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
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: NonLinearLeastSquares
25 .. _section_ref_algorithm_NonLinearLeastSquares:
27 Calculation algorithm "*NonLinearLeastSquares*"
28 -----------------------------------------------
33 This algorithm realizes a state estimation by variational minimization of the
34 classical :math:`J` function of weighted "Least Squares":
36 .. math:: J(\mathbf{x})=(\mathbf{y}^o-\mathbf{H}.\mathbf{x})^T.\mathbf{R}^{-1}.(\mathbf{y}^o-\mathbf{H}.\mathbf{x})
38 It is similar to the :ref:`section_ref_algorithm_3DVAR`, without its background
39 part. The background, required in the interface, is only used as an initial
40 point for the variational minimization.
42 In all cases, it is recommended to prefer the :ref:`section_ref_algorithm_3DVAR`
43 for its stability as for its behaviour during optimization.
45 Optional and required commands
46 ++++++++++++++++++++++++++++++
48 .. index:: single: Background
49 .. index:: single: Observation
50 .. index:: single: ObservationError
51 .. index:: single: ObservationOperator
52 .. index:: single: Minimizer
53 .. index:: single: Bounds
54 .. index:: single: MaximumNumberOfSteps
55 .. index:: single: CostDecrementTolerance
56 .. index:: single: ProjectedGradientTolerance
57 .. index:: single: GradientNormTolerance
58 .. index:: single: StoreInternalVariables
59 .. index:: single: StoreSupplementaryCalculations
61 The general required commands, available in the editing user interface, are the
65 *Required command*. This indicates the background or initial vector used,
66 previously noted as :math:`\mathbf{x}^b`. Its value is defined as a
67 "*Vector*" or a *VectorSerie*" type object.
70 *Required command*. This indicates the observation vector used for data
71 assimilation or optimization, previously noted as :math:`\mathbf{y}^o`. It
72 is defined as a "*Vector*" or a *VectorSerie* type object.
75 *Required command*. This indicates the observation error covariance matrix,
76 previously noted as :math:`\mathbf{R}`. It is defined as a "*Matrix*" type
77 object, a "*ScalarSparseMatrix*" type object, or a "*DiagonalSparseMatrix*"
81 *Required command*. This indicates the observation operator, previously
82 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
83 results :math:`\mathbf{y}` to be compared to observations
84 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
85 a "*Matrix*" type one. In the case of "*Function*" type, different
86 functional forms can be used, as described in the section
87 :ref:`section_ref_operator_requirements`. If there is some control :math:`U`
88 included in the observation, the operator has to be applied to a pair
91 The general optional commands, available in the editing user interface, are
92 indicated in :ref:`section_ref_assimilation_keywords`. In particular, the
93 optional command "*AlgorithmParameters*" allows to choose the specific options,
94 described hereafter, of the algorithm. See
95 :ref:`section_ref_options_AlgorithmParameters` for the good use of this command.
97 The options of the algorithm are the following:
100 This key allows to choose the optimization minimizer. The default choice is
101 "LBFGSB", and the possible ones are "LBFGSB" (nonlinear constrained
102 minimizer, see [Byrd95]_, [Morales11]_ and [Zhu97]_), "TNC" (nonlinear
103 constrained minimizer), "CG" (nonlinear unconstrained minimizer), "BFGS"
104 (nonlinear unconstrained minimizer), "NCG" (Newton CG minimizer). It is
105 strongly recommended to stay with the default.
107 Example : ``{"Minimizer":"LBFGSB"}``
110 This key allows to define upper and lower bounds for every state variable
111 being optimized. Bounds have to be given by a list of list of pairs of
112 lower/upper bounds for each variable, with possibly ``None`` every time
113 there is no bound. The bounds can always be specified, but they are taken
114 into account only by the constrained optimizers.
116 Example : ``{"Bounds":[[2.,5.],[1.e-2,10.],[-30.,None],[None,None]]}``
119 This key indicates the maximum number of iterations allowed for iterative
120 optimization. The default is 15000, which is very similar to no limit on
121 iterations. It is then recommended to adapt this parameter to the needs on
122 real problems. For some optimizers, the effective stopping step can be
123 slightly different due to algorithm internal control requirements.
125 Example : ``{"MaximumNumberOfSteps":100}``
127 CostDecrementTolerance
128 This key indicates a limit value, leading to stop successfully the
129 iterative optimization process when the cost function decreases less than
130 this tolerance at the last step. The default is 1.e-7, and it is
131 recommended to adapt it to the needs on real problems.
133 Example : ``{"CostDecrementTolerance":1.e-7}``
135 ProjectedGradientTolerance
136 This key indicates a limit value, leading to stop successfully the iterative
137 optimization process when all the components of the projected gradient are
138 under this limit. It is only used for constrained optimizers. The default is
139 -1, that is the internal default of each minimizer (generally 1.e-5), and it
140 is not recommended to change it.
142 Example : ``{"ProjectedGradientTolerance":-1}``
144 GradientNormTolerance
145 This key indicates a limit value, leading to stop successfully the
146 iterative optimization process when the norm of the gradient is under this
147 limit. It is only used for non-constrained optimizers. The default is
148 1.e-5 and it is not recommended to change it.
150 Example : ``{"GradientNormTolerance":1.e-5}``
152 StoreInternalVariables
153 This Boolean key allows to store default internal variables, mainly the
154 current state during iterative optimization process. Be careful, this can be
155 a numerically costly choice in certain calculation cases. The default is
158 Example : ``{"StoreInternalVariables":True}``
160 StoreSupplementaryCalculations
161 This list indicates the names of the supplementary variables that can be
162 available at the end of the algorithm. It involves potentially costly
163 calculations or memory consumptions. The default is a void list, none of
164 these variables being calculated and stored by default. The possible names
165 are in the following list: ["BMA", "OMA", "OMB", "Innovation"].
167 Example : ``{"StoreSupplementaryCalculations":["BMA","Innovation"]}``
169 *Tips for this algorithm:*
171 As the *"BackgroundError"* command is required for ALL the calculation
172 algorithms in the interface, you have to provide a value, even if this
173 command is not required for this algorithm, and will not be used. The
174 simplest way is to give "1" as a STRING.
179 References to other sections:
180 - :ref:`section_ref_algorithm_3DVAR`
182 Bibliographical references: