2 Copyright (C) 2008-2016 EDF R&D
4 This file is part of SALOME ADAO module.
6 This library is free software; you can redistribute it and/or
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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: DerivativeFreeOptimization
25 .. _section_ref_algorithm_DerivativeFreeOptimization:
27 Calculation algorithm "*DerivativeFreeOptimization*"
28 ----------------------------------------------------
32 in its present version, this algorithm is experimental, and so changes can be
33 required in forthcoming versions.
38 This algorithm realizes an estimation of the state of a dynamic system by
39 minimization of a cost function :math:`J` without gradient. It is a method that
40 doesn't use the derivatives of the cost function. It fall in the same category
41 then the :ref:`section_ref_algorithm_ParticleSwarmOptimization`.
43 This is an optimization method allowing for global minimum search of a general
44 error function :math:`J` of type :math:`L^1`, :math:`L^2` or :math:`L^{\infty}`,
45 with or without weights. The default error function is the augmented weighted
46 least squares function, classicaly used in data assimilation.
48 Optional and required commands
49 ++++++++++++++++++++++++++++++
51 .. index:: single: AlgorithmParameters
52 .. index:: single: Background
53 .. index:: single: BackgroundError
54 .. index:: single: Observation
55 .. index:: single: ObservationError
56 .. index:: single: ObservationOperator
57 .. index:: single: Minimizer
58 .. index:: single: MaximumNumberOfSteps
59 .. index:: single: MaximumNumberOfFunctionEvaluations
60 .. index:: single: StateVariationTolerance
61 .. index:: single: CostDecrementTolerance
62 .. index:: single: QualityCriterion
63 .. index:: single: StoreSupplementaryCalculations
65 The general required commands, available in the editing user interface, are the
69 *Required command*. This indicates the background or initial vector used,
70 previously noted as :math:`\mathbf{x}^b`. Its value is defined as a
71 "*Vector*" or a *VectorSerie*" type object.
74 *Required command*. This indicates the background error covariance matrix,
75 previously noted as :math:`\mathbf{B}`. Its value is defined as a "*Matrix*"
76 type object, a "*ScalarSparseMatrix*" type object, or a
77 "*DiagonalSparseMatrix*" type object.
80 *Required command*. This indicates the observation vector used for data
81 assimilation or optimization, previously noted as :math:`\mathbf{y}^o`. It
82 is defined as a "*Vector*" or a *VectorSerie* type object.
85 *Required command*. This indicates the observation error covariance matrix,
86 previously noted as :math:`\mathbf{R}`. It is defined as a "*Matrix*" type
87 object, a "*ScalarSparseMatrix*" type object, or a "*DiagonalSparseMatrix*"
91 *Required command*. This indicates the observation operator, previously
92 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
93 results :math:`\mathbf{y}` to be compared to observations
94 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
95 a "*Matrix*" type one. In the case of "*Function*" type, different
96 functional forms can be used, as described in the section
97 :ref:`section_ref_operator_requirements`. If there is some control :math:`U`
98 included in the observation, the operator has to be applied to a pair
101 The general optional commands, available in the editing user interface, are
102 indicated in :ref:`section_ref_assimilation_keywords`. Moreover, the parameters
103 of the command "*AlgorithmParameters*" allows to choose the specific options,
104 described hereafter, of the algorithm. See
105 :ref:`section_ref_options_Algorithm_Parameters` for the good use of this
108 The options of the algorithm are the following:
111 This key allows to choose the optimization minimizer. The default choice is
112 "POWELL", and the possible ones are "POWELL" (modified Powell unconstrained
113 minimizer, see [Powell]_), "SIMPLEX" (nonlinear constrained minimizer), "CG"
114 (simplex of Nelder-Mead unconstrained minimizer, see [Nelder]_). It is
115 recommended to stay with the default.
117 Example : ``{"Minimizer":"POWELL"}``
120 This key indicates the maximum number of iterations allowed for iterative
121 optimization. The default is 15000, which is very similar to no limit on
122 iterations. It is then recommended to adapt this parameter to the needs on
123 real problems. For some optimizers, the effective stopping step can be
124 slightly different of the limit due to algorithm internal control
127 Example : ``{"MaximumNumberOfSteps":50}``
129 MaximumNumberOfFunctionEvaluations
130 This key indicates the maximum number of evaluation of the cost function to
131 be optimized. The default is 15000, which is very similar to no limit on
132 iterations. The calculation can be over this limit when an outer
133 optimization loop has to be finished. It is strongly recommended to adapt
134 this parameter to the needs on real problems.
