Title: Preconditioning of conjugate-gradients in observation space with an application to 4D-Var data assimilation

Authors: Serge Gratton (Professor at ENSEEIHT, Toulouse, France)
Selime Gurol (PhD student at CERFACS, Toulouse, France)
Philippe Toint (Professor at University of Namur, Belgium)

We investigate the trucated Gauss-Newton (TGN) method for the solution of 4D-Var data assimilation problem. TGN relies on the approximate solution of the quadratic approximations of the nonlinear least-squares cost function. The well-known method to solve the quadratic approximations which can be formulated as linear least squares problems is via conjugate-gradients. This technique is known as incremental 4D-Var data assimilation among the community.

This study considers an alternative conjugate-gradient-like method called Restricted Preconditioned Conjugate Gradient (RPCG). RPCG performs the quadratic minimization in observation space and produces mathematically equivalent iterates to that of incremental 4D-Var approach in exact arithmetic. This algorithm is computationally interesting whenever the dimension of the observation space is significantly smaller than that of the state space since it reduces the computational cost. The relation between RPCG and the other well-known observation space solver called PSAS whose iterates does not provide monotonic decrease of the cost function in state space is explained.

The techniques used to have a better convergence like subspace preconditioning or re-orthogonalization considered to be expensive when used by incremental approach becomes possible to implement when using RPCG while keeping the convergence properties in state space. In this study, we focus on warm start Quasi-Newton preconditioners which are generated using by-products of RPCG. We derived the refined convergence bounds which results in a practical solution method.

We illustrate the results using a toy-problem based on heat equations.

GMAO Head: Michele Rienecker Global Modeling and Assimilation Office NASA Goddard Space Flight Center

Curator: Nikki Privé Last Updated: May 27 2011