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.