+.. _section_examples:
+
================================================================================
Examples on using the ADAO module
================================================================================
-This section presents some examples on using the ADAO module in SALOME.
+.. |eficas_new| image:: images/eficas_new.png
+ :align: middle
+ :scale: 50%
+.. |eficas_save| image:: images/eficas_save.png
+ :align: middle
+ :scale: 50%
+.. |eficas_yacs| image:: images/eficas_yacs.png
+ :align: middle
+ :scale: 50%
+
+This section presents some examples on using the ADAO module in SALOME. The
+first one shows how to build a simple data assimilation case defining
+explicitly all the required data through the GUI. The second one shows, on the
+same case, how to define data using external sources through scripts.
Building a simple estimation case with explicit data definition
---------------------------------------------------------------------------------
+---------------------------------------------------------------
This simple example is a demonstration one, and describes how to set a BLUE
estimation framework in order to get *weighted least square estimated state* of
We choose to operate in a 3-dimensional space. 3D is chosen in order to restrict
the size of numerical object to explicitly enter by the user, but the problem is
-not dependant of the dimension and can be set in dimension 1000... The observed
-state is of value 1 in each direction, so:
+not dependant of the dimension and can be set in dimension 1000... The
+observation :math:`\mathbf{y}^o` is of value 1 in each direction, so:
``Yo = [1 1 1]``
-The background state, wich represent some *a priori* knowledge or a
-regularization, is of value of 0 in each direction, which is:
+The background state :math:`\mathbf{x}^b`, which represent some *a priori*
+knowledge or a regularization, is of value of 0 in each direction, which is:
``Xb = [0 0 0]``
-Data assimilation requires information on errors covariances for observation and
-background variables. We choose here to have uncorrelated errors (that is,
-diagonal matrices) and to have the same variance of 1 for all variables (that
-is, identity matrices. We get:
+Data assimilation requires information on errors covariances :math:`\mathbf{R}`
+and :math:`\mathbf{B}` respectively for observation and background variables. We
+choose here to have uncorrelated errors (that is, diagonal matrices) and to have
+the same variance of 1 for all variables (that is, identity matrices). We get:
``B = R = [1 0 0 ; 0 1 0 ; 0 0 1]``
-Last, we need an observation operator to convert the background value in the
-space of observation value. Here, because the space dimensions are the same, we
-can choose the identity as the observation operator:
+Last, we need an observation operator :math:`\mathbf{H}` to convert the
+background value in the space of observation value. Here, because the space
+dimensions are the same, we can choose the identity as the observation
+operator:
``H = [1 0 0 ; 0 1 0 ; 0 0 1]``
-With such choices, the Best Linear Unbiased Estimator (BLUE) will be the
-average vector between ``Yo`` and ``Xb``, named the *analysis* and denoted by
-``Xa``:
+With such choices, the Best Linear Unbiased Estimator (BLUE) will be the average
+vector between :math:`\mathbf{y}^o` and :math:`\mathbf{x}^b`, named the
+*analysis* and denoted by :math:`\mathbf{x}^a`:
``Xa = [0.5 0.5 0.5]``
-As en extension of this example, one can change the variances for ``B`` or ``R``
-independantly, and the analysis will move to ``Yo`` or ``Xb`` in inverse
-proportion of the variances in ``B`` and ``R``. It is also equivalent to serach
-for the analysis thought a BLUE algorithm or a 3DVAR one.
+As en extension of this example, one can change the variances for
+:math:`\mathbf{B}` or :math:`\mathbf{R}` independently, and the analysis will
+move to :math:`\mathbf{y}^o` or to :math:`\mathbf{x}^b` in inverse proportion of
+the variances in :math:`\mathbf{B}` and :math:`\mathbf{R}`. It is also
+equivalent to search for the analysis thought a BLUE algorithm or a 3DVAR one.
Using the GUI to build the ADAO case
++++++++++++++++++++++++++++++++++++
First, you have to activate the ADAO module by choosing the appropriate module
button or menu of SALOME, and you will see:
- .. _adao_activate:
+ .. _adao_activate2:
.. image:: images/adao_activate.png
:align: center
+ :width: 100%
.. centered::
**Activating the module ADAO in SALOME**
-.. |eficas_new| image:: images/eficas_new.png
- :align: middle
-
Choose the "*New*" button in this window. You will directly get the EFICAS
interface for variables definition, along with the "*Object browser*". You can
then click on the "*New*" button |eficas_new| to create a new ADAO case, and you
To go further, we need now to generate the YACS scheme from the ADAO case
definition. In order to do that, right click on the name of the file case in the
-"*Object browser*" window, and choose the "*Export to YACS*" submenu as below:
+"*Object browser*" window, and choose the "*Export to YACS*" sub-menu (or the
+"*Export to YACS*" button |eficas_yacs|) as below:
.. _adao_exporttoyacs:
.. image:: images/adao_exporttoyacs.png
:align: center
:scale: 75%
.. centered::
- **"*Export to YACS*" submenu to generate the YACS scheme from the ADAO case**
+ **"Export to YACS" sub-menu to generate the YACS scheme from the ADAO case**
This command will generate the YACS scheme, activate YACS module in SALOME, and
-open the new scheme in the YACS module GUI [#]. After reordering the nodes by
-using the "*arrange local node*" submenu of the YACS graphical view of the
-scheme, you get the following representation of the generated ADAO scheme:
+open the new scheme in the GUI of the YACS module [#]_. After reordering the
+nodes by using the "*arrange local node*" sub-menu of the YACS graphical view of
+the scheme, you get the following representation of the generated ADAO scheme:
.. _yacs_generatedscheme:
.. image:: images/yacs_generatedscheme.png
After that point, all the modifications, executions and post-processing of the
data assimilation scheme will be done in YACS. In order to check the result in a
-simple way, we create here a new YACS node by using the "*inline script node*"
-submenu of the YACS graphical view, and we name it "*PostProcessing*".
+simple way, we create here a new YACS node by using the "*in-line script node*"
+sub-menu of the YACS graphical view, and we name it "*PostProcessing*".
This script will retrieve the data assimilation analysis from the
"*algoResults*" output port of the computation bloc (which gives access to a
SALOME Python Object), and will print it on the standard output.
-To obtain this, the inline script node need to have an input port of type
-"*pyobj*" named "*results*", that have to be linked graphically to the
-"*algoResults*" output port of the computation bloc. Then the code to fill in
-the script node is::
+To obtain this, the in-line script node need to have an input port of type
+"*pyobj*" named "*results*" for example, that have to be linked graphically to
+the "*algoResults*" output port of the computation bloc. Then the code to fill
+in the script node is::
Xa = results.ADD.get("Analysis").valueserie(-1)
print
The augmented YACS scheme can be saved (overwriting the generated scheme if the
-simple "*Save*" command or button is used, or with a new name). Then,
+simple "*Save*" command or button are used, or with a new name). Then,
classically in YACS, it have to be prepared for run, and then executed. After
completion, the printing on standard output is available in the "*YACS Container
-Log*", obtained throught the right click menu of the "*proc*" window in the YACS
+Log*", obtained through the right click menu of the "*proc*" window in the YACS
scheme as shown below:
.. _yacs_containerlog:
.. centered::
**YACS menu for Container Log, and dialog window showing the log**
-We verify that the result is correct by checking that the log contains the
-following line::
+We verify that the result is correct by checking that the log dialog window
+contains the following line::
Analysis = [0.5, 0.5, 0.5]
As a simple extension of this example, one can notice that the same problem
solved with a 3DVAR algorithm gives the same result. This algorithm can be
chosen at the ADAO case building step, before entering in YACS step. The
-ADAO 3DVAR case will look completly similar to the BLUE algoritmic case, as
+ADAO 3DVAR case will look completely similar to the BLUE algorithmic case, as
shown by the following figure:
- .. _adao_jdcexample01:
+ .. _adao_jdcexample02:
.. image:: images/adao_jdcexample02.png
:align: center
:width: 100%
.. centered::
- **Defining an ADAO 3DVAR case looks completly similar to a BLUE case**
+ **Defining an ADAO 3DVAR case looks completely similar to a BLUE case**
-There is only one keyword changing, with "*ThreeDVAR*" instead of "*Blue*".
+There is only one command changing, with "*3DVAR*" value instead of "*Blue*".
-Building a simple estimation case with external data definition by functions
---------------------------------------------------------------------------------
+Building a simple estimation case with external data definition by scripts
+--------------------------------------------------------------------------
It is useful to get parts or all of the data from external definition, using
Python script files to provide access to the data. As an example, we build here
First, we write the following script file, using conventional names for the
desired variables. Here, all the input variables are defined in the script, but
the user can choose to split the file in several ones, or to mix explicit data
-definition in the ADAO GUI and implicit data definition by external files. This
-script looks like::
+definition in the ADAO GUI and implicit data definition by external files. The
+present script looks like::
#-*-coding:iso-8859-1-*-
+ #
import numpy
#
# Definition of the Background as a vector
matrices, using list, string (as in Numpy or Octave), Numpy array type or Numpy
matrix type, and Numpy special functions. All of these syntaxes are valid.
-After saving this script somewhere in your path (named here
-"*test003_ADAO_example_scripts.py*" for the example), we use the GUI to build
-the ADAO case. The procedure to fill in the case is similar except that, instead
-of selecting the "*String*" option for the "*FROM*" keyword, we select the
-"*Script*" one. This leads to a "*SCRIPT_DATA/SCRIPT_FILE*" entry in the tree,
-allowing to choose a file as:
+After saving this script somewhere in your path (named here "*script.py*" for
+the example), we use the GUI to build the ADAO case. The procedure to fill in
+the case is similar except that, instead of selecting the "*String*" option for
+the "*FROM*" keyword, we select the "*Script*" one. This leads to a
+"*SCRIPT_DATA/SCRIPT_FILE*" entry in the tree, allowing to choose a file as:
.. _adao_scriptentry01:
.. image:: images/adao_scriptentry01.png
Other steps and results are exactly the same as in the `Building a simple
estimation case with explicit data definition`_ previous example.
-In fact, this script methodology allows to retrieve data from inline or previous
+In fact, this script methodology allows to retrieve data from in-line or previous
calculations, from static files, from database or from stream, all of them
outside of SALOME. It allows also to modify easily some input data, for example
-debug purpose or for repetitive process. **But be careful, it is not a "safe"
-process, in the sense that erroneous data, or errors in calculations, can be
-directly injected into the YACS scheme execution.**
+for debug purpose or for repetitive execution process, and it is the most
+versatile method in order to parametrize the input data. **But be careful,
+script methodology is not a "safe" procedure, in the sense that erroneous
+data, or errors in calculations, can be directly injected into the YACS scheme
+execution.**
Adding parameters to control the data assimilation algorithm
---------------------------------------------------------------------------------
+------------------------------------------------------------
One can add some optional parameters to control the data assimilation algorithm
calculation. This is done by using the "*AlgorithmParameters*" keyword in the
-definition of the ADAO case, which is an option of the algorithm. This keyword
-requires a Python dictionary containing some key/value pairs.
+definition of the ADAO case, which is an keyword of the ASSIMILATION_STUDY. This
+keyword requires a Python dictionary, containing some key/value pairs.
For example, with a 3DVAR algorithm, the possible keys are "*Minimizer*",
-"*MaximumNumberOfSteps*", "*Bounds*". Bounds can be given by a list of list of
-pairs of lower/upper bounds for each variable, with possibly ``None`` every time
-there is no bound. If no bounds at all are required on the control variables,
-then one can choose the "BFGS" or "CG" minimisation algorithm for the 3DVAR
-algorithm.
-
-This dictionary has to be defined in an external Python script file, using the
-mandatory variable name "*AlgorithmParameters*" for the dictionary. All the keys
-inside the dictionary are optional, they all have default values, and can exist
-without being used. For example::
+"*MaximumNumberOfSteps*", "ProjectedGradientTolerance", "GradientNormTolerance"
+and "*Bounds*":
+
+#. The "*Minimizer*" key allows to choose the optimisation minimizer. The
+ default choice is "LBFGSB", and the possible ones are "LBFGSB" (nonlinear
+ constrained minimizer, see [Byrd95] and [Zhu97]), "TNC" (nonlinear
+ constrained minimizer), "CG" (nonlinear unconstrained minimizer), "BFGS"
+ (nonlinear unconstrained minimizer), "NCG" (Newton CG minimizer).
+#. The "*MaximumNumberOfSteps*" key indicates the maximum number of iterations
+ allowed for iterative optimisation. The default is 15000, which very
+ similar of no limit on iterations. It is then recommended to adapt this
+ parameter to the needs on real problems.
+#. The "ProjectedGradientTolerance" key indicates a limit value, leading to
+ stop successfully the iterative optimisation process when all the components
+ of the projected gradient are under this limit.
+#. The "GradientNormTolerance" key indicates a limit value, leading to stop
+ successfully the iterative optimisation process when the norm of the
+ gradient is under this limit.
+#. The "*Bounds*" key allows to define upper and lower bounds for every
+ control variable being optimized. Bounds can be given by a list of list of
+ pairs of lower/upper bounds for each variable, with possibly ``None`` every
+ time there is no bound. The bounds can always be specified, but they are
+ taken into account only by the constrained minimizers.
+
+If no bounds at all are required on the control variables, then one can choose
+the "BFGS" or "CG" minimisation algorithm for the 3DVAR algorithm. For
+constrained optimisation, the minimizer "LBFGSB" is often more robust, but the
+"TNC" is always more performant.
+
+This dictionary has to be defined, for example, in an external Python script
+file, using the mandatory variable name "*AlgorithmParameters*" for the
+dictionary. All the keys inside the dictionary are optional, they all have
+default values, and can exist without being used. For example::
#-*-coding:iso-8859-1-*-
#
**Adding parameters to control the algorithm**
Other steps and results are exactly the same as in the `Building a simple
-estimation case with explicit data definition`_ previous example.
+estimation case with explicit data definition`_ previous example. The dictionary
+can also be directly given in the input field associated with the keyword.
+
+Building a complex case with external data definition by scripts
+----------------------------------------------------------------
+
+This more complex and complete example has to been considered as a framework for
+user inputs, that need to be tailored for each real application. Nevertheless,
+the file skeletons are sufficiently general to have been used for various
+applications in neutronic, fluid mechanics... Here, we will not focus on the
+results, but more on the user control of inputs and outputs in an ADAO case. As
+previously, all the numerical values of this example are arbitrary.
+
+The objective is to set up the input and output definitions of a physical case
+by external python scripts, using a general non-linear operator, adding control
+on parameters and so on... The complete framework scripts can be found in the
+ADAO examples standard directory.
+
+Experimental set up
++++++++++++++++++++
+
+We continue to operate in a 3-dimensional space, in order to restrict
+the size of numerical object shown in the scripts, but the problem is
+not dependant of the dimension.
+
+We choose a twin experiment context, using a known true state
+:math:`\mathbf{x}^t` of arbitrary values:
+
+ ``Xt = [1 2 3]``
+
+The background state :math:`\mathbf{x}^b`, which represent some *a priori*
+knowledge of the true state, is build as a normal random perturbation of 20% the
+true state :math:`\mathbf{x}^t` for each component, which is:
+
+ ``Xb = Xt + normal(0,20%*Xt)``
+
+To describe the background error covariances matrix :math:`\mathbf{B}`, we make
+as previously the hypothesis of uncorrelated errors (that is, a diagonal matrix,
+of size 3x3 because :math:`\mathbf{x}^b` is of lenght 3) and to have the same
+variance of 0.1 for all variables. We get:
+
+ ``B = 0.1 * diagonal( lenght(Xb) )``
+
+We suppose that there exist an observation operator :math:`\mathbf{H}`, which
+can be non linear. In real calibration procedure or inverse problems, the
+physical simulation codes are embedded in the observation operator. We need also
+to know its gradient with respect to each calibrated variable, which is a rarely
+known information with industrial codes. But we will see later how to obtain an
+approximated gradient in this case.
+
+Being in twin experiments, the observation :math:`\mathbf{y}^o` and its error
+covariances matrix :math:`\mathbf{R}` are generated by using the true state
+:math:`\mathbf{x}^t` and the observation operator :math:`\mathbf{H}`:
+
+ ``Yo = H( Xt )``
+
+and, with an arbitrary standard deviation of 1% on each error component:
+
+ ``R = 0.0001 * diagonal( lenght(Yo) )``
+
+All the required data assimilation informations are then defined.
+
+Skeletons of the scripts describing the setup
++++++++++++++++++++++++++++++++++++++++++++++
+
+We give here the essential parts of each script used afterwards to build the ADAO
+case. Remember that using these scripts in real Python files requires to
+correctly define the path to imported modules or codes (even if the module is in
+the same directory that the importing Python file ; we indicate the path
+adjustment using the mention ``"# INSERT PHYSICAL SCRIPT PATH"``), the encoding
+if necessary, etc. The indicated file names for the following scripts are
+arbitrary. Examples of complete file scripts are available in the ADAO examples
+standard directory.
+
+We first define the true state :math:`\mathbf{x}^t` and some convenient matrix
+building function, in a Python script file named
+``Physical_data_and_covariance_matrices.py``::
+
+ #-*-coding:iso-8859-1-*-
+ #
+ import numpy
+ #
+ def True_state():
+ """
+ Arbitrary values and names, as a tuple of two series of same length
+ """
+ return (numpy.array([1, 2, 3]), ['Para1', 'Para2', 'Para3'])
+ #
+ def Simple_Matrix( size, diagonal=None ):
+ """
+ Diagonal matrix, with either 1 or a given vector on the diagonal
+ """
+ if diagonal is not None:
+ S = numpy.diag( diagonal )
+ else:
+ S = numpy.matrix(numpy.identity(int(size)))
+ return S
+
+We can then define the background state :math:`\mathbf{x}^b` as a random
+perturbation of the true state, adding at the end of the script the definition
+of a *required ADAO variable* in order to export the defined value. It is done
+in a Python script file named ``Script_Background_xb.py``::
+
+ #-*-coding:iso-8859-1-*-
+ #
+ import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
+ from Physical_data_and_covariance_matrices import True_state
+ import numpy
+ #
+ xt, names = True_state()
+ #
+ Standard_deviation = 0.2*xt # 20% for each variable
+ #
+ xb = xt + abs(numpy.random.normal(0.,Standard_deviation,size=(len(xt),)))
+ #
+ # Creating the required ADAO variable
+ # -----------------------------------
+ Background = list(xb)
+
+In the same way, we define the background error covariance matrix
+:math:`\mathbf{B}` as a diagonal matrix of the same diagonal length as the
+background of the true state, using the convenient function already defined. It
+is done in a Python script file named ``Script_BackgroundError_B.py``::
+
+ #-*-coding:iso-8859-1-*-
+ #
+ import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
+ from Physical_data_and_covariance_matrices import True_state, Simple_Matrix
+ #
+ xt, names = True_state()
+ #
+ B = 0.1 * Simple_Matrix( size = len(xt) )
+ #
+ # Creating the required ADAO variable
+ # -----------------------------------
+ BackgroundError = B
+
+To continue, we need the observation operator :math:`\mathbf{H}` as a function
+of the state. It is here defined in an external file named
+``"Physical_simulation_functions.py"``, which should contain functions
+conveniently named here ``"FunctionH"`` and ``"AdjointH"``. These functions are
+user ones, representing as programming functions the :math:`\mathbf{H}` operator
+and its adjoint. We suppose these functions are given by the user (a simple
+skeleton is given in the Python script file ``Physical_simulation_functions.py``
+of the ADAO examples standard directory, not reproduced here).
+
+To operates in ADAO, it is required to define different types of operators: the
+(potentially non-linear) standard observation operator, named ``"Direct"``, its
+linearised approximation, named ``"Tangent"``, and the adjoint operator named
+``"Adjoint"``. The Python script have to retrieve an input parameter, found
+under the key "value", in a variable named ``"specificParameters"`` of the
+SALOME input data and parameters ``"computation"`` dictionary variable. If the
+operator is already linear, the ``"Direct"`` and ``"Tangent"`` functions are the
+same, as it is supposed here. The following example Python script file named
+``Script_ObservationOperator_H.py``, illustrates the case::
+
+ #-*-coding:iso-8859-1-*-
+ #
+ import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
+ import Physical_simulation_functions
+ import numpy, logging
+ #
+ # -----------------------------------------------------------------------
+ # SALOME input data and parameters: all information are the required input
+ # variable "computation", containing for example:
+ # {'inputValues': [[[[0.0, 0.0, 0.0]]]],
+ # 'inputVarList': ['adao_default'],
+ # 'outputVarList': ['adao_default'],
+ # 'specificParameters': [{'name': 'method', 'value': 'Direct'}]}
+ # -----------------------------------------------------------------------
+ #
+ # Recovering the type of computation: "Direct", "Tangent" or "Adjoint"
+ # --------------------------------------------------------------------
+ method = ""
+ for param in computation["specificParameters"]:
+ if param["name"] == "method":
+ method = param["value"]
+ logging.info("ComputationFunctionNode: Found method is \'%s\'"%method)
+ #
+ # Loading the H operator functions from external definitions
+ # ----------------------------------------------------------
+ logging.info("ComputationFunctionNode: Loading operator functions")
+ FunctionH = Physical_simulation_functions.FunctionH
+ AdjointH = Physical_simulation_functions.AdjointH
+ #
+ # Executing the possible computations
+ # -----------------------------------
+ if method == "Direct":
+ logging.info("ComputationFunctionNode: Direct computation")
+ Xcurrent = computation["inputValues"][0][0][0]
+ data = FunctionH(numpy.matrix( Xcurrent ).T)
+ #
+ if method == "Tangent":
+ logging.info("ComputationFunctionNode: Tangent computation")
+ Xcurrent = computation["inputValues"][0][0][0]
+ data = FunctionH(numpy.matrix( Xcurrent ).T)
+ #
+ if method == "Adjoint":
+ logging.info("ComputationFunctionNode: Adjoint computation")
+ Xcurrent = computation["inputValues"][0][0][0]
+ Ycurrent = computation["inputValues"][0][0][1]
+ data = AdjointH((numpy.matrix( Xcurrent ).T, numpy.matrix( Ycurrent ).T))
+ #
+ # Formatting the output
+ # ---------------------
+ logging.info("ComputationFunctionNode: Formatting the output")
+ it = data.flat
+ outputValues = [[[[]]]]
+ for val in it:
+ outputValues[0][0][0].append(val)
+ #
+ # Creating the required ADAO variable
+ # -----------------------------------
+ result = {}
+ result["outputValues"] = outputValues
+ result["specificOutputInfos"] = []
+ result["returnCode"] = 0
+ result["errorMessage"] = ""
+
+As output, this script has to define a nested list variable, as shown above with
+the ``"outputValues"`` variable, where the nested levels describe the different
+variables included in the state, then the different possible states at the same
+time, then the different time steps. In this case, because there is only one
+time step and one state, and all the variables are stored together, we only set
+the most inner level of the lists.
+
+In this twin experiments framework, the observation :math:`\mathbf{y}^o` and its
+error covariances matrix :math:`\mathbf{R}` can be generated. It is done in two
+Python script files, the first one being named ``Script_Observation_yo.py``::
+
+ #-*-coding:iso-8859-1-*-
+ #
+ import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
+ from Physical_data_and_covariance_matrices import True_state
+ from Physical_simulation_functions import FunctionH
+ #
+ xt, noms = True_state()
+ #
+ yo = FunctionH( xt )
+ #
+ # Creating the required ADAO variable
+ # -----------------------------------
+ Observation = list(yo)
+
+and the second one named ``Script_ObservationError_R.py``::
+
+ #-*-coding:iso-8859-1-*-
+ #
+ import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
+ from Physical_data_and_covariance_matrices import True_state, Simple_Matrix
+ from Physical_simulation_functions import FunctionH
+ #
+ xt, names = True_state()
+ #
+ yo = FunctionH( xt )
+ #
+ R = 0.0001 * Simple_Matrix( size = len(yo) )
+ #
+ # Creating the required ADAO variable
+ # -----------------------------------
+ ObservationError = R
+
+As in previous examples, it can be useful to define some parameters for the data
+assimilation algorithm. For example, if we use the standard 3DVAR algorithm, the
+following parameters can be defined in a Python script file named
+``Script_AlgorithmParameters.py``::
-.. [#] For more information on YACS, see the the *YACS User Guide* available in the main "*Help*" menu of SALOME GUI.
