2 Copyright (C) 2008-2015 EDF R&D
4 This file is part of SALOME ADAO module.
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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: Background
45 .. index:: single: BackgroundError
46 .. index:: single: Observation
47 .. index:: single: ObservationError
48 .. index:: single: ObservationOperator
49 .. index:: single: Minimizer
50 .. index:: single: Bounds
51 .. index:: single: MaximumNumberOfSteps
52 .. index:: single: CostDecrementTolerance
53 .. index:: single: ProjectedGradientTolerance
54 .. index:: single: GradientNormTolerance
55 .. index:: single: StoreInternalVariables
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`. In particular, the
100 optional command "*AlgorithmParameters*" allows to choose the specific options,
101 described hereafter, of the algorithm. See
102 :ref:`section_ref_options_AlgorithmParameters` for the good use of this command.
104 The options of the algorithm are the following:
107 This key allows to choose the optimization minimizer. The default choice is
108 "LBFGSB", and the possible ones are "LBFGSB" (nonlinear constrained
109 minimizer, see [Byrd95]_, [Morales11]_ and [Zhu97]_), "TNC" (nonlinear
110 constrained minimizer), "CG" (nonlinear unconstrained minimizer), "BFGS"
111 (nonlinear unconstrained minimizer), "NCG" (Newton CG minimizer). It is
112 strongly recommended to stay with the default.
114 Example : ``{"Minimizer":"LBFGSB"}``
117 This key allows to define upper and lower bounds for every state variable
118 being optimized. Bounds have to be given by a list of list of pairs of
119 lower/upper bounds for each variable, with possibly ``None`` every time
120 there is no bound. The bounds can always be specified, but they are taken
121 into account only by the constrained optimizers.
123 Example : ``{"Bounds":[[2.,5.],[1.e-2,10.],[-30.,None],[None,None]]}``
126 This key indicates the maximum number of iterations allowed for iterative
127 optimization. The default is 15000, which is very similar to no limit on
128 iterations. It is then recommended to adapt this parameter to the needs on
129 real problems. For some optimizers, the effective stopping step can be
130 slightly different of the limit due to algorithm internal control
133 Example : ``{"MaximumNumberOfSteps":100}``
135 CostDecrementTolerance
136 This key indicates a limit value, leading to stop successfully the
137 iterative optimization process when the cost function decreases less than
138 this tolerance at the last step. The default is 1.e-7, and it is
139 recommended to adapt it to the needs on real problems.
141 Example : ``{"CostDecrementTolerance":1.e-7}``
143 ProjectedGradientTolerance
144 This key indicates a limit value, leading to stop successfully the iterative
145 optimization process when all the components of the projected gradient are
146 under this limit. It is only used for constrained optimizers. The default is
147 -1, that is the internal default of each minimizer (generally 1.e-5), and it
148 is not recommended to change it.
150 Example : ``{"ProjectedGradientTolerance":-1}``
152 GradientNormTolerance
153 This key indicates a limit value, leading to stop successfully the
154 iterative optimization process when the norm of the gradient is under this
155 limit. It is only used for non-constrained optimizers. The default is
156 1.e-5 and it is not recommended to change it.
158 Example : ``{"GradientNormTolerance":1.e-5}``
160 StoreInternalVariables
161 This Boolean key allows to store default internal variables, mainly the
162 current state during iterative optimization process. Be careful, this can be
163 a numerically costly choice in certain calculation cases. The default is
166 Example : ``{"StoreInternalVariables":True}``
168 StoreSupplementaryCalculations
169 This list indicates the names of the supplementary variables that can be
170 available at the end of the algorithm. It involves potentially costly
171 calculations or memory consumptions. The default is a void list, none of
172 these variables being calculated and stored by default. The possible names
173 are in the following list: ["APosterioriCovariance", "BMA", "OMA", "OMB",
174 "Innovation", "SigmaObs2", "MahalanobisConsistency", "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 APosterioriCovariance
255 *List of matrices*. Each element is an *a posteriori* error covariance
256 matrix :math:`\mathbf{A}*` of the optimal state.
258 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
261 *List of vectors*. Each element is a vector of difference between the
262 background and the optimal state.
264 Example : ``bma = ADD.get("BMA")[-1]``
267 *List of vectors*. Each element is a usual state vector used during the
268 optimization algorithm procedure.
270 Example : ``Xs = ADD.get("CurrentState")[:]``
273 *List of vectors*. Each element is an innovation vector, which is in static
274 the difference between the optimal and the background, and in dynamic the
277 Example : ``d = ADD.get("Innovation")[-1]``
279 MahalanobisConsistency
280 *List of values*. Each element is a value of the Mahalanobis quality
283 Example : ``m = ADD.get("MahalanobisConsistency")[-1]``
286 *List of vectors*. Each element is a vector of difference between the
287 observation and the optimal state in the observation space.
289 Example : ``oma = ADD.get("OMA")[-1]``
292 *List of vectors*. Each element is a vector of difference between the
293 observation and the background state in the observation space.
295 Example : ``omb = ADD.get("OMB")[-1]``
298 *List of values*. Each element is a value of the quality indicator
299 :math:`(\sigma^o)^2` of the observation part.
301 Example : ``so2 = ADD.get("SigmaObs")[-1]``
304 *List of vectors*. Each element is a vector corresponding to the observed
305 state which realize the required quantile, in the same order than the
306 quantiles required by the user.
308 Example : ``sQuantiles = ADD.get("SimulationQuantiles")[:]``
313 References to other sections:
314 - :ref:`section_ref_algorithm_Blue`
315 - :ref:`section_ref_algorithm_ExtendedBlue`
316 - :ref:`section_ref_algorithm_LinearityTest`
318 Bibliographical references: