Introducing time parallelisation within data assimilation using parareal method

Forecasts made by 4-D Var involves forward integration of model before proceeding for the minimisation process. While one reaches saturation in space parallelisation we try to obtain some speedup by integrating our model using the Parareal method. Our setting ensures that the minimum is obtained by solving a linear symmetric system. We use a modified version of the inexact conjugate gradient method where the matrix-vector multiplication is supplied by the parareal. This helps us to determine a specific stopping criterion for the parareal. The results are demonstrated by considering a 1-D linear shallow water model. Our method produces a speedup factor of 3 using fewer parareal iterations as compared to the same results obtained by the exact conjugate gradient method