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Data assimilation and multigrid methods: Theoretical work

Task 2a, principal invertigator: LJK/MOISE.

This part of the project focuses on the use of multigrid methods for the solution of the optimality system of variational data assimilation methods. Both theoretical work and realistic applications are explored.

During Emilie Neveu Ph’D, several advances have been made in the understanding of the behavior of the multigrid methods when applied to the solution of the optimality system. First a rigourous analysis of the convergence properties has been addressed in a simplified linear framework. This work emphasizes the importance of the form of the objective function on the performance of usual multigrid algorithms. In a second part, extension to non linear problems have been studied. In that case, two different algorithms have been studied: the full approximation scheme (FAS) and the Gauss Newton multigrid (GN-MG) scheme. These two algorithms have been experimented in the context of a fully non linear shallow water models resulting in good results and given interesting perspectives for future improvements.

Left panel: Convergence of fully non linear mono and multigrid methods for the shallow water experiments. Right panel: Convergence of incremental mono and mutlgrid methods for the shallow water experiments.