doc/en/ref_algorithm_4DVAR.rst

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  20    See http://www.salome-platform.org/ or email : webmaster.salome@opencascade.com
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  22    Author: Jean-Philippe Argaud, jean-philippe.argaud@edf.fr, EDF R&D
  23
  24 .. index:: single: 4DVAR
  25 .. index:: single: 4D-Var
  26 .. _section_ref_algorithm_4DVAR:
  27
  28 Calculation algorithm "*4DVAR*"
  29 -------------------------------
  30
  31 .. ------------------------------------ ..
  32 .. include:: snippets/Header2Algo01.rst
  33
  34 This algorithm realizes an estimation of the state of a dynamic system, by a
  35 variational minimization method of the classical :math:`J` function in data
  36 assimilation:
  37
  38 .. math:: J(\mathbf{x})=(\mathbf{x}-\mathbf{x}^b)^T.\mathbf{B}^{-1}.(\mathbf{x}-\mathbf{x}^b)+\sum_{t\in T}(\mathbf{y^o}(t)-H(\mathbf{x},t))^T.\mathbf{R}^{-1}.(\mathbf{y^o}(t)-H(\mathbf{x},t))
  39
  40 which is usually designed as the "*4D-Var*" functional (see for example
  41 [Talagrand97]_). The terms "*4D-Var*", "*4D-VAR*" and "*4DVAR*" are equivalent.
  42 It is well suited in cases of non-linear observation and evolution operators,
  43 its application domain is similar to the one of Kalman filters, specially the
  44 :ref:`section_ref_algorithm_ExtendedKalmanFilter` or the
  45 :ref:`section_ref_algorithm_UnscentedKalmanFilter`.
  46
  47 .. ------------------------------------ ..
  48 .. include:: snippets/Header2Algo02.rst
  49
  50 .. include:: snippets/Background.rst
  51
  52 .. include:: snippets/BackgroundError.rst
  53
  54 .. include:: snippets/EvolutionError.rst
  55
  56 .. include:: snippets/EvolutionModel.rst
  57
  58 .. include:: snippets/Observation.rst
  59
  60 .. include:: snippets/ObservationError.rst
  61
  62 .. include:: snippets/ObservationOperator.rst
  63
  64 .. ------------------------------------ ..
  65 .. include:: snippets/Header2Algo03AdOp.rst
  66
  67 .. include:: snippets/BoundsWithNone.rst
  68
  69 .. include:: snippets/ConstrainedBy.rst
  70
  71 .. include:: snippets/CostDecrementTolerance.rst
  72
  73 .. include:: snippets/EstimationOf_State.rst
  74
  75 .. include:: snippets/GradientNormTolerance.rst
  76
  77 .. include:: snippets/InitializationPoint.rst
  78
  79 .. include:: snippets/MaximumNumberOfIterations.rst
  80
  81 .. include:: snippets/Minimizer_xDVAR.rst
  82
  83 .. include:: snippets/ProjectedGradientTolerance.rst
  84
  85 StoreSupplementaryCalculations
  86   .. index:: single: StoreSupplementaryCalculations
  87
  88   *List of names*. This list indicates the names of the supplementary
  89   variables, that can be available during or at the end of the algorithm, if
  90   they are initially required by the user. Their avalability involves,
  91   potentially, costly calculations or memory consumptions. The default is then
  92   a void list, none of these variables being calculated and stored by default
  93   (excepted the unconditionnal variables). The possible names are in the
  94   following list (the detailed description of each named variable is given in
  95   the following part of this specific algorithmic documentation, in the
  96   sub-section "*Information and variables available at the end of the
  97   algorithm*"): [
  98   "Analysis",
  99   "BMA",
 100   "CostFunctionJ",
 101   "CostFunctionJAtCurrentOptimum",
 102   "CostFunctionJb",
 103   "CostFunctionJbAtCurrentOptimum",
 104   "CostFunctionJo",
 105   "CostFunctionJoAtCurrentOptimum",
 106   "CurrentIterationNumber",
 107   "CurrentOptimum",
 108   "CurrentState",
 109   "IndexOfOptimum",
 110   ].
 111
 112   Example :
 113   ``{"StoreSupplementaryCalculations":["CurrentState", "Residu"]}``
 114
 115 .. ------------------------------------ ..
 116 .. include:: snippets/Header2Algo04.rst
 117
 118 .. include:: snippets/Analysis.rst
 119
 120 .. include:: snippets/CostFunctionJ.rst
 121
 122 .. include:: snippets/CostFunctionJb.rst
 123
 124 .. include:: snippets/CostFunctionJo.rst
 125
 126 .. ------------------------------------ ..
 127 .. include:: snippets/Header2Algo05.rst
 128
 129 .. include:: snippets/Analysis.rst
 130
 131 .. include:: snippets/BMA.rst
 132
 133 .. include:: snippets/CostFunctionJ.rst
 134
 135 .. include:: snippets/CostFunctionJAtCurrentOptimum.rst
 136
 137 .. include:: snippets/CostFunctionJb.rst
 138
 139 .. include:: snippets/CostFunctionJbAtCurrentOptimum.rst
 140
 141 .. include:: snippets/CostFunctionJo.rst
 142
 143 .. include:: snippets/CostFunctionJoAtCurrentOptimum.rst
 144
 145 .. include:: snippets/CurrentIterationNumber.rst
 146
 147 .. include:: snippets/CurrentOptimum.rst
 148
 149 .. include:: snippets/CurrentState.rst
 150
 151 .. include:: snippets/IndexOfOptimum.rst
 152
 153 .. ------------------------------------ ..
 154 .. _section_ref_algorithm_4DVAR_examples:
 155 .. include:: snippets/Header2Algo06.rst
 156
 157 - :ref:`section_ref_algorithm_3DVAR`
 158 - :ref:`section_ref_algorithm_KalmanFilter`
 159 - :ref:`section_ref_algorithm_ExtendedKalmanFilter`
 160 - :ref:`section_ref_algorithm_EnsembleKalmanFilter`
 161
 162 .. ------------------------------------ ..
 163 .. include:: snippets/Header2Algo07.rst
 164
 165 - [Byrd95]_
 166 - [Morales11]_
 167 - [Talagrand97]_
 168 - [Zhu97]_