In an ensemble Kalman filter, at each assimilation step, there is an intrinsic need of evaluating the covariance matrix of the propagated ensemble. In the "standard" ensemble Kalman, this is done with the empirical covariance matrix of the ensemble. This has strong drawbacks. The size of the ensemble being often smaller than the size of the system, there is an intrinsic lack of information. Usually, this is solved by inflation (increase of the overall value of the covariance matrix by a factor) and localisation (reduction of the off-diagonal elements of the matrix). The tuning of localisation and inflation is done empirically, and has a computational cost that can be huge.
The aim of this project is to develop and algorithm where inflation and localisation are optimally tuned by the algorithm itself. Pr. Marc Bocquet, in NPG (2015) expanded the ensemble Kalman filter using bayesien hierarchical methods. This leads to a natural emergence of inflation and its optimal value in the algorithm. We are now aiming with Dr. Alexis Hannart to use a shrinkage estimator for the covariance matrix (JMVA (2014)) that would replace localisation.
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The understanding of persistent atmospheric patterns on top of SST fronts is an important problem that can have implications for climate and synoptic weather dynamics. In this project, we adressed the problem using a simplified analytical model, that includes a realistic parametrization of turbulence.
On the one hand, we have been able to reproduce and explain the main features of numerical simulations performed with the Méso-NH model. On the other hand, the computation of the integrated wind divergence in the boundary layer revelad different regimes in its link with SST, that have not, to our knowledge, been discussed in the litterature.
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In an ultracold gas, a bright soliton is defined as a peak in particle density. The aim of this project is to understand better the dynamics and the collisions of such solitons, in an uni-dimensional gas, where analytical solutions to the multi-particles Schrödinger exist. The first step of this project has been to compute the single-particle density matrix of such a state, which gives interesting information about the quantum states in which the particles are, and the correlation between the states. This has invloved extensive research on the Bethe ansatz and its framework, the quantum inverse scattering method.

We have developed a formalism that will serve as a basis for further computations of this type. As an exemple, we have studied the scaling of different quantities such as the condensed fraction with the number of particles.

[1] Alex Ayet and Joachim Brand, "The single-particle density matrix of a quantum bright soliton from the coordinate Bethe Ansatz", J. Stat. Mech. Submitted.