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The Maximum Likelihood Ensemble Filter as a non‐differentiable minimization algorithm
Author(s) -
Zupanski Milija,
Navon I. Michael,
Zupanski Dusanka
Publication year - 2008
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1002/qj.251
Subject(s) - hessian matrix , differentiable function , conjugate gradient method , mathematics , minification , generalization , algorithm , nonlinear conjugate gradient method , matrix (chemical analysis) , gradient method , directional derivative , function (biology) , mathematical optimization , gradient descent , computer science , mathematical analysis , artificial neural network , materials science , machine learning , composite material , evolutionary biology , biology
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 © 2008 Royal Meteorological Society