Title: Using a Bi-Conjugate Gradient minimization algorithm for variational data assimilation

Authors: Amal El Akkraoui, Ricardo Todling, and Yannick Tremolet

Presently, a preferred minimization for strong constraint 4dvar uses a Lanczos-based CG algorithm. This requires the availability of the square-root of the background error covariance matrix. In the context of weak constraint 4dvar, this requirement might be too restrictive for the formulations of the model error term. It might therefore be desirable to avoid a square-root decomposition of the augmented background term. In this respect, an appealing minimization scheme is the double CG minimization employed, for example, in the Grid-point Statistical Interpolation analysis (GSI). However, this procedure cannot easily take advantage of the preconditioning using the eigenvectors of the Hessian, readily available in the Lanczos-based CG. Alternatively, this could be exploited by considering a bi-conjugate gradient (Bi-CG) algorithm, which by construction is suitable to non-symmetric matrices.

The present work introduces the mathematical formulation of the Bi-CG and a Lanczos-biorthogonalization procedure by which the left- and right-Hessian eigenvectors are calculated for use in the preconditioning. Implementation of the scheme is done in the context of GSI, and preliminary studies are presented for the 3dvar. Also, three minimization strategies are compared in GSI: the double CG (with an added re-orthogonalization step for improved performance), the square-root(B) Lanczos-based CG, and the new Bi-CG algorithm. Results show that the three algorithms converge with the same rate and to the same solution. Despite the additional computational cost, the importance of the re-orthogonalization step is shown to be fundamental, especially for the BiCG scheme. Furthermore, when using the eigenvectors for preconditioning, the BiCG behavior is comparable to that of the Lanczos-CG algorithm. Both schemes construct the same approximation of the Hessian with the same number of eigenvectors, and benefit in the same way of the reduction of the condition number. The efficiency, computational cost,and stability of the three algorithms are discussed.

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

Curator: Nikki Privé Last Updated: May 27 2011