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 This mono-objective optimization algorithm is naturally written for a single
  43 estimate on a time window for the simulation. It is well suited in cases of
  44 non-linear observation and evolution operators, its application domain is
  45 similar to the one of Kalman filters, specially the
  46 :ref:`section_ref_algorithm_ExtendedKalmanFilter` or the
  47 :ref:`section_ref_algorithm_UnscentedKalmanFilter`.
  48
  49 .. ------------------------------------ ..
  50 .. include:: snippets/Header2Algo12.rst
  51
  52 .. include:: snippets/FeaturePropNonLocalOptimization.rst
  53
  54 .. include:: snippets/FeaturePropDerivativeNeeded.rst
  55
  56 .. include:: snippets/FeaturePropParallelDerivativesOnly.rst
  57
  58 .. include:: snippets/FeaturePropConvergenceOnBoth.rst
  59
  60 .. ------------------------------------ ..
  61 .. include:: snippets/Header2Algo02.rst
  62
  63 .. include:: snippets/Background.rst
  64
  65 .. include:: snippets/BackgroundError.rst
  66
  67 .. include:: snippets/EvolutionError.rst
  68
  69 .. include:: snippets/EvolutionModel.rst
  70
  71 .. include:: snippets/Observation.rst
  72
  73 .. include:: snippets/ObservationError.rst
  74
  75 .. include:: snippets/ObservationOperator.rst
  76
  77 .. ------------------------------------ ..
  78 .. include:: snippets/Header2Algo03AdOp.rst
  79
  80 .. include:: snippets/BoundsWithNone.rst
  81
  82 .. include:: snippets/ConstrainedBy.rst
  83
  84 .. include:: snippets/CostDecrementTolerance.rst
  85
  86 .. include:: snippets/EstimationOf_State.rst
  87
  88 .. include:: snippets/GradientNormTolerance.rst
  89
  90 .. include:: snippets/InitializationPoint.rst
  91
  92 .. include:: snippets/MaximumNumberOfIterations.rst
  93
  94 .. include:: snippets/Minimizer_xDVAR.rst
  95
  96 .. include:: snippets/ProjectedGradientTolerance.rst
  97
  98 StoreSupplementaryCalculations
  99   .. index:: single: StoreSupplementaryCalculations
 100
 101   *List of names*. This list indicates the names of the supplementary
 102   variables, that can be available during or at the end of the algorithm, if
 103   they are initially required by the user. Their availability involves,
 104   potentially, costly calculations or memory consumptions. The default is then
 105   a void list, none of these variables being calculated and stored by default
 106   (excepted the unconditional variables). The possible names are in the
 107   following list (the detailed description of each named variable is given in
 108   the following part of this specific algorithmic documentation, in the
 109   sub-section "*Information and variables available at the end of the
 110   algorithm*"): [
 111   "Analysis",
 112   "BMA",
 113   "CostFunctionJ",
 114   "CostFunctionJAtCurrentOptimum",
 115   "CostFunctionJb",
 116   "CostFunctionJbAtCurrentOptimum",
 117   "CostFunctionJo",
 118   "CostFunctionJoAtCurrentOptimum",
 119   "CurrentIterationNumber",
 120   "CurrentOptimum",
 121   "CurrentState",
 122   "IndexOfOptimum",
 123   ].
 124
 125   Example :
 126   ``{"StoreSupplementaryCalculations":["CurrentState", "Residu"]}``
 127
 128 .. ------------------------------------ ..
 129 .. include:: snippets/Header2Algo04.rst
 130
 131 .. include:: snippets/Analysis.rst
 132
 133 .. include:: snippets/CostFunctionJ.rst
 134
 135 .. include:: snippets/CostFunctionJb.rst
 136
 137 .. include:: snippets/CostFunctionJo.rst
 138
 139 .. ------------------------------------ ..
 140 .. include:: snippets/Header2Algo05.rst
 141
 142 .. include:: snippets/Analysis.rst
 143
 144 .. include:: snippets/BMA.rst
 145
 146 .. include:: snippets/CostFunctionJ.rst
 147
 148 .. include:: snippets/CostFunctionJAtCurrentOptimum.rst
 149
 150 .. include:: snippets/CostFunctionJb.rst
 151
 152 .. include:: snippets/CostFunctionJbAtCurrentOptimum.rst
 153
 154 .. include:: snippets/CostFunctionJo.rst
 155
 156 .. include:: snippets/CostFunctionJoAtCurrentOptimum.rst
 157
 158 .. include:: snippets/CurrentIterationNumber.rst
 159
 160 .. include:: snippets/CurrentOptimum.rst
 161
 162 .. include:: snippets/CurrentState.rst
 163
 164 .. include:: snippets/IndexOfOptimum.rst
 165
 166 .. ------------------------------------ ..
 167 .. _section_ref_algorithm_4DVAR_examples:
 168
 169 .. include:: snippets/Header2Algo06.rst
 170
 171 - :ref:`section_ref_algorithm_3DVAR`
 172 - :ref:`section_ref_algorithm_KalmanFilter`
 173 - :ref:`section_ref_algorithm_ExtendedKalmanFilter`
 174 - :ref:`section_ref_algorithm_EnsembleKalmanFilter`
 175
 176 .. ------------------------------------ ..
 177 .. include:: snippets/Header2Algo07.rst
 178
 179 - [Byrd95]_
 180 - [Morales11]_
 181 - [Talagrand97]_
 182 - [Zhu97]_