpler scheme based on alternative maximum a posteriori estimation is also presented. The new schemes are shown to provide more accurate estimates of the angles between the subspaces, compared to singular value decomposition based independent estimation of the two subspaces.