2 Copyright (C) 2008-2018 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: LinearLeastSquares
25 .. _section_ref_algorithm_LinearLeastSquares:
27 Calculation algorithm "*LinearLeastSquares*"
28 --------------------------------------------
33 This algorithm realizes a "Least Squares" linear type estimation of the state of
34 a system. It is similar to the :ref:`section_ref_algorithm_Blue`, without its
37 This algorithm is always the fastest of all the optimization algorithms of ADAO.
38 It is theoretically reserved for observation operator cases which are linear,
39 even if it sometimes works in "slightly" non-linear cases. One can verify the
40 linearity of the observation operator with the help of the
41 :ref:`section_ref_algorithm_LinearityTest`.
43 In all cases, it is recommanded to prefer at least the
44 :ref:`section_ref_algorithm_Blue`, or the
45 :ref:`section_ref_algorithm_ExtendedBlue` or the
46 :ref:`section_ref_algorithm_3DVAR`.
48 Optional and required commands
49 ++++++++++++++++++++++++++++++
51 .. index:: single: AlgorithmParameters
52 .. index:: single: Observation
53 .. index:: single: ObservationError
54 .. index:: single: ObservationOperator
55 .. index:: single: StoreSupplementaryCalculations
57 The general required commands, available in the editing user interface, are the
61 *Required command*. This indicates the observation vector used for data
62 assimilation or optimization, previously noted as :math:`\mathbf{y}^o`. It
63 is defined as a "*Vector*" or a *VectorSerie* type object.
66 *Required command*. This indicates the observation error covariance matrix,
67 previously noted as :math:`\mathbf{R}`. It is defined as a "*Matrix*" type
68 object, a "*ScalarSparseMatrix*" type object, or a "*DiagonalSparseMatrix*"
72 *Required command*. This indicates the observation operator, previously
73 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
74 results :math:`\mathbf{y}` to be compared to observations
75 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
76 a "*Matrix*" type one. In the case of "*Function*" type, different
77 functional forms can be used, as described in the section
78 :ref:`section_ref_operator_requirements`. If there is some control :math:`U`
79 included in the observation, the operator has to be applied to a pair
82 The general optional commands, available in the editing user interface, are
83 indicated in :ref:`section_ref_assimilation_keywords`. Moreover, the parameters
84 of the command "*AlgorithmParameters*" allows to choose the specific options,
85 described hereafter, of the algorithm. See
86 :ref:`section_ref_options_Algorithm_Parameters` for the good use of this
89 The options of the algorithm are the following:
91 StoreSupplementaryCalculations
92 This list indicates the names of the supplementary variables that can be
93 available at the end of the algorithm. It involves potentially costly
94 calculations or memory consumptions. The default is a void list, none of
95 these variables being calculated and stored by default. The possible names
96 are in the following list: ["OMA", "CurrentState", "CostFunctionJ",
97 "CostFunctionJb", "CostFunctionJo", "SimulatedObservationAtCurrentState",
98 "SimulatedObservationAtOptimum"].
100 Example : ``{"StoreSupplementaryCalculations":["OMA", "CurrentState"]}``
102 *Tips for this algorithm:*
104 As the *"Background"* and *"BackgroundError"* commands are required for ALL
105 the calculation algorithms in the interface, you have to provide a value,
106 even if these commands are not required for this algorithm, and will not be
107 used. The simplest way is to give "1" as a STRING for both.
109 Information and variables available at the end of the algorithm
110 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
112 At the output, after executing the algorithm, there are variables and
113 information originating from the calculation. The description of
114 :ref:`section_ref_output_variables` show the way to obtain them by the method
115 named ``get`` of the variable "*ADD*" of the post-processing. The input
116 variables, available to the user at the output in order to facilitate the
117 writing of post-processing procedures, are described in the
118 :ref:`subsection_r_o_v_Inventaire`.
120 The unconditional outputs of the algorithm are the following:
123 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
124 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
126 Example : ``Xa = ADD.get("Analysis")[-1]``
129 *List of values*. Each element is a value of the error function :math:`J`.
131 Example : ``J = ADD.get("CostFunctionJ")[:]``
134 *List of values*. Each element is a value of the error function :math:`J^b`,
135 that is of the background difference part.
137 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
140 *List of values*. Each element is a value of the error function :math:`J^o`,
141 that is of the observation difference part.
143 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
145 The conditional outputs of the algorithm are the following:
148 *List of vectors*. Each element is a vector of difference between the
149 observation and the optimal state in the observation space.
151 Example : ``oma = ADD.get("OMA")[-1]``
153 SimulatedObservationAtOptimum
154 *List of vectors*. Each element is a vector of observation simulated from
155 the analysis or optimal state :math:`\mathbf{x}^a`.
157 Example : ``hxa = ADD.get("SimulatedObservationAtOptimum")[-1]``
162 References to other sections:
163 - :ref:`section_ref_algorithm_Blue`
164 - :ref:`section_ref_algorithm_ExtendedBlue`
165 - :ref:`section_ref_algorithm_3DVAR`
166 - :ref:`section_ref_algorithm_LinearityTest`