On the use of the saddle formulation in weakly-constrained 4D-VAR data assimilation

This paper discusses the practical use of the saddle variational formulation
for the weakly-constrained 4D-VAR method in data assimilation. It is shown
that the method, in its original form, may produce erratic results or diverge
because of the inherent lack of monotonicity of the produced objective
function values. Convergent, variationaly coherent variants of the algorithm
are then proposed whose practical performance is compared to that of other
formulations. This comparison is conducted on two data assimilation instances
(Burgers equation and the Quasi-Geostrophic model), using two different
assumptions on parallel computing environment. Because these
variants essentially retain the parallelization advantages of the original
proposal, they often --- but not always --- perform best, even for moderate
numbers of computing processes.