Comparison of optimization methods for the maximum likelihood ensemble filter

The Newton method, which requires the Hessian matrix, is prohibitively expensive in adjoint-based variational data assimilation (VAR). It may be rather attractive for ensemble-based VAR because the ensemble size is usually several orders of magnitude smaller than that of the state size. In the present paper the Newton method is compared against the conjugate-gradient (CG) method, which is one of the most popular choices in adjoint-based VAR. To make comparisons, the maximum likelihood ensemble filter (MLEF) is used as a framework for data assimilation experiments. The Hessian preconditioning is used with CG as formulated in the original MLEF. Alternatively we propose to use the Hessian in the Newton method. In the exact Newton (EN) method, the Newton equation is solved exactly, i.e. the step size is fixed to unity avoiding a line search. In the 1000-member wind-speed assimilation test, CG is stagnated early in iteration and terminated due to a line search error while EN converges quadratically. This behaviour is consistent with the workings of the EN and CG in the minimization of the Rosenbrock function. In the repetitive cycled experiments using the Korteweg-de Vries-Burgers (KdVB) model with a quadratic observation operator, EN performs competitively in accuracy to CG with significantly enhanced stability. These idealized experiments indicate the benefit of adopting EN for the optimization in MLEF.