2 Copyright (C) 2008-2014 EDF R&D
4 This file is part of SALOME ADAO module.
6 This library is free software; you can redistribute it and/or
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
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12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
16 You should have received a copy of the GNU Lesser General Public
17 License along with this library; if not, write to the Free Software
18 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
20 See http://www.salome-platform.org/ or email : webmaster.salome@opencascade.com
22 Author: Jean-Philippe Argaud, jean-philippe.argaud@edf.fr, EDF R&D
24 .. index:: single: 3DVAR
25 .. _section_ref_algorithm_3DVAR:
27 Calculation algorithm "*3DVAR*"
28 -------------------------------
33 This algorithm performs a state estimation by variational minimization of the
34 classical :math:`J` function in static data assimilation:
36 .. math:: J(\mathbf{x})=(\mathbf{x}-\mathbf{x}^b)^T.\mathbf{B}^{-1}.(\mathbf{x}-\mathbf{x}^b)+(\mathbf{y}^o-\mathbf{H}.\mathbf{x})^T.\mathbf{R}^{-1}.(\mathbf{y}^o-\mathbf{H}.\mathbf{x})
38 which is usually designed as the "*3D-VAR*" function (see for example
41 Optional and required commands
42 ++++++++++++++++++++++++++++++
44 .. index:: single: Background
45 .. index:: single: BackgroundError
46 .. index:: single: Observation
47 .. index:: single: ObservationError
48 .. index:: single: ObservationOperator
49 .. index:: single: Minimizer
50 .. index:: single: Bounds
51 .. index:: single: MaximumNumberOfSteps
52 .. index:: single: CostDecrementTolerance
53 .. index:: single: ProjectedGradientTolerance
54 .. index:: single: GradientNormTolerance
55 .. index:: single: StoreInternalVariables
56 .. index:: single: StoreSupplementaryCalculations
58 The general required commands, available in the editing user interface, are the
62 *Required command*. This indicates the background or initial vector used,
63 previously noted as :math:`\mathbf{x}^b`. Its value is defined as a
64 "*Vector*" or a *VectorSerie*" type object.
67 *Required command*. This indicates the background error covariance matrix,
68 previously noted as :math:`\mathbf{B}`. Its value is defined as a "*Matrix*"
69 type object, a "*ScalarSparseMatrix*" type object, or a
70 "*DiagonalSparseMatrix*" type object.
73 *Required command*. This indicates the observation vector used for data
74 assimilation or optimization, previously noted as :math:`\mathbf{y}^o`. It
75 is defined as a "*Vector*" or a *VectorSerie* type object.
78 *Required command*. This indicates the observation error covariance matrix,
79 previously noted as :math:`\mathbf{R}`. It is defined as a "*Matrix*" type
80 object, a "*ScalarSparseMatrix*" type object, or a "*DiagonalSparseMatrix*"
84 *Required command*. This indicates the observation operator, previously
85 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
86 results :math:`\mathbf{y}` to be compared to observations
87 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
88 a "*Matrix*" type one. In the case of "*Function*" type, different
89 functional forms can be used, as described in the section
90 :ref:`section_ref_operator_requirements`. If there is some control :math:`U`
91 included in the observation, the operator has to be applied to a pair
94 The general optional commands, available in the editing user interface, are
95 indicated in :ref:`section_ref_assimilation_keywords`. In particular, the
96 optional command "*AlgorithmParameters*" allows to choose the specific options,
97 described hereafter, of the algorithm. See
98 :ref:`section_ref_options_AlgorithmParameters` for the good use of this command.
100 The options of the algorithm are the following:
103 This key allows to choose the optimization minimizer. The default choice is
104 "LBFGSB", and the possible ones are "LBFGSB" (nonlinear constrained
105 minimizer, see [Byrd95]_, [Morales11]_ and [Zhu97]_), "TNC" (nonlinear
106 constrained minimizer), "CG" (nonlinear unconstrained minimizer), "BFGS"
107 (nonlinear unconstrained minimizer), "NCG" (Newton CG minimizer). It is
108 strongly recommended to stay with the default.
111 This key allows to define upper and lower bounds for every state variable
112 being optimized. Bounds have to be given by a list of list of pairs of
113 lower/upper bounds for each variable, with possibly ``None`` every time
114 there is no bound. The bounds can always be specified, but they are taken
115 into account only by the constrained optimizers.
118 This key indicates the maximum number of iterations allowed for iterative
119 optimization. The default is 15000, which is very similar to no limit on
120 iterations. It is then recommended to adapt this parameter to the needs on
121 real problems. For some optimizers, the effective stopping step can be
122 slightly different of the limit due to algorithm internal control
125 CostDecrementTolerance
126 This key indicates a limit value, leading to stop successfully the
127 iterative optimization process when the cost function decreases less than
128 this tolerance at the last step. The default is 1.e-7, and it is
129 recommended to adapt it to the needs on real problems.
131 ProjectedGradientTolerance
132 This key indicates a limit value, leading to stop successfully the iterative
133 optimization process when all the components of the projected gradient are
134 under this limit. It is only used for constrained optimizers. The default is
135 -1, that is the internal default of each minimizer (generally 1.e-5), and it
136 is not recommended to change it.
138 GradientNormTolerance
139 This key indicates a limit value, leading to stop successfully the
140 iterative optimization process when the norm of the gradient is under this
141 limit. It is only used for non-constrained optimizers. The default is
142 1.e-5 and it is not recommended to change it.
144 StoreInternalVariables
145 This Boolean key allows to store default internal variables, mainly the
146 current state during iterative optimization process. Be careful, this can be
147 a numerically costly choice in certain calculation cases. The default is
150 StoreSupplementaryCalculations
151 This list indicates the names of the supplementary variables that can be
152 available at the end of the algorithm. It involves potentially costly
153 calculations. The default is a void list, none of these variables being
154 calculated and stored by default. The possible names are in the following
155 list: ["APosterioriCovariance", "BMA", "OMA", "OMB", "Innovation",
156 "SigmaObs2", "MahalanobisConsistency"].
161 References to other sections:
162 - :ref:`section_ref_algorithm_Blue`
163 - :ref:`section_ref_algorithm_ExtendedBlue`
164 - :ref:`section_ref_algorithm_LinearityTest`
166 Bibliographical references: