2 Copyright (C) 2008-2024 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
9 version 2.1 of the License, or (at your option) any later version.
11 This library is distributed in the hope that it will be useful,
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: 4DVAR
25 .. index:: single: 4D-Var
26 .. _section_ref_algorithm_4DVAR:
28 Calculation algorithm "*4DVAR*"
29 -------------------------------
31 .. ------------------------------------ ..
32 .. include:: snippets/Header2Algo01.rst
34 This algorithm realizes an estimation of the state of a dynamic system, by a
35 variational minimization method of the classical :math:`J` function in data
38 .. math:: J(\mathbf{x})=(\mathbf{x}-\mathbf{x}^b)^T.\mathbf{B}^{-1}.(\mathbf{x}-\mathbf{x}^b)+\sum_{t\in T}(\mathbf{y^o}(t)-H(\mathbf{x},t))^T.\mathbf{R}^{-1}.(\mathbf{y^o}(t)-H(\mathbf{x},t))
40 which is usually designed as the "*4D-Var*" functional (see for example
41 [Talagrand97]_). The terms "*4D-Var*", "*4D-VAR*" and "*4DVAR*" are equivalent.
42 This mono-objective optimization algorithm is naturally written for a single
43 estimate on a time window for the simulation. It is well suited in cases of
44 non-linear observation and evolution operators, its application domain is
45 similar to the one of Kalman filters, specially the
46 :ref:`section_ref_algorithm_ExtendedKalmanFilter` or the
47 :ref:`section_ref_algorithm_UnscentedKalmanFilter`.
49 .. ------------------------------------ ..
50 .. include:: snippets/Header2Algo12.rst
52 .. include:: snippets/FeaturePropNonLocalOptimization.rst
54 .. include:: snippets/FeaturePropDerivativeNeeded.rst
56 .. include:: snippets/FeaturePropParallelDerivativesOnly.rst
58 .. ------------------------------------ ..
59 .. include:: snippets/Header2Algo02.rst
61 .. include:: snippets/Background.rst
63 .. include:: snippets/BackgroundError.rst
65 .. include:: snippets/EvolutionError.rst
67 .. include:: snippets/EvolutionModel.rst
69 .. include:: snippets/Observation.rst
71 .. include:: snippets/ObservationError.rst
73 .. include:: snippets/ObservationOperator.rst
75 .. ------------------------------------ ..
76 .. include:: snippets/Header2Algo03AdOp.rst
78 .. include:: snippets/BoundsWithNone.rst
80 .. include:: snippets/ConstrainedBy.rst
82 .. include:: snippets/CostDecrementTolerance.rst
84 .. include:: snippets/EstimationOf_State.rst
86 .. include:: snippets/GradientNormTolerance.rst
88 .. include:: snippets/InitializationPoint.rst
90 .. include:: snippets/MaximumNumberOfIterations.rst
92 .. include:: snippets/Minimizer_xDVAR.rst
94 .. include:: snippets/ProjectedGradientTolerance.rst
96 StoreSupplementaryCalculations
97 .. index:: single: StoreSupplementaryCalculations
99 *List of names*. This list indicates the names of the supplementary
100 variables, that can be available during or at the end of the algorithm, if
101 they are initially required by the user. Their availability involves,
102 potentially, costly calculations or memory consumptions. The default is then
103 a void list, none of these variables being calculated and stored by default
104 (excepted the unconditional variables). The possible names are in the
105 following list (the detailed description of each named variable is given in
106 the following part of this specific algorithmic documentation, in the
107 sub-section "*Information and variables available at the end of the
112 "CostFunctionJAtCurrentOptimum",
114 "CostFunctionJbAtCurrentOptimum",
116 "CostFunctionJoAtCurrentOptimum",
117 "CurrentIterationNumber",
124 ``{"StoreSupplementaryCalculations":["CurrentState", "Residu"]}``
126 .. ------------------------------------ ..
127 .. include:: snippets/Header2Algo04.rst
129 .. include:: snippets/Analysis.rst
131 .. include:: snippets/CostFunctionJ.rst
133 .. include:: snippets/CostFunctionJb.rst
135 .. include:: snippets/CostFunctionJo.rst
137 .. ------------------------------------ ..
138 .. include:: snippets/Header2Algo05.rst
140 .. include:: snippets/Analysis.rst
142 .. include:: snippets/BMA.rst
144 .. include:: snippets/CostFunctionJ.rst
146 .. include:: snippets/CostFunctionJAtCurrentOptimum.rst
148 .. include:: snippets/CostFunctionJb.rst
150 .. include:: snippets/CostFunctionJbAtCurrentOptimum.rst
152 .. include:: snippets/CostFunctionJo.rst
154 .. include:: snippets/CostFunctionJoAtCurrentOptimum.rst
156 .. include:: snippets/CurrentIterationNumber.rst
158 .. include:: snippets/CurrentOptimum.rst
160 .. include:: snippets/CurrentState.rst
162 .. include:: snippets/IndexOfOptimum.rst
164 .. ------------------------------------ ..
165 .. _section_ref_algorithm_4DVAR_examples:
167 .. include:: snippets/Header2Algo06.rst
169 - :ref:`section_ref_algorithm_3DVAR`
170 - :ref:`section_ref_algorithm_KalmanFilter`
171 - :ref:`section_ref_algorithm_ExtendedKalmanFilter`
172 - :ref:`section_ref_algorithm_EnsembleKalmanFilter`
174 .. ------------------------------------ ..
175 .. include:: snippets/Header2Algo07.rst