2 Copyright (C) 2008-2017 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: 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 system by minimization
39 of a cost function :math:`J` without gradient. It is a method that doesn't use
40 the derivatives of the cost function. It fall, for example, in the same category
41 than 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 "BOBYQA", and the possible ones are
113 "BOBYQA" (minimization with or without contraints by quadratic approximation [Powell09]_),
114 "COBYLA" (minimization with or without contraints by linear approximation [Powell94]_ [Powell98]_).
115 "NEWUOA" (minimization with or without contraints by iterative quadratic approximation [Powell04]_),
116 "POWELL" (minimization unconstrained using conjugate directions [Powell64]_),
117 "SIMPLEX" (minimization with or without contraints using Nelder-Mead simplex algorithm [Nelder65]_),
118 "SUBPLEX" (minimization with or without contraints using Nelder-Mead on a sequence of subspaces [Rowan90]_).
119 Remark: the "POWELL" method perform a dual outer/inner loops optimization,
120 leading then to less control on the cost function evaluation number because
121 it is the outer loop limit than is controled. If precise control on this
122 cost function evaluation number is required, choose an another minimizer.
124 Example : ``{"Minimizer":"BOBYQA"}``
127 This key indicates the maximum number of iterations allowed for iterative
128 optimization. The default is 15000, which is very similar to no limit on
129 iterations. It is then recommended to adapt this parameter to the needs on
130 real problems. For some optimizers, the effective stopping step can be
131 slightly different of the limit due to algorithm internal control
134 Example : ``{"MaximumNumberOfSteps":50}``
136 MaximumNumberOfFunctionEvaluations
137 This key indicates the maximum number of evaluation of the cost function to
138 be optimized. The default is 15000, which is an arbitrary limit. It is then
139 recommended to adapt this parameter to the needs on real problems. For some
140 optimizers, the effective number of function evaluations can be slightly
141 different of the limit due to algorithm internal control requirements.
143 Example : ``{"MaximumNumberOfFunctionEvaluations":50}``
145 StateVariationTolerance
146 This key indicates the maximum relative variation of the state for stopping
147 by convergence on the state. The default is 1.e-4, and it is recommended to
148 adapt it to the needs on real problems.
150 Example : ``{"StateVariationTolerance":1.e-4}``
152 CostDecrementTolerance
153 This key indicates a limit value, leading to stop successfully the
154 iterative optimization process when the cost function decreases less than
155 this tolerance at the last step. The default is 1.e-7, and it is
156 recommended to adapt it to the needs on real problems.
158 Example : ``{"CostDecrementTolerance":1.e-7}``
161 This key indicates the quality criterion, minimized to find the optimal
162 state estimate. The default is the usual data assimilation criterion named
163 "DA", the augmented weighted least squares. The possible criteria has to be
164 in the following list, where the equivalent names are indicated by the sign
165 "=": ["AugmentedWeightedLeastSquares"="AWLS"="DA",
166 "WeightedLeastSquares"="WLS", "LeastSquares"="LS"="L2",
167 "AbsoluteValue"="L1", "MaximumError"="ME"].
169 Example : ``{"QualityCriterion":"DA"}``
171 StoreSupplementaryCalculations
172 This list indicates the names of the supplementary variables that can be
173 available at the end of the algorithm. It involves potentially costly
174 calculations or memory consumptions. The default is a void list, none of
175 these variables being calculated and stored by default. The possible names
176 are in the following list: ["CurrentState", "CostFunctionJ",
177 "CostFunctionJb", "CostFunctionJo", "CostFunctionJAtCurrentOptimum",
178 "CurrentOptimum", "IndexOfOptimum", "InnovationAtCurrentState", "BMA",
179 "OMA", "OMB", "SimulatedObservationAtBackground",
180 "SimulatedObservationAtCurrentOptimum",
181 "SimulatedObservationAtCurrentState", "SimulatedObservationAtOptimum"].
183 Example : ``{"StoreSupplementaryCalculations":["BMA", "Innovation"]}``
185 Information and variables available at the end of the algorithm
186 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
188 At the output, after executing the algorithm, there are variables and
189 information originating from the calculation. The description of
190 :ref:`section_ref_output_variables` show the way to obtain them by the method
191 named ``get`` of the variable "*ADD*" of the post-processing. The input
192 variables, available to the user at the output in order to facilitate the
193 writing of post-processing procedures, are described in the
194 :ref:`subsection_r_o_v_Inventaire`.
196 The unconditional outputs of the algorithm are the following:
199 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
200 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
202 Example : ``Xa = ADD.get("Analysis")[-1]``
205 *List of values*. Each element is a value of the error function :math:`J`.
207 Example : ``J = ADD.get("CostFunctionJ")[:]``
210 *List of values*. Each element is a value of the error function :math:`J^b`,
211 that is of the background difference part.
213 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
216 *List of values*. Each element is a value of the error function :math:`J^o`,
217 that is of the observation difference part.
219 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
222 *List of vectors*. Each element is a usual state vector used during the
223 optimization algorithm procedure.
225 Example : ``Xs = ADD.get("CurrentState")[:]``
227 The conditional outputs of the algorithm are the following:
229 SimulatedObservationAtBackground
230 *List of vectors*. Each element is a vector of observation simulated from
231 the background :math:`\mathbf{x}^b`.
233 Example : ``hxb = ADD.get("SimulatedObservationAtBackground")[-1]``
235 SimulatedObservationAtCurrentState
236 *List of vectors*. Each element is an observed vector at the current state,
237 that is, in the observation space.
239 Example : ``Ys = ADD.get("SimulatedObservationAtCurrentState")[-1]``
241 SimulatedObservationAtOptimum
242 *List of vectors*. Each element is a vector of observation simulated from
243 the analysis or optimal state :math:`\mathbf{x}^a`.
245 Example : ``hxa = ADD.get("SimulatedObservationAtOptimum")[-1]``
250 References to other sections:
251 - :ref:`section_ref_algorithm_ParticleSwarmOptimization`
253 Bibliographical references:
