ABSTRACT
This chapter presents minimization algorithms that are best suited for uncon-
strained and constrained minimization of large-scale systems such as four-
dimensional variational data assimilation, referred to as “4-D VAR,” in
weather prediction and similar applications in the geophysical sciences. The
operational implementation of the 4-D VAR method hinges crucially upon fast
convergence of efficient gradient-based large-scale unconstrained minimization
algorithms. These algorithms generally minimize a cost function which at-
tempts to quantify the discrepancies between forecast and observations in a
window of assimilation, where the model is used as a strong constraint. Data
assimilation problems in oceanography and meteorology contain many degrees
of freedom (≈ 107). Consequently, conjugate-gradient methods and limitedmemory quasi-Newton (LMQN) methods come into consideration since they
require storage of only a few vectors, containing information from a few it-
erations. Studies (e.g., refs. [74] and [126]) indicate that “Limited Memory
Broyden Fletcher-Goldfarb and Shanno” (L-BFGS) and its French equivalent
M1QN3 are among the best LMQN methods available to date.
