2 Copyright (C) 2008-2014 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: GradientTest
25 .. _section_ref_algorithm_GradientTest:
27 Checking algorithm "*GradientTest*"
28 -----------------------------------
33 This algorithm allows to check the quality of the adjoint operator, by
34 calculating a residue with known theoretical properties. Different residue
35 formula are available.
37 In any cases, one take :math:`\mathbf{dx}_0=Normal(0,\mathbf{x})` and
38 :math:`\mathbf{dx}=\alpha*\mathbf{dx}_0`. :math:`F` is the calculation code.
43 One observe the residue coming from the Taylor development of the :math:`F`
44 function, normalized by the value at the nominal point:
46 .. math:: R(\alpha) = \frac{|| F(\mathbf{x}+\alpha*\mathbf{dx}) - F(\mathbf{x}) - \alpha * \nabla_xF(\mathbf{dx}) ||}{|| F(\mathbf{x}) ||}
48 If the residue is decreasing and the decrease change in :math:`\alpha^2` with
49 respect to :math:`\alpha`, until a certain level after which the residue remains
50 small and constant, it signifies that the :math:`F` is linear and that the
51 residue is decreasing due to the error coming from :math:`\nabla_xF` term
57 One observe the residue based on the gradient approximation:
59 .. math:: R(\alpha) = \frac{|| F(\mathbf{x}+\alpha*\mathbf{dx}) - F(\mathbf{x}) ||}{\alpha}
61 which has to remain stable until the calculation precision is reached.
63 Optional and required commands
64 ++++++++++++++++++++++++++++++
66 .. index:: single: CheckingPoint
67 .. index:: single: ObservationOperator
68 .. index:: single: AmplitudeOfInitialDirection
69 .. index:: single: EpsilonMinimumExponent
70 .. index:: single: InitialDirection
71 .. index:: single: ResiduFormula
72 .. index:: single: SetSeed
74 The general required commands, available in the editing user interface, are the
78 *Required command*. This indicates the vector used as the state around which
79 to perform the required check, noted :math:`\mathbf{x}` and similar to the
80 background :math:`\mathbf{x}^b`. It is defined as a "*Vector*" type object.
83 *Required command*. This indicates the observation operator, previously
84 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
85 results :math:`\mathbf{y}` to be compared to observations
86 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
87 a "*Matrix*" type one. In the case of "*Function*" type, different
88 functional forms can be used, as described in the section
89 :ref:`section_ref_operator_requirements`. If there is some control
90 :math:`U` included in the observation, the operator has to be applied to a
93 The general optional commands, available in the editing user interface, are
94 indicated in :ref:`section_ref_assimilation_keywords`. In particular, the
95 optional command "*AlgorithmParameters*" allows to choose the specific options,
96 described hereafter, of the algorithm. See
97 :ref:`section_ref_options_AlgorithmParameters` for the good use of this command.
99 The options of the algorithm are the following:
101 AmplitudeOfInitialDirection
102 This key indicates the scaling of the initial perturbation build as a vector
103 used for the directional derivative around the nominal checking point. The
104 default is 1, that means no scaling.
106 Example : ``{"AmplitudeOfInitialDirection":0.5}``
108 EpsilonMinimumExponent
109 This key indicates the minimal exponent value of the power of 10 coefficient
110 to be used to decrease the increment multiplier. The default is -8, and it
111 has to be between 0 and -20. For example, its default value leads to
112 calculate the residue of the scalar product formula with a fixed increment
113 multiplied from 1.e0 to 1.e-8.
115 Example : ``{"EpsilonMinimumExponent":-12}``
118 This key indicates the vector direction used for the directional derivative
119 around the nominal checking point. It has to be a vector. If not specified,
120 this direction defaults to a random perturbation around zero of the same
121 vector size than the checking point.
123 Example : ``{"InitialDirection":[0.1,0.1,100.,3}``
126 This key indicates the residue formula that has to be used for the test. The
127 default choice is "Taylor", and the possible ones are "Taylor" (residue of
128 the Taylor development of the operator, which has to decrease with the
129 square power of the perturbation) and "Norm" (residue obtained by taking the
130 norm of the Taylor development at zero order approximation, which
131 approximate the gradient, and which has to remain constant).
133 Example : ``{"ResiduFormula":"Taylor"}``
136 This key allow to give an integer in order to fix the seed of the random
137 generator used to generate the ensemble. A convenient value is for example
138 1000. By default, the seed is left uninitialized, and so use the default
139 initialization from the computer.
141 Example : ``{"SetSeed":1000}``
146 References to other sections:
147 - :ref:`section_ref_algorithm_FunctionTest`
148 - :ref:`section_ref_algorithm_TangentTest`
149 - :ref:`section_ref_algorithm_AdjointTest`