2 Copyright (C) 2008-2015 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: 3DVAR
25 .. _section_ref_algorithm_3DVAR:
27 Calculation algorithm "*3DVAR*"
28 -------------------------------
33 This algorithm performs a state estimation by variational minimization of the
34 classical :math:`J` function in static data assimilation:
36 .. math:: J(\mathbf{x})=(\mathbf{x}-\mathbf{x}^b)^T.\mathbf{B}^{-1}.(\mathbf{x}-\mathbf{x}^b)+(\mathbf{y}^o-H(\mathbf{x}))^T.\mathbf{R}^{-1}.(\mathbf{y}^o-H(\mathbf{x}))
38 which is usually designed as the "*3D-VAR*" function (see for example
41 Optional and required commands
42 ++++++++++++++++++++++++++++++
44 .. index:: single: AlgorithmParameters
45 .. index:: single: Background
46 .. index:: single: BackgroundError
47 .. index:: single: Observation
48 .. index:: single: ObservationError
49 .. index:: single: ObservationOperator
50 .. index:: single: Minimizer
51 .. index:: single: Bounds
52 .. index:: single: MaximumNumberOfSteps
53 .. index:: single: CostDecrementTolerance
54 .. index:: single: ProjectedGradientTolerance
55 .. index:: single: GradientNormTolerance
56 .. index:: single: StoreSupplementaryCalculations
57 .. index:: single: Quantiles
58 .. index:: single: SetSeed
59 .. index:: single: NumberOfSamplesForQuantiles
60 .. index:: single: SimulationForQuantiles
62 The general required commands, available in the editing user interface, are the
66 *Required command*. This indicates the background or initial vector used,
67 previously noted as :math:`\mathbf{x}^b`. Its value is defined as a
68 "*Vector*" or a *VectorSerie*" type object.
71 *Required command*. This indicates the background error covariance matrix,
72 previously noted as :math:`\mathbf{B}`. Its value is defined as a "*Matrix*"
73 type object, a "*ScalarSparseMatrix*" type object, or a
74 "*DiagonalSparseMatrix*" type object.
77 *Required command*. This indicates the observation vector used for data
78 assimilation or optimization, previously noted as :math:`\mathbf{y}^o`. It
79 is defined as a "*Vector*" or a *VectorSerie* type object.
82 *Required command*. This indicates the observation error covariance matrix,
83 previously noted as :math:`\mathbf{R}`. It is defined as a "*Matrix*" type
84 object, a "*ScalarSparseMatrix*" type object, or a "*DiagonalSparseMatrix*"
88 *Required command*. This indicates the observation operator, previously
89 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
90 results :math:`\mathbf{y}` to be compared to observations
91 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
92 a "*Matrix*" type one. In the case of "*Function*" type, different
93 functional forms can be used, as described in the section
94 :ref:`section_ref_operator_requirements`. If there is some control :math:`U`
95 included in the observation, the operator has to be applied to a pair
98 The general optional commands, available in the editing user interface, are
99 indicated in :ref:`section_ref_assimilation_keywords`. Moreover, the parameters
100 of the command "*AlgorithmParameters*" allows to choose the specific options,
101 described hereafter, of the algorithm. See
102 :ref:`section_ref_options_Algorithm_Parameters` for the good use of this
105 The options of the algorithm are the following:
108 This key allows to choose the optimization minimizer. The default choice is
109 "LBFGSB", and the possible ones are "LBFGSB" (nonlinear constrained
110 minimizer, see [Byrd95]_, [Morales11]_ and [Zhu97]_), "TNC" (nonlinear
111 constrained minimizer), "CG" (nonlinear unconstrained minimizer), "BFGS"
112 (nonlinear unconstrained minimizer), "NCG" (Newton CG minimizer). It is
113 strongly recommended to stay with the default.
115 Example : ``{"Minimizer":"LBFGSB"}``
118 This key allows to define upper and lower bounds for every state variable
119 being optimized. Bounds have to be given by a list of list of pairs of
120 lower/upper bounds for each variable, with possibly ``None`` every time
121 there is no bound. The bounds can always be specified, but they are taken
122 into account only by the constrained optimizers.
124 Example : ``{"Bounds":[[2.,5.],[1.e-2,10.],[-30.,None],[None,None]]}``
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":100}``
136 CostDecrementTolerance
137 This key indicates a limit value, leading to stop successfully the
138 iterative optimization process when the cost function decreases less than
139 this tolerance at the last step. The default is 1.e-7, and it is
140 recommended to adapt it to the needs on real problems.
142 Example : ``{"CostDecrementTolerance":1.e-7}``
144 ProjectedGradientTolerance
145 This key indicates a limit value, leading to stop successfully the iterative
146 optimization process when all the components of the projected gradient are
147 under this limit. It is only used for constrained optimizers. The default is
148 -1, that is the internal default of each minimizer (generally 1.e-5), and it
149 is not recommended to change it.
