Efficient implementation of covariance multiplication for data assimilation with the representer method

The application of Markov time correlation within the representer method is revised. Usually, the Markov time correlation is implemented by solving a two-point boundary-value problem that is split into two Langevin equations, one forward and the other backward in time. A new splitting of the two-point boundary-value problem is proposed. An examination of time and memory consumption is carried out when portions of the trajectory are written and read from available disk space, and when checkpoints are used. It is shown that the new splitting will reduce the computation time and the core computer memory required to solve the variational data assimilation problem through the representer method.