This paper uses two simple variational data assimilation problems with the 1D viscous Burgers' equation on a periodic domain to investigate the impact of various diagonal-preconditioner update and scaling strategies, both on the L-BFGS inverse Hessian approximation and on the minimization performance. These simple problems share some charasteristics with the large-scale variational data assimilation problems commonly dealt with in meteorology and oceanography.
The update formulae studied are those proposed by Gilbert and Lemaréchal (1989, section 4.2) and the quasi-Cauchy formula of Zhu et al. (1999). Which information should be used for updating the diagonal preconditioner, the one to be forgotten or the most recent one, is considered first. Then, following the former authors, a scaling of the diagonal preconditioner is introduced for the corresponding formulae in order to improve the minimization performance. The large negative impact of such a scaling on the quality of the L-BFGS inverse Hessian approximation led us to propose an alternate updating and scaling strategy, that provides a good inverse Hessian approximation and gives the best minimization performance for the problems considered. With this approach the quality of the inverse Hessian approximation improves steadily during the minimization process. Moreover, this quality and the L-BFGS minimization performance improves when the amount of stored information is increased.