Minimal Residual Method (MINRES) for R4D-Var and 4D-PSAS

[arango]

A new C-preprocessing option was added MINRES to accelerate the convergence of the dual formulation data assimilation algorithms (R4D-Var and 4D-PSAS) using the Minimal Residual Method (MinRes) proposed by El Akkraoui and Gauthier (2010, QJRMS, 136, 107-115). This method is described by Paige and Saunders (Sparse Indefinite Systems of Linear Equations, 1975, SIAM Journal on Numerical Analysis, 617-619). Specifically we use equations 6.10 and 6.11 of this paper.

This is a remarkable algorithm and makes R4D-Var and 4D-PSAS practical in strong- and weak-constraint applications. The convergence rate is similar to that of the primal form I4D-Var. This is illustrated in the following plot:

This algorithm also works for the associated observation sensitivity and observation impact. We still need to work on the posterior analysis routines posterior.F and posterior_var.F. These routines will be updated to MINRES later.

Many thanks to Andy Moore for his help in coding and testing this new algorithm. Also many thanks to Amal El Akkraoui for bringing this minimization algorithm to our attention at the last International Adjoint Workshop.