A preconditioning algorithm for large-scale minimization problems

Abstract

A new preconditioning algorithm is proposed, employing a Taylor series expansion of the cost-function, and the relation between the adjustment of the control variable and the computed gradient norm. The preconditioning matrix is a positive definite diagonal matrix, being a product of two positive definite diagonal matrices. One is the weight matrix related to the Hessian matrix definition in the case of identity model operator (“rough” scaling), and the other matrix is interpreted as a refined scaling of the control variable. The procedure is quite easy to implement, and the computer time and space requirements are negligible. The algorithm was tested in two cases of realistic four-dimensional variational data assimilation experiments, performed using an adiabatic version of the NMC's new regional forecast model and operationally obtained optimal interpolation analyses. Test results show a significant improvement in the decrease of the cost-function and the gradient norm when using the new preconditioning procedure. The preconditioning was applied to a memoryless quasi-Newton method, however, the technique should be applicable to other minimization algorithms.