# Parallel in time algorithms for data assimilation

#### Speciality : Mathématiques Appliquées

23/11/2023 - 14:00 Rishabh Bhatt IMAG Salle de séminaire 2

Keywords :

- Data Assimilation
- Optimisation
- Parallelisation in time

Four dimensional variational data assimilation (4DVAR) which is based on optimisation algorithms is used by the leading meterological institutions as a means of initialising the numerical climate models. The optimal initial condition is found by minimisng a cost function which accounts for the misfits of the model trajectory with the observations of the system over a given time window. In its incremental formulation, the integration of the forward and adjoint version of the original model is required in order to compute the gradient. A common issue in the retrieval of the initial condition is the enormous size of the state variable (10^9 ) which makes the minimisation an expensive and time consuming task. Moreover 4DVAR is an inherently sequential algorithm and to use it in parallel architectures, the models are clasically parallelised only in the spatial dimension. This limits the scope of further speed up once spatial saturation is reached and also the maximum number of computing cores that one can use. The objective of this PhD is to introduce an additional time-paralllelisation in the data assimilation framework by using the well known parareal method. Our approach is used here for running the forward integration. We use a modified version of the inexact conjugate gradient method where the matrix-vector multiplications are supplied through the parareal and thus are not exact. The associated convergence conditions of the inexact conjugate gradient allows us to use parareal adaptively by monitoring the errors in the matrix-vector product and obtaining the same levels of accuracy as with the usual conjugate gradient method at the same time. To ensure the feasibility and a practical implementation, all the norms which are hard to compute are replaced by the easily computable approximations. The results are demonstrated by considering the one and two 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. For the more complex two dimensional model we use a Krylov enhanced subspace parareal version which accelerates the convergence of the parareal and brings down the number of iterations. In the end, the ways to time-parallelise the adjoint version is also discussed as a further avenue for research.

#### Directors:

- Laurent Debreu
*(INRIA-LJK )* - Arthur Vidard
*(INRIA-LJK )*

#### Raporteurs:

- Antoine Rousseau
*(INRIA Cote-d'Azur )* - Julien Salomon
*(INRIA Paris )*

#### Examinators:

- Caroline Japhet
*(Universite Sorborne Paris Nord )* - Martin Schreiber
*(Universite Grenoble Alpes )* - Ehouarn Simon
*(INP-ENSEEIHT Toulouse )*