151 Example : ``{"ProjectedGradientTolerance":-1}``
153 GradientNormTolerance
154 This key indicates a limit value, leading to stop successfully the
155 iterative optimization process when the norm of the gradient is under this
156 limit. It is only used for non-constrained optimizers. The default is
157 1.e-5 and it is not recommended to change it.
159 Example : ``{"GradientNormTolerance":1.e-5}``
161 StoreSupplementaryCalculations
162 This list indicates the names of the supplementary variables that can be
163 available at the end of the algorithm. It involves potentially costly
164 calculations or memory consumptions. The default is a void list, none of
165 these variables being calculated and stored by default. The possible names
166 are in the following list: ["APosterioriCorrelations",
167 "APosterioriCovariance", "APosterioriStandardDeviations",
168 "APosterioriVariances", "BMA", "CostFunctionJ",
169 "CostFunctionJAtCurrentOptimum", "CurrentOptimum", "CurrentState",
170 "IndexOfOptimum", "Innovation", "InnovationAtCurrentState",
171 "MahalanobisConsistency", "OMA", "OMB", "SigmaObs2",
172 "SimulatedObservationAtBackground", "SimulatedObservationAtCurrentOptimum",
173 "SimulatedObservationAtCurrentState", "SimulatedObservationAtOptimum",
174 "SimulationQuantiles"].
176 Example : ``{"StoreSupplementaryCalculations":["BMA", "Innovation"]}``
179 This list indicates the values of quantile, between 0 and 1, to be estimated
180 by simulation around the optimal state. The sampling uses a multivariate
181 gaussian random sampling, directed by the *a posteriori* covariance matrix.
182 This option is useful only if the supplementary calculation
183 "SimulationQuantiles" has been chosen. The default is a void list.
185 Example : ``{"Quantiles":[0.1,0.9]}``
188 This key allow to give an integer in order to fix the seed of the random
189 generator used to generate the ensemble. A convenient value is for example
190 1000. By default, the seed is left uninitialized, and so use the default
191 initialization from the computer.
193 Example : ``{"SetSeed":1000}``
195 NumberOfSamplesForQuantiles
196 This key indicates the number of simulation to be done in order to estimate
197 the quantiles. This option is useful only if the supplementary calculation
198 "SimulationQuantiles" has been chosen. The default is 100, which is often
199 sufficient for correct estimation of common quantiles at 5%, 10%, 90% or
202 Example : ``{"NumberOfSamplesForQuantiles":100}``
204 SimulationForQuantiles
205 This key indicates the type of simulation, linear (with the tangent
206 observation operator applied to perturbation increments around the optimal
207 state) or non-linear (with standard observation operator applied to
208 perturbated states), one want to do for each perturbation. It changes mainly
209 the time of each elementary calculation, usually longer in non-linear than
210 in linear. This option is useful only if the supplementary calculation
211 "SimulationQuantiles" has been chosen. The default value is "Linear", and
212 the possible choices are "Linear" and "NonLinear".
214 Example : ``{"SimulationForQuantiles":"Linear"}``
216 Information and variables available at the end of the algorithm
217 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
219 At the output, after executing the algorithm, there are variables and
220 information originating from the calculation. The description of
221 :ref:`section_ref_output_variables` show the way to obtain them by the method
222 named ``get`` of the variable "*ADD*" of the post-processing. The input
223 variables, available to the user at the output in order to facilitate the
224 writing of post-processing procedures, are described in the
225 :ref:`subsection_r_o_v_Inventaire`.
227 The unconditional outputs of the algorithm are the following:
230 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
231 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
233 Example : ``Xa = ADD.get("Analysis")[-1]``
236 *List of values*. Each element is a value of the error function :math:`J`.
238 Example : ``J = ADD.get("CostFunctionJ")[:]``
241 *List of values*. Each element is a value of the error function :math:`J^b`,
242 that is of the background difference part.
244 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
247 *List of values*. Each element is a value of the error function :math:`J^o`,
248 that is of the observation difference part.
250 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
252 The conditional outputs of the algorithm are the following:
254 APosterioriCorrelations
255 *List of matrices*. Each element is an *a posteriori* error correlations
256 matrix of the optimal state, coming from the :math:`\mathbf{A}*` covariance
259 Example : ``C = ADD.get("APosterioriCorrelations")[-1]``
261 APosterioriCovariance
262 *List of matrices*. Each element is an *a posteriori* error covariance
263 matrix :math:`\mathbf{A}*` of the optimal state.
265 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
267 APosterioriStandardDeviations
268 *List of matrices*. Each element is an *a posteriori* error standard
269 errors diagonal matrix of the optimal state, coming from the
270 :math:`\mathbf{A}*` covariance matrix.
