2 Copyright (C) 2008-2015 EDF R&D
4 This file is part of SALOME ADAO module.
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14 Lesser General Public License for more details.
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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-\mathbf{H}.\mathbf{x})^T.\mathbf{R}^{-1}.(\mathbf{y}^o-\mathbf{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", "CurrentState", "OMA",
169 "OMB", "Innovation", "SigmaObs2", "MahalanobisConsistency",
170 "SimulatedObservationAtBackground", "SimulatedObservationAtCurrentState",
171 "SimulatedObservationAtOptimum", "SimulationQuantiles"].
173 Example : ``{"StoreSupplementaryCalculations":["BMA","Innovation"]}``
176 This list indicates the values of quantile, between 0 and 1, to be estimated
177 by simulation around the optimal state. The sampling uses a multivariate
178 gaussian random sampling, directed by the *a posteriori* covariance matrix.
179 This option is useful only if the supplementary calculation
180 "SimulationQuantiles" has been chosen. The default is a void list.
182 Example : ``{"Quantiles":[0.1,0.9]}``
185 This key allow to give an integer in order to fix the seed of the random
186 generator used to generate the ensemble. A convenient value is for example
187 1000. By default, the seed is left uninitialized, and so use the default
188 initialization from the computer.
190 Example : ``{"SetSeed":1000}``
192 NumberOfSamplesForQuantiles
193 This key indicates the number of simulation to be done in order to estimate
194 the quantiles. This option is useful only if the supplementary calculation
195 "SimulationQuantiles" has been chosen. The default is 100, which is often
196 sufficient for correct estimation of common quantiles at 5%, 10%, 90% or
199 Example : ``{"NumberOfSamplesForQuantiles":100}``
201 SimulationForQuantiles
202 This key indicates the type of simulation, linear (with the tangent
203 observation operator applied to perturbation increments around the optimal
204 state) or non-linear (with standard observation operator applied to
205 perturbated states), one want to do for each perturbation. It changes mainly
206 the time of each elementary calculation, usually longer in non-linear than
207 in linear. This option is useful only if the supplementary calculation
208 "SimulationQuantiles" has been chosen. The default value is "Linear", and
209 the possible choices are "Linear" and "NonLinear".
211 Example : ``{"SimulationForQuantiles":"Linear"}``
213 Information and variables available at the end of the algorithm
214 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
216 At the output, after executing the algorithm, there are variables and
217 information originating from the calculation. The description of
218 :ref:`section_ref_output_variables` show the way to obtain them by the method
219 named ``get`` of the variable "*ADD*" of the post-processing. The input
220 variables, available to the user at the output in order to facilitate the
221 writing of post-processing procedures, are described in the
222 :ref:`subsection_r_o_v_Inventaire`.
224 The unconditional outputs of the algorithm are the following:
227 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
228 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
230 Example : ``Xa = ADD.get("Analysis")[-1]``
233 *List of values*. Each element is a value of the error function :math:`J`.
235 Example : ``J = ADD.get("CostFunctionJ")[:]``
238 *List of values*. Each element is a value of the error function :math:`J^b`,
239 that is of the background difference part.
241 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
244 *List of values*. Each element is a value of the error function :math:`J^o`,
245 that is of the observation difference part.
247 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
249 The conditional outputs of the algorithm are the following:
251 APosterioriCorrelations
252 *List of matrices*. Each element is an *a posteriori* error correlation
253 matrix of the optimal state.
255 Example : ``C = ADD.get("APosterioriCorrelations")[-1]``
257 APosterioriCovariance
258 *List of matrices*. Each element is an *a posteriori* error covariance
259 matrix :math:`\mathbf{A}*` of the optimal state.
261 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
263 APosterioriStandardDeviations
264 *List of matrices*. Each element is an *a posteriori* error standard
265 deviation matrix of the optimal state.
267 Example : ``E = ADD.get("APosterioriStandardDeviations")[-1]``
270 *List of matrices*. Each element is an *a posteriori* error variance matrix
271 of the optimal state.
273 Example : ``V = ADD.get("APosterioriVariances")[-1]``
276 *List of vectors*. Each element is a vector of difference between the
277 background and the optimal state.
279 Example : ``bma = ADD.get("BMA")[-1]``
282 *List of vectors*. Each element is a usual state vector used during the
283 optimization algorithm procedure.
285 Example : ``Xs = ADD.get("CurrentState")[:]``
288 *List of vectors*. Each element is an innovation vector, which is in static
289 the difference between the optimal and the background, and in dynamic the
292 Example : ``d = ADD.get("Innovation")[-1]``
294 MahalanobisConsistency
295 *List of values*. Each element is a value of the Mahalanobis quality
298 Example : ``m = ADD.get("MahalanobisConsistency")[-1]``
301 *List of vectors*. Each element is a vector of difference between the
302 observation and the optimal state in the observation space.
304 Example : ``oma = ADD.get("OMA")[-1]``
307 *List of vectors*. Each element is a vector of difference between the
308 observation and the background state in the observation space.
310 Example : ``omb = ADD.get("OMB")[-1]``
313 *List of values*. Each element is a value of the quality indicator
314 :math:`(\sigma^o)^2` of the observation part.
316 Example : ``so2 = ADD.get("SigmaObs")[-1]``
318 SimulatedObservationAtBackground
319 *List of vectors*. Each element is a vector of observation simulated from
320 the background :math:`\mathbf{x}^b`.
322 Example : ``hxb = ADD.get("SimulatedObservationAtBackground")[-1]``
324 SimulatedObservationAtCurrentState
325 *List of vectors*. Each element is an observed vector at the current state,
326 that is, in the observation space.
328 Example : ``Ys = ADD.get("SimulatedObservationAtCurrentState")[-1]``
330 SimulatedObservationAtOptimum
331 *List of vectors*. Each element is a vector of observation simulated from
332 the analysis or optimal state :math:`\mathbf{x}^a`.
334 Example : ``hxa = ADD.get("SimulatedObservationAtOptimum")[-1]``
337 *List of vectors*. Each element is a vector corresponding to the observed
338 state which realize the required quantile, in the same order than the
339 quantiles required by the user.
341 Example : ``sQuantiles = ADD.get("SimulationQuantiles")[:]``
346 References to other sections:
347 - :ref:`section_ref_algorithm_Blue`
348 - :ref:`section_ref_algorithm_ExtendedBlue`
349 - :ref:`section_ref_algorithm_LinearityTest`
351 Bibliographical references: