multifidelity_polynomial_chaos
Multifidelity uncertainty quantification using polynomial chaos expansions
Specification
Alias: None
Arguments: None
Child Keywords:
Required/Optional 
Description of Group 
Dakota Keyword 
Dakota Keyword Description 

Optional 
Automatic polynomial order refinement 

Optional 
Maximum number of expansion refinement iterations 

Optional 
Stopping criterion based on objective function or statistics convergence 

Optional 
define scaling of statistical metrics when adapting UQ surrogates 

Optional 
type of statistical metric rollup for multifidelity UQ methods 

Optional 
Sample allocation approach for multifidelity expansions 

Optional 
Formulation for emulation of model discrepancies. 

Required (Choose One) 
Chaos Coefficient Estimation Approach 
Sequence of quadrature orders used in a multistage expansion 

Sequence of sparse grid levels used in a multistage expansion 

Sequence of expansion orders used in a multistage expansion 

Build a polynomial chaos expansion from simulation samples using orthogonal least interpolation. 

Optional (Choose One) 
Basis Polynomial Family 
Select the standardized random variables (and associated basis polynomials) from the Askey family that best match the userspecified random variables. 

Use standard normal random variables (along with Hermite orthogonal basis polynomials) when transforming to a standardized probability space. 

Optional 
The normalized specification requests output of PCE coefficients that correspond to normalized orthogonal basis polynomials 

Optional 
Export the coefficients and multiindex of a Polynomial Chaos Expansion (PCE) to a file 

Optional 
Number of samples at which to evaluate an emulator (surrogate) 

Optional 
Selection of sampling strategy 

Optional 
Selection of a random number generator 

Optional 
Allow refinement of probability and generalized reliability results using importance sampling 

Optional 
Output moments of the specified type and include them within the set of final statistics. 

Optional 
Values at which to estimate desired statistics for each response 

Optional 
Specify probability levels at which to estimate the corresponding response value 

Optional 
Specify reliability levels at which the response values will be estimated 

Optional 
Specify generalized relability levels at which to estimate the corresponding response value 

Optional 
Selection of cumulative or complementary cumulative functions 

Optional 
Activates global sensitivity analysis based on decomposition of response variance into main, interaction, and total effects 

Optional (Choose One) 
Covariance Type 
Display only the diagonal terms of the covariance matrix 

Display the full covariance matrix 

Optional 
Filename for points at which to evaluate the PCE/SC surrogate 

Optional 
Output file for surrogate model value evaluations 

Optional 
Sequence of seed values for multistage random sampling 

Optional 
Reuses the same seed value for multiple random sampling sets 

Optional 
Identifier for model block to be used by a method 
Description
As described in polynomial_chaos
, the polynomial chaos
expansion (PCE) is a general framework for the approximate representation
of random response functions in terms of series expansions in standardized
random variables:
where \(\alpha_i\) is a deterministic coefficient, \(\Psi_i\) is a multidimensional orthogonal polynomial and \(\xi\) is a vector of standardized random variables.
In the multilevel and multifidelity cases, we decompose this expansion into several constituent expansions, one per model form or solution control level. In a bifidelity case with lowfidelity (LF) and highfidelity (HF) models and an additive discrepancy approach, we have:
where \(\delta_i\) is a coefficient for the discrepancy expansion.
The same specification options are available as described in
polynomial_chaos
with one key difference: many of the
coefficient estimation inputs change from a scalar input for a single
expansion to a <i>sequence</i> specification for a lowfidelity expansion
followed by multiple discrepancy expansions.
To obtain the coefficients \(\alpha_i\) and \(\delta_i\) for each of the expansions, the following options are provided:
multidimensional integration by a tensorproduct of Gaussian quadrature rules (specified with
quadrature_order_sequence
, and, optionally,dimension_preference
).multidimensional integration by the Smolyak sparse grid method (specified with
sparse_grid_level_sequence
and, optionally,dimension_preference
)multidimensional integration by Latin hypercube sampling (specified with
expansion_order_sequence
andexpansion_samples_sequence
).linear regression (specified with
expansion_order_sequence
and eithercollocation_points_sequence
orcollocation_ratio
), using either overdetermined (least squares) or underdetermined (compressed sensing) approaches.orthogonal least interpolation (specified with
orthogonal_least_interpolation
andcollocation_points_sequence
)
It is important to note that, while quadrature_order_sequence
,
sparse_grid_level_sequence
, expansion_order_sequence
,
expansion_samples_sequence
, and collocation_points_sequence
are
array inputs, only one scalar from these arrays is active at a time
for a particular expansion estimation. In order to specify anisotropy
in resolution across the random variable set, a dimension_preference
specification can be used to augment scalar specifications for
quadrature order, sparse grid level, and expansion order.
Multifidelity UQ using PCE requires that the model selected for
iteration by the method specification is an ensemble surrogate model
(see ensemble
), which defines an ordered
sequence of model fidelities or resolutions. Two types of hierarchies are
supported: (i) a hierarchy of model forms composed from more than one
model within the ordered_model_fidelities
specification, or (ii) a
hierarchy of discretization levels comprised from a single model
(either from a truth_model_pointer
specification or a single entry
within an ordered_model_fidelities
specification) which in turn
specifies a solution_level_control
(see
solution_level_control
).
In both cases, an expansion will first be formed for the low fidelity
model or coarse discretization, using the first value within the
coefficient estimation sequence, along with any specified refinement
strategy. Second, expansions are formed for one or more model
discrepancies (the difference between response results if additive
correction
or the ratio of results if multiplicative
correction
), using all subsequent values in the coefficient estimation
sequence (if the sequence does not provide a new value, then the
previous value is reused) along with any specified refinement
strategy. The number of discrepancy expansions is determined by the
number of model forms or discretization levels in the hierarchy.
After formation and refinement of the constituent expansions, each of
the expansions is combined (added or multiplied) into an expansion
that approximates the high fidelity model, from which the final set of
statistics are generated. For polynomial chaos expansions, this high
fidelity expansion can differ significantly in form from the low
fidelity and discrepancy expansions, particularly in the
multiplicative
case where it is expanded to include all of the basis
products.
Additional Resources
Dakota provides access to multifidelity PCE methods through the NonDMultilevelPolynomialChaos class. Refer to the Stochastic Expansion Methods chapter of the Theory Manual [ABD+23] for additional information on the Multifidelity PCE algorithm.
Expected HDF5 Output
If Dakota was built with HDF5 support and run with the
hdf5
keyword, this method
writes the following results to HDF5:
Integration and Expansion Moments (expansion moments only)
In addition, the execution group has the attribute equiv_hf_evals
, which
records the equivalent number of highfidelity evaluations.
Examples
method,
multifidelity_polynomial_chaos
model_pointer = 'HIERARCH'
sparse_grid_level_sequence = 4 3 2
model,
id_model = 'HIERARCH'
surrogate ensemble
ordered_model_fidelities = 'LF' 'MF' 'HF'
correction additive zeroth_order