Breakdown of the equivalence between two common preconditionnings in multi-incremental variational data assimilation

Variational Data Assimilation (DA) schemes are often used to adress high dimensional non-linear problems in operational applications in the NWP domain. Because of the high computational cost of such minimization problems, various methods can be applied to improve the convergence at a reasonable numearical cost. One of these methods currently applied in operational DA schemes is the multi-incremental approach that consists in solving a succession of linearized versions of the original non-linear problem in several outer loops, by using well known algorithms (such as Lanczos) to ensure the convergence of the linear problem at the inner loop level, and using the solution of the inner loops to update the problem at each outer loop. In order to save computational cost, the multi-incremental multi-resolution method consists in starting the minimization at a lower resolution than the original one, and increasing it at the outer loop level until the full resolution of the problem. On the other hand, the conditionning of NWP problems is often poor, and one can use preconditionning techniques in order to improve the convergence. We have applied the multi-incremental multi-resolution scheme to a simplified problem in order to study the equivalence of two well known preconditionnings (”full” or ”square root”) in such a scheme and also present a new alternative method to update the problem at the outer loop level. We illustrate the differences with the standard method currently used and compare those two methods to the theoretical result. Some equivalence conditions between the updating methods Speaker Corresponding author: selime.gurol@cerfacs.fr Corresponding author: benjamin.menetrier@irit.fr Corresponding author: yann.michel@meteo.fr sciencesconf.org:symp-bonn2021:356951 and the two preconditionnings are drawn according to the way the resolution change is realised at the outer loop level.