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Dakota Reference Manual
Version 6.15
Explore and Predict with Confidence
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Basis adaptation model
Alias: none
Argument(s): none
Child Keywords:
Required/Optional | Description of Group | Dakota Keyword | Dakota Keyword Description | |
---|---|---|---|---|
Required | actual_model_pointer | Pointer to specify a "truth" model, from which to construct a surrogate | ||
Optional | truncation_tolerance | Convergence tolerance used to identify the (reduced) dimension of the rotation matrix |
A model that transforms the original model (given by actual_model_pointer) to one with a rotated set of variables. The current implementation does not support the reduction of the dimension for the new model and only transform the variable set in a rotated one in which the variables are arranged in a decreasing importance order.
An initial PCE representation is built with a sparse grid method and the PCE coefficients are then used to build a rotation matrix. This matrix is also reported as a result of the code execution. The new model can subsequently be used for an UQ workflow.
Expected Output A basis adaptation model will perform an initial sparse grid design to identify the rotation matrix.
Perform an initial sparse grid design (level 1) to evaluate the PCE expansion and evaluate the rotation matrix. Afterwards, 100 samples are generated for the model in the rotated space to obtain its statistics.
method, sampling model_pointer = 'SUBSPACE' samples = 100 seed = 1234567 model id_model = 'SUBSPACE' adapted_basis actual_model_pointer = 'FULLSPACE' sparse_grid_level = 1
The idea behind the Basis Adaptation method is to generate a PCE representation and rotate it such that the new set of rotated variables are organized in a decreasing importance ordering. Subsequently this rotation matrix can be truncated according to some criterion to only include the significant directions.
The first step of the Basis Adaptation is to compute a PCE expansion (we assume here to have a standard multivariate Gaussian distribution)
where with
is multi-index of dimension
and order up to
.
Afterwards, a multivariate Gaussian distribution is sought such that
where is an isometry such that
.
The basis adaptation model is obtained by expressing the original model with respect to the rotated set of variables as
Since the basis for both the original and the adapted basis model span the same space, we know that and therefore a relationships between the PCE coefficients exists
The linear adaptation strategy is used at this time in Dakota to obtain the rotation matrix . The steps to obtain the matrix
are the following:
See the theory manual for more details.
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