136 Example : ``{"MaximumNumberOfFunctionEvaluations":50}``
138 StateVariationTolerance
139 This key indicates the maximum relative variation of the state for stopping
140 by convergence on the state. The default is 1.e-4, and it is recommended to
141 adapt it to the needs on real problems.
143 Example : ``{"StateVariationTolerance":1.e-4}``
145 CostDecrementTolerance
146 This key indicates a limit value, leading to stop successfully the
147 iterative optimization process when the cost function decreases less than
148 this tolerance at the last step. The default is 1.e-7, and it is
149 recommended to adapt it to the needs on real problems.
151 Example : ``{"CostDecrementTolerance":1.e-7}``
154 This key indicates the quality criterion, minimized to find the optimal
155 state estimate. The default is the usual data assimilation criterion named
156 "DA", the augmented weighted least squares. The possible criteria has to be
157 in the following list, where the equivalent names are indicated by the sign
158 "=": ["AugmentedWeightedLeastSquares"="AWLS"="DA",
159 "WeightedLeastSquares"="WLS", "LeastSquares"="LS"="L2",
160 "AbsoluteValue"="L1", "MaximumError"="ME"].
162 Example : ``{"QualityCriterion":"DA"}``
164 StoreSupplementaryCalculations
165 This list indicates the names of the supplementary variables that can be
166 available at the end of the algorithm. It involves potentially costly
167 calculations or memory consumptions. The default is a void list, none of
168 these variables being calculated and stored by default. The possible names
169 are in the following list: ["CurrentState", "CostFunctionJ",
170 "CostFunctionJAtCurrentOptimum", "CurrentOptimum", "IndexOfOptimum",
171 "InnovationAtCurrentState", "BMA", "OMA", "OMB",
172 "SimulatedObservationAtBackground", "SimulatedObservationAtCurrentOptimum",
173 "SimulatedObservationAtCurrentState", "SimulatedObservationAtOptimum"].
175 Example : ``{"StoreSupplementaryCalculations":["BMA", "Innovation"]}``
177 Information and variables available at the end of the algorithm
178 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
180 At the output, after executing the algorithm, there are variables and
181 information originating from the calculation. The description of
182 :ref:`section_ref_output_variables` show the way to obtain them by the method
183 named ``get`` of the variable "*ADD*" of the post-processing. The input
184 variables, available to the user at the output in order to facilitate the
185 writing of post-processing procedures, are described in the
186 :ref:`subsection_r_o_v_Inventaire`.
188 The unconditional outputs of the algorithm are the following:
191 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
192 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
194 Example : ``Xa = ADD.get("Analysis")[-1]``
197 *List of values*. Each element is a value of the error function :math:`J`.
199 Example : ``J = ADD.get("CostFunctionJ")[:]``
202 *List of values*. Each element is a value of the error function :math:`J^b`,
203 that is of the background difference part.
205 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
208 *List of values*. Each element is a value of the error function :math:`J^o`,
209 that is of the observation difference part.
211 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
214 *List of vectors*. Each element is a usual state vector used during the
215 optimization algorithm procedure.
217 Example : ``Xs = ADD.get("CurrentState")[:]``
219 The conditional outputs of the algorithm are the following:
221 SimulatedObservationAtBackground
222 *List of vectors*. Each element is a vector of observation simulated from
223 the background :math:`\mathbf{x}^b`.
225 Example : ``hxb = ADD.get("SimulatedObservationAtBackground")[-1]``
227 SimulatedObservationAtCurrentState
228 *List of vectors*. Each element is an observed vector at the current state,
229 that is, in the observation space.
231 Example : ``Ys = ADD.get("SimulatedObservationAtCurrentState")[-1]``
233 SimulatedObservationAtOptimum
234 *List of vectors*. Each element is a vector of observation simulated from
235 the analysis or optimal state :math:`\mathbf{x}^a`.
237 Example : ``hxa = ADD.get("SimulatedObservationAtOptimum")[-1]``
242 References to other sections:
243 - :ref:`section_ref_algorithm_ParticleSwarmOptimization`
245 Bibliographical references: