Block Krylov methods for accelerating ensembles of variational data assimilations.

We consider the problem of efficiently solving ensembles of variational data assimilations in the context of numerical weather prediction. Running several assimilations notably allows us to initialize ensemble prediction systems and to more accurately represent background‐error statistics, but is computationally expensive, limiting ensemble size. We propose a new class of algorithms for speeding up the minimization of the ensemble of data assimilations. It consists of using block Krylov methods to simultaneously perform the minimization for all members of the ensemble, instead of performing each minimization separately. We develop preconditioned block Krylov versions of the Full Orthogonal Method and of the Lanczos algorithm in both primal and dual space. The latter works in observation space that is usually of smaller dimension than the state space, thus giving an advantage in terms of memory requirements and computational cost. We describe and compare several parallelization strategies for speeding up the minimization and limiting the communications. These methods have been tested on a quasi‐geostrophic system, consisting of a simplified atmospheric circulation model equipped with an ensemble of 3D‐Var schemes tuned to mimic some features of a limited‐area numerical weather prediction system. Experimentation shows that the number of iterations needed to converge is drastically reduced by the block Krylov approaches. We indicate that, while working in primal space does not save significant computational time, working in the dual space may reduce the computational time by a factor of 2 to 5 (depending on ensemble size) compared to standard Krylov methods, making our approach attractive for operational use. Illustration of the two workload distributions implemented in this work on the AROME‐France domain for four MPI processes and two members in the ensemble. (a) Workload geographical distribution (no distribution by member), as used in the SEQ version of block B‐FOM. (b) Workload distribution by member, combined with an underlying geographical distribution, as used in the MPI and MPIstored versions of block B‐FOM, and in B‐FOM. [ABSTRACT FROM AUTHOR]