2 Copyright (C) 2008-2015 EDF R&D
4 This file is part of SALOME ADAO module.
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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: AlgorithmParameters
116 .. index:: single: CheckingPoint
117 .. index:: single: ObservationOperator
118 .. index:: single: AmplitudeOfInitialDirection
119 .. index:: single: EpsilonMinimumExponent
120 .. index:: single: InitialDirection
121 .. index:: single: ResiduFormula
122 .. index:: single: SetSeed
124 The general required commands, available in the editing user interface, are the
128 *Required command*. This indicates the vector used as the state around which
129 to perform the required check, noted :math:`\mathbf{x}` and similar to the
130 background :math:`\mathbf{x}^b`. It is defined as a "*Vector*" type object.
133 *Required command*. This indicates the observation operator, previously
134 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
135 results :math:`\mathbf{y}` to be compared to observations
136 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
137 a "*Matrix*" type one. In the case of "*Function*" type, different
138 functional forms can be used, as described in the section
139 :ref:`section_ref_operator_requirements`. If there is some control
140 :math:`U` included in the observation, the operator has to be applied to a
143 The general optional commands, available in the editing user interface, are
144 indicated in :ref:`section_ref_assimilation_keywords`. Moreover, the parameters
145 of the command "*AlgorithmParameters*" allow to choose the specific options,
146 described hereafter, of the algorithm. See
147 :ref:`section_ref_options_Algorithm_Parameters` for the good use of this
150 The options of the algorithm are the following:
152 AmplitudeOfInitialDirection
153 This key indicates the scaling of the initial perturbation build as a vector
154 used for the directional derivative around the nominal checking point. The
155 default is 1, that means no scaling.
157 Example : ``{"AmplitudeOfInitialDirection":0.5}``
159 EpsilonMinimumExponent
160 This key indicates the minimal exponent value of the power of 10 coefficient
161 to be used to decrease the increment multiplier. The default is -8, and it
162 has to be between 0 and -20. For example, its default value leads to
163 calculate the residue of the scalar product formula with a fixed increment
164 multiplied from 1.e0 to 1.e-8.
166 Example : ``{"EpsilonMinimumExponent":-12}``
169 This key indicates the vector direction used for the directional derivative
170 around the nominal checking point. It has to be a vector. If not specified,
171 this direction defaults to a random perturbation around zero of the same
172 vector size than the checking point.
174 Example : ``{"InitialDirection":[0.1,0.1,100.,3}``
177 This key indicates the residue formula that has to be used for the test. The
178 default choice is "CenteredDL", and the possible ones are "CenteredDL"
179 (residue of the difference between the function at nominal point and the
180 values with positive and negative increments, which has to stay very small),
181 "Taylor" (residue of the Taylor development of the operator normalized by
182 the nominal value, which has to stay very small), "NominalTaylor" (residue
183 of the order 1 approximations of the operator, normalized to the nominal
184 point, which has to stay close to 1), and "NominalTaylorRMS" (residue of the
185 order 1 approximations of the operator, normalized by RMS to the nominal
186 point, which has to stay close to 0).
188 Example : ``{"ResiduFormula":"CenteredDL"}``
191 This key allow to give an integer in order to fix the seed of the random
192 generator used to generate the ensemble. A convenient value is for example
193 1000. By default, the seed is left uninitialized, and so use the default
194 initialization from the computer.
196 Example : ``{"SetSeed":1000}``
201 References to other sections:
202 - :ref:`section_ref_algorithm_FunctionTest`