Dakota Reference Manual
Version 6.12
Explore and Predict with Confidence

Multifidelity uncertainty quantification using polynomial chaos expansions
Alias: none
Argument(s): none
Child Keywords:
Required/Optional  Description of Group  Dakota Keyword  Dakota Keyword Description  

Optional  p_refinement  Automatic polynomial order refinement  
Optional  max_refinement_iterations  Maximum number of expansion refinement iterations  
Optional  allocation_control  Sample allocation approach for multifidelity expansions  
Optional  discrepancy_emulation  Formulation for emulation of model discrepancies.  
Required (Choose One)  Chaos Coefficient Estimation Approach (Group 1)  quadrature_order_sequence  Sequence of quadrature orders used in a multistage expansion  
sparse_grid_level_sequence  Sequence of sparse grid levels used in a multistage expansion  
expansion_order_sequence  Sequence of expansion orders used in a multistage expansion  
orthogonal_least_interpolation  Build a polynomial chaos expansion from simulation samples using orthogonal least interpolation.  
Optional (Choose One)  Basis Polynomial Family (Group 2)  askey  Select the standardized random variables (and associated basis polynomials) from the Askey family that best match the userspecified random variables.  
wiener  Use standard normal random variables (along with Hermite orthogonal basis polynomials) when transforming to a standardized probability space.  
Optional  normalized  The normalized specification requests output of PCE coefficients that correspond to normalized orthogonal basis polynomials  
Optional  export_expansion_file  Export the coefficients and multiindex of a Polynomial Chaos Expansion (PCE) to a file  
Optional  samples_on_emulator  Number of samples at which to evaluate an emulator (surrogate)  
Optional  sample_type  Selection of sampling strategy  
Optional  rng  Selection of a random number generator  
Optional  probability_refinement  Allow refinement of probability and generalized reliability results using importance sampling  
Optional  final_moments  Output moments of the specified type and include them within the set of final statistics.  
Optional  response_levels  Values at which to estimate desired statistics for each response  
Optional  probability_levels  Specify probability levels at which to estimate the corresponding response value  
Optional  reliability_levels  Specify reliability levels at which the response values will be estimated  
Optional  gen_reliability_levels  Specify generalized relability levels at which to estimate the corresponding response value  
Optional  distribution  Selection of cumulative or complementary cumulative functions  
Optional  variance_based_decomp  Activates global sensitivity analysis based on decomposition of response variance into main, interaction, and total effects  
Optional (Choose One)  Covariance Type (Group 3)  diagonal_covariance  Display only the diagonal terms of the covariance matrix  
full_covariance  Display the full covariance matrix  
Optional  convergence_tolerance  Stopping criterion based on objective function or statistics convergence  
Optional  import_approx_points_file  Filename for points at which to evaluate the PCE/SC surrogate  
Optional  export_approx_points_file  Output file for evaluations of a surrogate model  
Optional  seed_sequence  Sequence of seed values for a multistage random sampling  
Optional  fixed_seed  Reuses the same seed value for multiple random sampling sets  
Optional  model_pointer  Identifier for model block to be used by a method 
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 is a deterministic coefficient, is a multidimensional orthogonal polynomial and 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 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 sequence specification for a lowfidelity expansion followed by multiple discrepancy expansions.
To obtain the coefficients and for each of the expansions, the following options are provided:
quadrature_order_sequence
, and, optionally, dimension_preference
). sparse_grid_level_sequence
and, optionally, dimension_preference
) expansion_order_sequence
and expansion_samples_sequence
). expansion_order_sequence
and either collocation_points_sequence
or collocation_ratio
), using either overdetermined (least squares) or underdetermined (compressed sensing) approaches. orthogonal least interpolation (specified with orthogonal_least_interpolation
and collocation_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 a multifidelity surrogate model (see hierarchical), which defines an ordered_model_sequence
(see hierarchical). Two types of hierarchies are supported: (i) a hierarchy of model forms composed from more than one model within the ordered_model_sequence
, or (ii) a hierarchy of discretization levels comprised from a single model within the ordered_model_sequence
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[4] 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:
In addition, the execution group has the attribute equiv_hf_evals
, which records the equivalent number of highfidelity evaluations.
method, multifidelity_polynomial_chaos model_pointer = 'HIERARCH' sparse_grid_level_sequence = 4 3 2 model, id_model = 'HIERARCH' surrogate hierarchical ordered_model_fidelities = 'LF' 'MF' 'HF' correction additive zeroth_order
These keywords may also be of interest: