3 ================================================================================
4 Examples on using the ADAO module
5 ================================================================================
7 .. |eficas_new| image:: images/eficas_new.png
10 .. |eficas_save| image:: images/eficas_save.png
13 .. |eficas_yacs| image:: images/eficas_yacs.png
17 This section presents some examples on using the ADAO module in SALOME. The
18 first one shows how to build a simple data assimilation case defining
19 explicitly all the required data through the GUI. The second one shows, on the
20 same case, how to define data using external sources through scripts.
22 Building a simple estimation case with explicit data definition
23 ---------------------------------------------------------------
25 This simple example is a demonstration one, and describes how to set a BLUE
26 estimation framework in order to get *weighted least square estimated state* of
27 a system from an observation of the state and from an *a priori* knowledge (or
28 background) of this state. In other words, we look for the weighted middle
29 between the observation and the background vectors. All the numerical values of
30 this example are arbitrary.
35 We choose to operate in a 3-dimensional space. 3D is chosen in order to restrict
36 the size of numerical object to explicitly enter by the user, but the problem is
37 not dependant of the dimension and can be set in dimension 1000... The
38 observation :math:`\mathbf{y}^o` is of value 1 in each direction, so:
42 The background state :math:`\mathbf{x}^b`, which represent some *a priori*
43 knowledge or a regularization, is of value of 0 in each direction, which is:
47 Data assimilation requires information on errors covariances :math:`\mathbf{R}`
48 and :math:`\mathbf{B}` respectively for observation and background variables. We
49 choose here to have uncorrelated errors (that is, diagonal matrices) and to have
50 the same variance of 1 for all variables (that is, identity matrices). We get:
52 ``B = R = [1 0 0 ; 0 1 0 ; 0 0 1]``
54 Last, we need an observation operator :math:`\mathbf{H}` to convert the
55 background value in the space of observation value. Here, because the space
56 dimensions are the same, we can choose the identity as the observation
59 ``H = [1 0 0 ; 0 1 0 ; 0 0 1]``
61 With such choices, the Best Linear Unbiased Estimator (BLUE) will be the average
62 vector between :math:`\mathbf{y}^o` and :math:`\mathbf{x}^b`, named the
63 *analysis* and denoted by :math:`\mathbf{x}^a`:
65 ``Xa = [0.5 0.5 0.5]``
67 As en extension of this example, one can change the variances for
68 :math:`\mathbf{B}` or :math:`\mathbf{R}` independently, and the analysis will
69 move to :math:`\mathbf{y}^o` or to :math:`\mathbf{x}^b` in inverse proportion of
70 the variances in :math:`\mathbf{B}` and :math:`\mathbf{R}`. It is also
71 equivalent to search for the analysis thought a BLUE algorithm or a 3DVAR one.
73 Using the GUI to build the ADAO case
74 ++++++++++++++++++++++++++++++++++++
76 First, you have to activate the ADAO module by choosing the appropriate module
77 button or menu of SALOME, and you will see:
80 .. image:: images/adao_activate.png
84 **Activating the module ADAO in SALOME**
86 Choose the "*New*" button in this window. You will directly get the EFICAS
87 interface for variables definition, along with the "*Object browser*". You can
88 then click on the "*New*" button |eficas_new| to create a new ADAO case, and you
92 .. image:: images/adao_viewer.png
96 **The EFICAS viewer for cases definition in module ADAO**
98 Then fill in the variables to build the ADAO case by using the experimental set
99 up described above. All the technical information given above will be directly
100 inserted in the ADAO case definition, by using the *String* type for all the
101 variables. When the case definition is ready, save it to a "*JDC (\*.comm)*"
102 native file somewhere in your path. Remember that other files will be also
103 created near this first one, so it is better to make a specific directory for
104 your case, and to save the file inside. The name of the file will appear in the
105 "*Object browser*" window, under the "*ADAO*" menu. The final case definition
108 .. _adao_jdcexample01:
109 .. image:: images/adao_jdcexample01.png
113 **Definition of the experimental set up chosen for the ADAO case**
115 To go further, we need now to generate the YACS scheme from the ADAO case
116 definition. In order to do that, right click on the name of the file case in the
117 "*Object browser*" window, and choose the "*Export to YACS*" sub-menu (or the
118 "*Export to YACS*" button |eficas_yacs|) as below:
120 .. _adao_exporttoyacs:
121 .. image:: images/adao_exporttoyacs.png
125 **"Export to YACS" sub-menu to generate the YACS scheme from the ADAO case**
127 This command will generate the YACS scheme, activate YACS module in SALOME, and
128 open the new scheme in the GUI of the YACS module [#]_. After reordering the
129 nodes by using the "*arrange local node*" sub-menu of the YACS graphical view of
130 the scheme, you get the following representation of the generated ADAO scheme:
132 .. _yacs_generatedscheme:
133 .. image:: images/yacs_generatedscheme.png
137 **YACS generated scheme from the ADAO case**
139 After that point, all the modifications, executions and post-processing of the
140 data assimilation scheme will be done in YACS. In order to check the result in a
141 simple way, we create here a new YACS node by using the "*in-line script node*"
142 sub-menu of the YACS graphical view, and we name it "*PostProcessing*".
144 This script will retrieve the data assimilation analysis from the
145 "*algoResults*" output port of the computation bloc (which gives access to a
146 SALOME Python Object), and will print it on the standard output.
148 To obtain this, the in-line script node need to have an input port of type
149 "*pyobj*" named "*results*" for example, that have to be linked graphically to
150 the "*algoResults*" output port of the computation bloc. Then the code to fill
151 in the script node is::
153 Xa = results.ADD.get("Analysis").valueserie(-1)
156 print "Analysis =",Xa
159 The augmented YACS scheme can be saved (overwriting the generated scheme if the
160 simple "*Save*" command or button are used, or with a new name). Then,
161 classically in YACS, it have to be prepared for run, and then executed. After
162 completion, the printing on standard output is available in the "*YACS Container
163 Log*", obtained through the right click menu of the "*proc*" window in the YACS
164 scheme as shown below:
166 .. _yacs_containerlog:
167 .. image:: images/yacs_containerlog.png
171 **YACS menu for Container Log, and dialog window showing the log**
173 We verify that the result is correct by checking that the log dialog window
174 contains the following line::
176 Analysis = [0.5, 0.5, 0.5]
178 as shown in the image above.
180 As a simple extension of this example, one can notice that the same problem
181 solved with a 3DVAR algorithm gives the same result. This algorithm can be
182 chosen at the ADAO case building step, before entering in YACS step. The
183 ADAO 3DVAR case will look completely similar to the BLUE algorithmic case, as
184 shown by the following figure:
186 .. _adao_jdcexample02:
187 .. image:: images/adao_jdcexample02.png
191 **Defining an ADAO 3DVAR case looks completely similar to a BLUE case**
193 There is only one command changing, with "*3DVAR*" value instead of "*Blue*".
195 Building a simple estimation case with external data definition by scripts
196 --------------------------------------------------------------------------
198 It is useful to get parts or all of the data from external definition, using
199 Python script files to provide access to the data. As an example, we build here
200 an ADAO case representing the same experimental set up as in the above example
201 `Building a simple estimation case with explicit data definition`_, but using
202 data form a single one external Python script file.
204 First, we write the following script file, using conventional names for the
205 desired variables. Here, all the input variables are defined in the script, but
206 the user can choose to split the file in several ones, or to mix explicit data
207 definition in the ADAO GUI and implicit data definition by external files. The
208 present script looks like::
210 #-*-coding:iso-8859-1-*-
214 # Definition of the Background as a vector
215 # ----------------------------------------
216 Background = [0, 0, 0]
218 # Definition of the Observation as a vector
219 # -----------------------------------------
220 Observation = "1 1 1"
222 # Definition of the Background Error covariance as a matrix
223 # ---------------------------------------------------------
224 BackgroundError = numpy.array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]])
226 # Definition of the Observation Error covariance as a matrix
227 # ----------------------------------------------------------
228 ObservationError = numpy.matrix("1 0 0 ; 0 1 0 ; 0 0 1")
230 # Definition of the Observation Operator as a matrix
231 # --------------------------------------------------
232 ObservationOperator = numpy.identity(3)
234 The names of the Python variables above are mandatory, in order to define the
235 right variables, but the Python script can be bigger and define classes,
236 functions, etc. with other names. It shows different ways to define arrays and
237 matrices, using list, string (as in Numpy or Octave), Numpy array type or Numpy
238 matrix type, and Numpy special functions. All of these syntaxes are valid.
240 After saving this script somewhere in your path (named here "*script.py*" for
241 the example), we use the GUI to build the ADAO case. The procedure to fill in
242 the case is similar except that, instead of selecting the "*String*" option for
243 the "*FROM*" keyword, we select the "*Script*" one. This leads to a
244 "*SCRIPT_DATA/SCRIPT_FILE*" entry in the tree, allowing to choose a file as:
246 .. _adao_scriptentry01:
247 .. image:: images/adao_scriptentry01.png
251 **Defining an input value using an external script file**
253 Other steps and results are exactly the same as in the `Building a simple
254 estimation case with explicit data definition`_ previous example.
256 In fact, this script methodology allows to retrieve data from in-line or previous
257 calculations, from static files, from database or from stream, all of them
258 outside of SALOME. It allows also to modify easily some input data, for example
259 for debug purpose or for repetitive execution process, and it is the most
260 versatile method in order to parametrize the input data. **But be careful,
261 script methodology is not a "safe" procedure, in the sense that erroneous
262 data, or errors in calculations, can be directly injected into the YACS scheme
265 Adding parameters to control the data assimilation algorithm
266 ------------------------------------------------------------
268 One can add some optional parameters to control the data assimilation algorithm
269 calculation. This is done by using the "*AlgorithmParameters*" keyword in the
270 definition of the ADAO case, which is an keyword of the ASSIMILATION_STUDY. This
271 keyword requires a Python dictionary, containing some key/value pairs.
273 For example, with a 3DVAR algorithm, the possible keys are "*Minimizer*",
274 "*MaximumNumberOfSteps*", and "*Bounds*":
276 #. The "*Minimizer*" key allows to choose the optimisation minimizer, the
277 default choice being "LBFGSB", and the possible ones "LBFGSB" (nonlinear
278 constrained minimizer, see [Byrd95] and [Zhu97]), "TNC" (nonlinear
279 constrained minimizer), "CG" (nonlinear unconstrained minimizer), "BFGS"
280 (non-linear unconstrained minimizer).
281 #. The "*MaximumNumberOfSteps*" key indicates the maximum number of iterations
282 allowed for iterative optimisation. The default is 15000, which very
283 similar of no limit on iterations. It is then recommended to adapt this
284 parameter to the needs on real problems.
285 #. The "*Bounds*" key allows to define upper and lower bounds for every
286 control variable being optimized. Bounds can be given by a list of list of
287 pairs of lower/upper bounds for each variable, with possibly ``None`` every
288 time there is no bound.
290 If no bounds at all are required on the control variables, then one can choose
291 the "BFGS" or "CG" minimisation algorithm for the 3DVAR algorithm.
293 This dictionary has to be defined, for example, in an external Python script
294 file, using the mandatory variable name "*AlgorithmParameters*" for the
295 dictionary. All the keys inside the dictionary are optional, they all have
296 default values, and can exist without being used. For example::
298 #-*-coding:iso-8859-1-*-
300 AlgorithmParameters = {
301 "Minimizer" : "CG", # Possible choice : "LBFGSB", "TNC", "CG", "BFGS"
302 "MaximumNumberOfSteps" : 10,
305 Then the script can be added to the ADAO case, in a file entry describing the
306 "*AlgorithmParameters*" keyword, as follows:
308 .. _adao_scriptentry02:
309 .. image:: images/adao_scriptentry02.png
313 **Adding parameters to control the algorithm**
315 Other steps and results are exactly the same as in the `Building a simple
316 estimation case with explicit data definition`_ previous example. The dictionary
317 can also be directly given in the input field associated with the keyword.
319 Building a complex case with external data definition by scripts
320 ----------------------------------------------------------------
322 This more complex and complete example has to been considered as a framework for
323 user inputs, that need to be tailored for each real application. Nevertheless,
324 the file skeletons are sufficiently general to have been used for various
325 applications in neutronic, fluid mechanics... Here, we will not focus on the
326 results, but more on the user control of inputs and outputs in an ADAO case. As
327 previously, all the numerical values of this example are arbitrary.
329 The objective is to set up the input and output definitions of a physical case
330 by external python scripts, using a general non-linear operator, adding control
331 on parameters and so on... The complete framework scripts can be found in the
332 ADAO examples standard directory.
337 We continue to operate in a 3-dimensional space, in order to restrict
338 the size of numerical object shown in the scripts, but the problem is
339 not dependant of the dimension.
341 We choose a twin experiment context, using a known true state
342 :math:`\mathbf{x}^t` of arbitrary values:
346 The background state :math:`\mathbf{x}^b`, which represent some *a priori*
347 knowledge of the true state, is build as a normal random perturbation of 20% the
348 true state :math:`\mathbf{x}^t` for each component, which is:
350 ``Xb = Xt + normal(0,20%*Xt)``
352 To describe the background error covariances matrix :math:`\mathbf{B}`, we make
353 as previously the hypothesis of uncorrelated errors (that is, a diagonal matrix,
354 of size 3x3 because :math:`\mathbf{x}^b` is of lenght 3) and to have the same
355 variance of 0.1 for all variables. We get:
357 ``B = 0.1 * diagonal( lenght(Xb) )``
359 We suppose that there exist an observation operator :math:`\mathbf{H}`, which
360 can be non linear. In real calibration procedure or inverse problems, the
361 physical simulation codes are embedded in the observation operator. We need also
362 to know its gradient with respect to each calibrated variable, which is a rarely
363 known information with industrial codes. But we will see later how to obtain an
364 approximated gradient in this case.
366 Being in twin experiments, the observation :math:`\mathbf{y}^o` and its error
367 covariances matrix :math:`\mathbf{R}` are generated by using the true state
368 :math:`\mathbf{x}^t` and the observation operator :math:`\mathbf{H}`:
372 and, with an arbitrary standard deviation of 1% on each error component:
374 ``R = 0.0001 * diagonal( lenght(Yo) )``
376 All the required data assimilation informations are then defined.
378 Skeletons of the scripts describing the setup
379 +++++++++++++++++++++++++++++++++++++++++++++
381 We give here the essential parts of each script used afterwards to build the ADAO
382 case. Remember that using these scripts in real Python files requires to
383 correctly define the path to imported modules or codes (even if the module is in
384 the same directory that the importing Python file ; we indicate the path
385 adjustment using the mention ``"# INSERT PHYSICAL SCRIPT PATH"``), the encoding
386 if necessary, etc. The indicated file names for the following scripts are
387 arbitrary. Examples of complete file scripts are available in the ADAO examples
390 We first define the true state :math:`\mathbf{x}^t` and some convenient matrix
391 building function, in a Python script file named
392 ``Physical_data_and_covariance_matrices.py``::
394 #-*-coding:iso-8859-1-*-
400 Arbitrary values and names, as a tuple of two series of same length
402 return (numpy.array([1, 2, 3]), ['Para1', 'Para2', 'Para3'])
404 def Simple_Matrix( size, diagonal=None ):
406 Diagonal matrix, with either 1 or a given vector on the diagonal
408 if diagonal is not None:
409 S = numpy.diag( diagonal )
411 S = numpy.matrix(numpy.identity(int(size)))
414 We can then define the background state :math:`\mathbf{x}^b` as a random
415 perturbation of the true state, adding at the end of the script the definition
416 of a *required ADAO variable* in order to export the defined value. It is done
417 in a Python script file named ``Script_Background_xb.py``::
419 #-*-coding:iso-8859-1-*-
421 import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
422 from Physical_data_and_covariance_matrices import True_state
425 xt, names = True_state()
427 Standard_deviation = 0.2*xt # 20% for each variable
429 xb = xt + abs(numpy.random.normal(0.,Standard_deviation,size=(len(xt),)))
431 # Creating the required ADAO variable
432 # -----------------------------------
433 Background = list(xb)
435 In the same way, we define the background error covariance matrix
436 :math:`\mathbf{B}` as a diagonal matrix of the same diagonal length as the
437 background of the true state, using the convenient function already defined. It
438 is done in a Python script file named ``Script_BackgroundError_B.py``::
440 #-*-coding:iso-8859-1-*-
442 import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
443 from Physical_data_and_covariance_matrices import True_state, Simple_Matrix
445 xt, names = True_state()
447 B = 0.1 * Simple_Matrix( size = len(xt) )
449 # Creating the required ADAO variable
450 # -----------------------------------
453 To continue, we need the observation operator :math:`\mathbf{H}` as a function
454 of the state. It is here defined in an external file named
455 ``"Physical_simulation_functions.py"``, which should contain functions
456 conveniently named here ``"FunctionH"`` and ``"AdjointH"``. These functions are
457 user ones, representing as programming functions the :math:`\mathbf{H}` operator
458 and its adjoint. We suppose these functions are given by the user (a simple
459 skeleton is given in the Python script file ``Physical_simulation_functions.py``
460 of the ADAO examples standard directory, not reproduced here).
462 To operates in ADAO, it is required to define different types of operators: the
463 (potentially non-linear) standard observation operator, named ``"Direct"``, its
464 linearised approximation, named ``"Tangent"``, and the adjoint operator named
465 ``"Adjoint"``. The Python script have to retrieve an input parameter, found
466 under the key "value", in a variable named ``"specificParameters"`` of the
467 SALOME input data and parameters ``"computation"`` dictionary variable. If the
468 operator is already linear, the ``"Direct"`` and ``"Tangent"`` functions are the
469 same, as it is supposed here. The following example Python script file named
470 ``Script_ObservationOperator_H.py``, illustrates the case::
472 #-*-coding:iso-8859-1-*-
474 import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
475 import Physical_simulation_functions
476 import numpy, logging
478 # -----------------------------------------------------------------------
479 # SALOME input data and parameters: all information are the required input
480 # variable "computation", containing for example:
481 # {'inputValues': [[[[0.0, 0.0, 0.0]]]],
482 # 'inputVarList': ['adao_default'],
483 # 'outputVarList': ['adao_default'],
484 # 'specificParameters': [{'name': 'method', 'value': 'Direct'}]}
485 # -----------------------------------------------------------------------
487 # Recovering the type of computation: "Direct", "Tangent" or "Adjoint"
488 # --------------------------------------------------------------------
490 for param in computation["specificParameters"]:
491 if param["name"] == "method":
492 method = param["value"]
493 logging.info("ComputationFunctionNode: Found method is \'%s\'"%method)
495 # Loading the H operator functions from external definitions
496 # ----------------------------------------------------------
497 logging.info("ComputationFunctionNode: Loading operator functions")
498 FunctionH = Physical_simulation_functions.FunctionH
499 AdjointH = Physical_simulation_functions.AdjointH
501 # Executing the possible computations
502 # -----------------------------------
503 if method == "Direct":
504 logging.info("ComputationFunctionNode: Direct computation")
505 Xcurrent = computation["inputValues"][0][0][0]
506 data = FunctionH(numpy.matrix( Xcurrent ).T)
508 if method == "Tangent":
509 logging.info("ComputationFunctionNode: Tangent computation")
510 Xcurrent = computation["inputValues"][0][0][0]
511 data = FunctionH(numpy.matrix( Xcurrent ).T)
513 if method == "Adjoint":
514 logging.info("ComputationFunctionNode: Adjoint computation")
515 Xcurrent = computation["inputValues"][0][0][0]
516 Ycurrent = computation["inputValues"][0][0][1]
517 data = AdjointH((numpy.matrix( Xcurrent ).T, numpy.matrix( Ycurrent ).T))
519 # Formatting the output
520 # ---------------------
521 logging.info("ComputationFunctionNode: Formatting the output")
523 outputValues = [[[[]]]]
525 outputValues[0][0][0].append(val)
527 # Creating the required ADAO variable
528 # -----------------------------------
530 result["outputValues"] = outputValues
531 result["specificOutputInfos"] = []
532 result["returnCode"] = 0
533 result["errorMessage"] = ""
535 As output, this script has to define a nested list variable, as shown above with
536 the ``"outputValues"`` variable, where the nested levels describe the different
537 variables included in the state, then the different possible states at the same
538 time, then the different time steps. In this case, because there is only one
539 time step and one state, and all the variables are stored together, we only set
540 the most inner level of the lists.
542 In this twin experiments framework, the observation :math:`\mathbf{y}^o` and its
543 error covariances matrix :math:`\mathbf{R}` can be generated. It is done in two
544 Python script files, the first one being named ``Script_Observation_yo.py``::
546 #-*-coding:iso-8859-1-*-
548 import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
549 from Physical_data_and_covariance_matrices import True_state
550 from Physical_simulation_functions import FunctionH
552 xt, noms = True_state()
556 # Creating the required ADAO variable
557 # -----------------------------------
558 Observation = list(yo)
560 and the second one named ``Script_ObservationError_R.py``::
562 #-*-coding:iso-8859-1-*-
564 import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
565 from Physical_data_and_covariance_matrices import True_state, Simple_Matrix
566 from Physical_simulation_functions import FunctionH
568 xt, names = True_state()
572 R = 0.0001 * Simple_Matrix( size = len(yo) )
574 # Creating the required ADAO variable
575 # -----------------------------------
578 As in previous examples, it can be useful to define some parameters for the data
579 assimilation algorithm. For example, if we use the standard 3DVAR algorithm, the
580 following parameters can be defined in a Python script file named
581 ``Script_AlgorithmParameters.py``::
583 #-*-coding:iso-8859-1-*-
585 # Creating the required ADAO variable
586 # -----------------------------------
587 AlgorithmParameters = {
588 "Minimizer" : "TNC", # Possible : "LBFGSB", "TNC", "CG", "BFGS"
589 "MaximumNumberOfSteps" : 15, # Number of global iterative steps
591 [ None, None ], # Bound on the first parameter
592 [ 0., 4. ], # Bound on the second parameter
593 [ 0., None ], # Bound on the third parameter
597 Finally, it is common to post-process the results, retrieving them after the
598 data assimilation phase in order to analyse, print or show them. It requires to
599 use a intermediary Python script file in order to extract these results. The
600 following example Python script file named ``Script_UserPostAnalysis.py``,
601 illustrates the fact::
603 #-*-coding:iso-8859-1-*-
605 import sys ; sys.path.insert(0,"# INSERT PHYSICAL SCRIPT PATH")
606 from Physical_data_and_covariance_matrices import True_state
609 xt, names = True_state()
610 xa = ADD.get("Analysis").valueserie(-1)
611 x_series = ADD.get("CurrentState").valueserie()
612 J = ADD.get("CostFunctionJ").valueserie()
614 # Verifying the results by printing
615 # ---------------------------------
618 print "xa = %s"%numpy.array(xa)
620 for i in range( len(x_series) ):
621 print "Step %2i : J = %.5e et X = %s"%(i, J[i], x_series[i])
624 At the end, we get a description of the whole case setup through a set of files
627 #. ``Physical_data_and_covariance_matrices.py``
628 #. ``Physical_simulation_functions.py``
629 #. ``Script_AlgorithmParameters.py``
630 #. ``Script_BackgroundError_B.py``
631 #. ``Script_Background_xb.py``
632 #. ``Script_ObservationError_R.py``
633 #. ``Script_ObservationOperator_H.py``
634 #. ``Script_Observation_yo.py``
635 #. ``Script_UserPostAnalysis.py``
637 We insist here that all these scripts are written by the user and can not be
638 automatically tested. So the user is required to verify the scripts (and in
639 particular their input/output) in order to limit the difficulty of debug. We
640 recall: **script methodology is not a "safe" procedure, in the sense that
641 erroneous data, or errors in calculations, can be directly injected into the
642 YACS scheme execution.**
644 Building the case with external data definition by scripts
645 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
647 All these scripts can then be used to define the ADAO case with external data
648 definition by Python script files. It is entirely similar to the method
649 described in the `Building a simple estimation case with external data
650 definition by scripts`_ previous section. For each variable to be defined, we
651 select the "*Script*" option of the "*FROM*" keyword, which leads to a
652 "*SCRIPT_DATA/SCRIPT_FILE*" entry in the tree.
654 The other steps to build the ADAO case are exactly the same as in the `Building
655 a simple estimation case with explicit data definition`_ previous section.
657 Using the simple linear operator :math:`\mathbf{H}` from the Python script file
658 ``Physical_simulation_functions.py`` in the ADAO examples standard directory,
659 the results will look like::
662 xa = [ 1.000014 2.000458 3.000390]
664 Step 0 : J = 1.81750e+03 et X = [1.014011, 2.459175, 3.390462]
665 Step 1 : J = 1.81750e+03 et X = [1.014011, 2.459175, 3.390462]
666 Step 2 : J = 1.79734e+01 et X = [1.010771, 2.040342, 2.961378]
667 Step 3 : J = 1.79734e+01 et X = [1.010771, 2.040342, 2.961378]
668 Step 4 : J = 1.81909e+00 et X = [1.000826, 2.000352, 3.000487]
669 Step 5 : J = 1.81909e+00 et X = [1.000826, 2.000352, 3.000487]
670 Step 6 : J = 1.81641e+00 et X = [1.000247, 2.000651, 3.000156]
671 Step 7 : J = 1.81641e+00 et X = [1.000247, 2.000651, 3.000156]
672 Step 8 : J = 1.81569e+00 et X = [1.000015, 2.000432, 3.000364]
673 Step 9 : J = 1.81569e+00 et X = [1.000015, 2.000432, 3.000364]
674 Step 10 : J = 1.81568e+00 et X = [1.000013, 2.000458, 3.000390]
677 The state at the first step is the randomly generated background state
678 :math:`\mathbf{x}^b`. After completion, these printing on standard output is
679 available in the "*YACS Container Log*", obtained through the right click menu
680 of the "*proc*" window in the YACS scheme.
682 .. [#] For more information on YACS, see the the *YACS module User's Guide* available in the main "*Help*" menu of SALOME GUI.