Introducing Time Parallelization Within Data Assimilation

Four-dimensional variational data assimilation (4D-Var), in its incremental formulation, is based on optimization algorithms which require the integration of the forward and adjoint versions of the original model in order to compute the gradient. For their use on parallel computers, these models are classically parallelized only in spatial dimension and this is a limiting factor on the maximum number of cores that can be utilized. We present here a novel approach of introducing additional time parallelization using the Parareal algorithm. This approach is used here for integration of the forward model. We use a modified version of the inexact conjugate gradient method where the matrix-vector multiplication is supplied through Parareal. The use of this inexact conjugate gradient and the associated convergence conditions allows us to precisely determine the stopping criterion of the Parareal iterations. The results are demonstrated by considering a one-dimensional shallow water model. They are presented in terms of the accuracy (in comparison with the original exact conjugate gradient) and in terms of the number of required iterations of the Parareal algorithm.