272 Exemple : ``S = ADD.get("APosterioriStandardDeviations")[-1]``
275 *List of matrices*. Each element is an *a posteriori* error variance
276 errors diagonal matrix of the optimal state, coming from the
277 :math:`\mathbf{A}*` covariance matrix.
279 Example : ``V = ADD.get("APosterioriVariances")[-1]``
282 *List of vectors*. Each element is a vector of difference between the
283 background and the optimal state.
285 Example : ``bma = ADD.get("BMA")[-1]``
287 CostFunctionJAtCurrentOptimum
288 *List of values*. Each element is a value of the error function :math:`J`.
289 At each step, the value corresponds to the optimal state found from the
292 Example : ``JACO = ADD.get("CostFunctionJAtCurrentOptimum")[:]``
294 CostFunctionJbAtCurrentOptimum
295 *List of values*. Each element is a value of the error function :math:`J^b`,
296 that is of the background difference part. At each step, the value
297 corresponds to the optimal state found from the beginning.
299 Example : ``JbACO = ADD.get("CostFunctionJbAtCurrentOptimum")[:]``
301 CostFunctionJoAtCurrentOptimum
302 *List of values*. Each element is a value of the error function :math:`J^o`,
303 that is of the observation difference part. At each step, the value
304 corresponds to the optimal state found from the beginning.
306 Example : ``JoACO = ADD.get("CostFunctionJoAtCurrentOptimum")[:]``
309 *List of vectors*. Each element is the optimal state obtained at the current
310 step of the optimization algorithm. It is not necessarely the last state.
312 Exemple : ``Xo = ADD.get("CurrentOptimum")[:]``
315 *List of vectors*. Each element is a usual state vector used during the
316 optimization algorithm procedure.
318 Example : ``Xs = ADD.get("CurrentState")[:]``
321 *List of integers*. Each element is the iteration index of the optimum
322 obtained at the current step the optimization algorithm. It is not
323 necessarely the number of the last iteration.
325 Exemple : ``i = ADD.get("IndexOfOptimum")[-1]``
328 *List of vectors*. Each element is an innovation vector, which is in static
329 the difference between the optimal and the background, and in dynamic the
332 Example : ``d = ADD.get("Innovation")[-1]``
334 InnovationAtCurrentState
335 *List of vectors*. Each element is an innovation vector at current state.
337 Example : ``ds = ADD.get("InnovationAtCurrentState")[-1]``
339 MahalanobisConsistency
340 *List of values*. Each element is a value of the Mahalanobis quality
343 Example : ``m = ADD.get("MahalanobisConsistency")[-1]``
346 *List of vectors*. Each element is a vector of difference between the
347 observation and the optimal state in the observation space.
349 Example : ``oma = ADD.get("OMA")[-1]``
352 *List of vectors*. Each element is a vector of difference between the
353 observation and the background state in the observation space.
355 Example : ``omb = ADD.get("OMB")[-1]``
358 *List of values*. Each element is a value of the quality indicator
359 :math:`(\sigma^o)^2` of the observation part.
361 Example : ``so2 = ADD.get("SigmaObs")[-1]``
363 SimulatedObservationAtBackground
364 *List of vectors*. Each element is a vector of observation simulated from
365 the background :math:`\mathbf{x}^b`.
367 Example : ``hxb = ADD.get("SimulatedObservationAtBackground")[-1]``
369 SimulatedObservationAtCurrentOptimum
370 *List of vectors*. Each element is a vector of observation simulated from
371 the optimal state obtained at the current step the optimization algorithm,
372 that is, in the observation space.
374 Exemple : ``hxo = ADD.get("SimulatedObservationAtCurrentOptimum")[-1]``
376 SimulatedObservationAtCurrentState
377 *List of vectors*. Each element is an observed vector at the current state,
378 that is, in the observation space.
380 Example : ``hxs = ADD.get("SimulatedObservationAtCurrentState")[-1]``
382 SimulatedObservationAtOptimum
383 *List of vectors*. Each element is a vector of observation simulated from
384 the analysis or optimal state :math:`\mathbf{x}^a`.
386 Example : ``hxa = ADD.get("SimulatedObservationAtOptimum")[-1]``
389 *List of vectors*. Each element is a vector corresponding to the observed
390 state which realize the required quantile, in the same order than the
391 quantiles required by the user.
393 Example : ``sQuantiles = ADD.get("SimulationQuantiles")[:]``
398 References to other sections:
399 - :ref:`section_ref_algorithm_Blue`
400 - :ref:`section_ref_algorithm_ExtendedBlue`
401 - :ref:`section_ref_algorithm_LinearityTest`
403 Bibliographical references: