Pierre Allain Obelix team

Post doctoral researcher at LETG-Costel and Irisa Rennes in the Obelix team. My work currently focuses on time series classification in satellite images for crop and grassland detection.

This work has been done during my PhD thesis under the supervision of Nicolas Courty and Thomas Corpetti.

Pachinko experiment

In a Lagrangian framework, adjoint method for variational assimilation yields very different solutions wether the gradient descent is local or global. In the case there exists repulsion forces between particles, the gradient descent by the time integration of the derived model (local descent) will reach a local minima. This is due to the tangling issue, consequence of repulsion which conserve relative position between particles.

a - Local descent

b - Local descent with time fractioning

c - Global descent

The above Figures represents the trajectory of a particle going straight forward from left to right and bumping into other particles. In each case, the trajectory has to pass by three constraint points at determined moments. The control aims at finding the less costful path abiding by these constraints. Sequences evolve with control iterations, and therefore show trajectory evolution. One can see in Sequence a that the path at convergence is not the shortest. This is because local descent is unable to make the particle switch the side it goes over another one. On the contrary, the global descent used in Sequence c shows that the obtained trajectory matches the shortest path abiding by the constraints. This is due to the time integration of the model during control iterations in place of the derived model in the local descent approach. Indeed, the model is able to switch the overtaking side of particles, not the dervied one.

In Sequence c, a hybrid approach combining both local and global descent is used. The trajectory time is divided in three parts corresponding to the number of position constraints. The first part is solved using local descent, while the rest of the trajectory is simply driven by the time-integration of the model. After convergence is reached for the first constraint, this part is blocked and the same procedure is repeated to the second part, and so on for the third.

Control of physical particles

Using optimal control theory, we manage to control a swarm of particles driven by a second order dynamic model (i.e. including acceleration and mass). With respect with this dynamic, it is therefore possible to make the considered swarm to take different shapes either defined at particles level (like configurations) or at environment level (like locally defined particles density). This approach even allows to design higher level constraints such as swarm vorticity.

Control of the swarm model is performed using the adjoint method for data assimilation.

Particle swarm control

Related references :

1. Riccardo Poli, James Kennedy, and Tim Blackwell. Particle swarm optimization. Swarm Intelligence, 1:33-57, 2007.
2. François-Xavier Le Dimet and Olivier Talagrand. Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A, 38A(2):97--110, 1986.
3. J.L. Lions. Optimal control of systems governed by partial differential equations. Springer-Verlag, 1971.