The Maximum Likelihood Ensemble Filter as a non-differentiable minimization algorithm

The Maximum Likelihood Ensemble Filter (MLEF) equations are derived without the differentiability requirement for the prediction model and for the observation operators. The derivation reveals that a new non-differentiable minimization method can be defined as a generalization of the gradient-based unconstrained methods, such as the preconditioned conjugate-gradient and quasi-Newton methods. In the new minimization algorithm the vector of first-order increments of the cost function is defined as a generalized gradient, while the symmetric matrix of second-order increments of the cost function is defined as a generalized Hessian matrix. In the case of differentiable observation operators, the minimization algorithm reduces to the standard gradient-based form. The non-differentiable aspect of the MLEF algorithm is illustrated in an example with one-dimensional Burgers model and simulated observations. The MLEF algorithm has a robust performance, producing satisfactory results for tested non-differentiable observation operators. Copyright (C) 2008 Royal Meteorological Society.