Preconditioning of variational data assimilation and the use of a bi‐conjugate gradient method

Abstract

Presently, a preferred minimization for strong-constraint four-dimensional variational (4D-Var) assimilation uses a Lanczos-based conjugate gradient (CG) algorithm. This requires the availability of a square-root of the backgrounderror covariance matrix (B). In the context of weak-constraint 4D-Var, 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. An appealing minimization scheme is the double CG minimization employed, for example, in the grid-point statistical interpolation (GSI) analysis. Realizing the double CG algorithm is a special case of the more general bi-conjugate gradient (BiCG) method for solving non-symmetric problems, the present work introduces a Lanczos-based preconditioning strategy when B, instead of its square-root, is used initially. Implementation of the scheme is done in the context of the GSI analysis system, and preliminary experiments are presented using its 3D-Var version. Comparison of the Lanczos-based CG and the BiCG shows that the algorithms converge at the same rate and to the same solution. Despite the additional computational cost, the importance of the re-orthogonalization step is also shown to be fundamental to any of these CG algorithms. Furthermore, when using the Hessian eigenvectors for preconditioning, the BiCG behaviour is shown to be 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 from the reduction of the condition number. The efficiency, computational cost, and stability of the three algorithms are discussed. Copyright

A review of operational methods of variational and ensemble‐variational data assimilation

“…EVIL is based on the fact that conjugate gradient (CG)‐based minimization algorithms are closely related to Lanczos methods (Paige and Saunders, ; Fisher and Courtier, ; ElAkkraoui et al , ). In outline, the gradient descent vectors from q iterations of the CG procedure form a Krylov sub‐space in which the Hessian of the cost function is tridiagonal ( q here takes the role of N ).…”

Assessing the impact of observations in a multi‐year reanalysis

“…Furthermore, the MERRA-2 GSI replaces the weak-constraint balance operator of the MERRA GSI with the strong-constraint tangent linear normal mode formulation of Kleist et al (2009a). The MERRA-2 GSI also invokes the bi-conjugate gradient procedure of El Akkraoui et al (2013); a three-dimensional variational data assimilation (3D-Var) algorithm that incorporates a middle-loop strategy -linearization of the observation operator -with two such middle loops, each with 100 inner iterations. The variational procedure operates at a resolution of 0.5 • on 72 vertical hybrid eta levels and uses a first-guess at the appropriate time (FGAT) strategy and a climatological background-error covariance derived on the basis of the NMC method (Parrish and Derber, 1992), which is unchanged throughout the reanalysis period.…”

A Brief Assessment of the Impact of Nearly 40 Years of Assimilated Observations Over the Amazon Basin

“…The analysis operates on a regular latitude‐longitude 361 576 grid with a horizontal resolution comparable to the atmospheric model, roughly 50 km, and the 72 model vertical levels. Key to this analysis system is its reliance on the Community Radiative Transfer Model (Release 2.1.3: Chen et al, ; Han et al, ); the use of the Tangent Linear Normal Mode Constraint of Kleist et al () for incremental balance adjustment; and the employment of the biconjugate gradient minimization procedure of El Akkraoui et al () in its 3D‐Var form, using two middle‐loop iterations to accommodate nonlinearities in the GSI observation operators, and 100 inner iterations in each outer loop.…”