+ #-*-coding:iso-8859-1-*-
+ #
+ # Creating the required ADAO variable
+ # -----------------------------------
+ AlgorithmParameters = {
+ "Minimizer" : "TNC", # Possible : "LBFGSB", "TNC", "CG", "BFGS"
+ "MaximumNumberOfSteps" : 15, # Number of global iterative steps
+ "Bounds" : [
+ [ None, None ], # Bound on the first parameter
+ [ 0., 4. ], # Bound on the second parameter
+ [ 0., None ], # Bound on the third parameter
+ ],
+ }
+
+Finally, it is common to post-process the results, retrieving them after the
+data assimilation phase in order to analyse, print or show them. It requires to
+use a intermediary Python script file in order to extract these results. The
+following example Python script file named ``Script_UserPostAnalysis.py``,
+illustrates the fact::
+
+ #-*-coding:iso-8859-1-*-
+ #
+ import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
+ from Physical_data_and_covariance_matrices import True_state
+ import numpy
+ #
+ xt, names = True_state()
+ xa = ADD.get("Analysis").valueserie(-1)
+ x_series = ADD.get("CurrentState").valueserie()
+ J = ADD.get("CostFunctionJ").valueserie()
+ #
+ # Verifying the results by printing
+ # ---------------------------------
+ print
+ print "xt = %s"%xt
+ print "xa = %s"%numpy.array(xa)
+ print
+ for i in range( len(x_series) ):
+ print "Step %2i : J = %.5e et X = %s"%(i, J[i], x_series[i])
+ print
+At the end, we get a description of the whole case setup through a set of files
+listed here:
+
+#. ``Physical_data_and_covariance_matrices.py``
+#. ``Physical_simulation_functions.py``
+#. ``Script_AlgorithmParameters.py``
+#. ``Script_BackgroundError_B.py``
+#. ``Script_Background_xb.py``
+#. ``Script_ObservationError_R.py``
+#. ``Script_ObservationOperator_H.py``
+#. ``Script_Observation_yo.py``
+#. ``Script_UserPostAnalysis.py``
+
+We insist here that all these scripts are written by the user and can not be
+automatically tested. So the user is required to verify the scripts (and in
+particular their input/output) in order to limit the difficulty of debug. We
+recall: **script methodology is not a "safe" procedure, in the sense that
+erroneous data, or errors in calculations, can be directly injected into the
+YACS scheme execution.**
+
+Building the case with external data definition by scripts
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+All these scripts can then be used to define the ADAO case with external data
+definition by Python script files. It is entirely similar to the method
+described in the `Building a simple estimation case with external data
+definition by scripts`_ previous section. For each variable to be defined, we
+select the "*Script*" option of the "*FROM*" keyword, which leads to a
+"*SCRIPT_DATA/SCRIPT_FILE*" entry in the tree.
+
+The other steps to build the ADAO case are exactly the same as in the `Building
+a simple estimation case with explicit data definition`_ previous section.
+
+Using the simple linear operator :math:`\mathbf{H}` from the Python script file
+``Physical_simulation_functions.py`` in the ADAO examples standard directory,
+the results will look like::
+
+ xt = [1 2 3]
+ xa = [ 1.000014 2.000458 3.000390]
+
+ Step 0 : J = 1.81750e+03 et X = [1.014011, 2.459175, 3.390462]
+ Step 1 : J = 1.81750e+03 et X = [1.014011, 2.459175, 3.390462]
+ Step 2 : J = 1.79734e+01 et X = [1.010771, 2.040342, 2.961378]
+ Step 3 : J = 1.79734e+01 et X = [1.010771, 2.040342, 2.961378]
+ Step 4 : J = 1.81909e+00 et X = [1.000826, 2.000352, 3.000487]
+ Step 5 : J = 1.81909e+00 et X = [1.000826, 2.000352, 3.000487]
+ Step 6 : J = 1.81641e+00 et X = [1.000247, 2.000651, 3.000156]
+ Step 7 : J = 1.81641e+00 et X = [1.000247, 2.000651, 3.000156]
+ Step 8 : J = 1.81569e+00 et X = [1.000015, 2.000432, 3.000364]
+ Step 9 : J = 1.81569e+00 et X = [1.000015, 2.000432, 3.000364]
+ Step 10 : J = 1.81568e+00 et X = [1.000013, 2.000458, 3.000390]
+ ...
+
+The state at the first step is the randomly generated background state
+:math:`\mathbf{x}^b`. After completion, these printing on standard output is
+available in the "*YACS Container Log*", obtained through the right click menu
+of the "*proc*" window in the YACS scheme.
+
+.. [#] For more information on YACS, see the the *YACS module User's Guide* available in the main "*Help*" menu of SALOME GUI.
