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
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13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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: LinearityTest
25 .. _section_ref_algorithm_LinearityTest:
27 Checking algorithm "*LinearityTest*"
28 ------------------------------------
33 This algorithm allows to check the linear quality of the 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 following residue, coming from the centered difference of the
44 :math:`F` values at nominal point and at perturbed points, normalized by the
45 value at the nominal point:
47 .. math:: R(\alpha) = \frac{|| F(\mathbf{x}+\alpha*\mathbf{dx}) + F(\mathbf{x}-\alpha*\mathbf{dx}) - 2*F(\mathbf{x}) ||}{|| F(\mathbf{x}) ||}
49 If it stays constantly really small with respect to 1, the linearity hypothesis
50 of :math:`F` is verified.
52 If the residue is varying, or if it is of order 1 or more, and it is small only
53 at a certain order of increment, the linearity hypothesis of :math:`F` is not
56 If the residue is decreasing and the decrease change in :math:`\alpha^2` with
57 respect to :math:`\alpha`, it signifies that the gradient is correctly
58 calculated until the stopping level of the quadratic decrease.
63 One observe the residue coming from the Taylor development of the :math:`F`
64 function, normalized by the value at the nominal point:
66 .. math:: R(\alpha) = \frac{|| F(\mathbf{x}+\alpha*\mathbf{dx}) - F(\mathbf{x}) - \alpha * \nabla_xF(\mathbf{dx}) ||}{|| F(\mathbf{x}) ||}
68 If it stay constantly really small with respect to 1, the linearity hypothesis
69 of :math:`F` is verified.
71 If the residue is varying, or if it is of order 1 or more, and it is small only
72 at a certain order of increment, the linearity hypothesis of :math:`F` is not
75 If the residue is decreasing and the decrease change in :math:`\alpha^2` with
76 respect to :math:`\alpha`, it signifies that the gradient is correctly
77 calculated until the stopping level of the quadratic decrease.
79 "NominalTaylor" residue
80 ***********************
82 One observe the residue build from two approximations of order 1 of
83 :math:`F(\mathbf{x})`, normalized by the value at the nominal point:
85 .. math:: R(\alpha) = \max(|| F(\mathbf{x}+\alpha*\mathbf{dx}) - \alpha * F(\mathbf{dx}) || / || F(\mathbf{x}) ||,|| F(\mathbf{x}-\alpha*\mathbf{dx}) + \alpha * F(\mathbf{dx}) || / || F(\mathbf{x}) ||)
87 If the residue stays constant equal to 1 at less than 2 or 3 percents (that that
88 :math:`|R-1|` stays equal to 2 or 3 percents), the linearity hypothesis of
89 :math:`F` is verified.
91 If it is equal to 1 only on part of the variation domain of increment
92 :math:`\alpha`, it is on this sub-domain that the linearity hypothesis of
93 :math:`F` is verified.
95 "NominalTaylorRMS" residue
96 **************************
98 One observe the residue build from two approximations of order 1 of
99 :math:`F(\mathbf{x})`, normalized by the value at the nominal point, on which
100 one estimate the quadratic root mean square (RMS) with the value at the nominal
103 .. math:: R(\alpha) = \max(RMS( F(\mathbf{x}), F(\mathbf{x}+\alpha*\mathbf{dx}) - \alpha * F(\mathbf{dx}) ) / || F(\mathbf{x}) ||,RMS( F(\mathbf{x}), F(\mathbf{x}-\alpha*\mathbf{dx}) + \alpha * F(\mathbf{dx}) ) / || F(\mathbf{x}) ||)
105 If it stay constantly equal to 0 at less than 1 or 2 percents, the linearity
106 hypothesis of :math:`F` is verified.
108 If it is equal to 0 only on part of the variation domain of increment
109 :math:`\alpha`, it is on this sub-domain that the linearity hypothesis of
110 :math:`F` is verified.
112 Optional and required commands
113 ++++++++++++++++++++++++++++++
115 .. index:: single: CheckingPoint
116 .. index:: single: ObservationOperator
117 .. index:: single: AmplitudeOfInitialDirection
118 .. index:: single: EpsilonMinimumExponent
119 .. index:: single: InitialDirection
120 .. index:: single: ResiduFormula
121 .. index:: single: SetSeed
123 The general required commands, available in the editing user interface, are the
127 *Required command*. This indicates the vector used as the state around which
128 to perform the required check, noted :math:`\mathbf{x}` and similar to the
129 background :math:`\mathbf{x}^b`. It is defined as a "*Vector*" type object.
132 *Required command*. This indicates the observation operator, previously
133 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
134 results :math:`\mathbf{y}` to be compared to observations
135 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
136 a "*Matrix*" type one. In the case of "*Function*" type, different
137 functional forms can be used, as described in the section
138 :ref:`section_ref_operator_requirements`. If there is some control
139 :math:`U` included in the observation, the operator has to be applied to a
142 The general optional commands, available in the editing user interface, are
143 indicated in :ref:`section_ref_assimilation_keywords`. In particular, the
144 optional command "*AlgorithmParameters*" allows to choose the specific options,
145 described hereafter, of the algorithm. See
146 :ref:`section_ref_options_AlgorithmParameters` for the good use of this command.
148 The options of the algorithm are the following:
150 AmplitudeOfInitialDirection
151 This key indicates the scaling of the initial perturbation build as a vector
152 used for the directional derivative around the nominal checking point. The
153 default is 1, that means no scaling.
155 EpsilonMinimumExponent
156 This key indicates the minimal exponent value of the power of 10 coefficient
157 to be used to decrease the increment multiplier. The default is -8, and it
158 has to be between 0 and -20. For example, its default value leads to
159 calculate the residue of the scalar product formula with a fixed increment
160 multiplied from 1.e0 to 1.e-8.
163 This key indicates the vector direction used for the directional derivative
164 around the nominal checking point. It has to be a vector. If not specified,
165 this direction defaults to a random perturbation around zero of the same
166 vector size than the checking point.
169 This key indicates the residue formula that has to be used for the test. The
170 default choice is "CenteredDL", and the possible ones are "CenteredDL"
171 (residue of the difference between the function at nominal point and the
172 values with positive and negative increments, which has to stay very small),
173 "Taylor" (residue of the Taylor development of the operator normalized by
174 the nominal value, which has to stay very small), "NominalTaylor" (residue
175 of the order 1 approximations of the operator, normalized to the nominal
176 point, which has to stay close to 1), and "NominalTaylorRMS" (residue of the
177 order 1 approximations of the operator, normalized by RMS to the nominal
178 point, which has to stay close to 0).
181 This key allow to give an integer in order to fix the seed of the random
182 generator used to generate the ensemble. A convenient value is for example
183 1000. By default, the seed is left uninitialized, and so use the default
184 initialization from the computer.
189 References to other sections:
190 - :ref:`section_ref_algorithm_FunctionTest`