Dakota Reference Manual
Version 6.12
Explore and Predict with Confidence

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

Optional  max_iterations  Number of iterations allowed for optimizers and adaptive UQ methods  
Optional  allocation_control  Sample allocation approach for multilevel expansions  
Optional  discrepancy_emulation  Formulation for emulation of model discrepancies.  
Required (Choose One)  Coefficient Computation Approach (Group 1)  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, we have:
where is a coefficient for the discrepancy expansion.
For the case of regressionbased PCE (least squares, compressed sensing, or orthogonal least interpolation), an optimal sample allocation procedure can be applied for the resolution of each level within a multilevel sampling procedure as in multilevel_sampling. The core difference is that a Monte Carlo estimator of the statistics is replaced with a PCEbased estimator of the statistics, requiring approximation of the variance of these estimators.
Initial prototypes for multilevel PCE can be explored using dakota/share/dakota/test/dakota_uq_diffusion_mlpce.in
, and will be stabilized in future releases.
Additional Resources
Dakota provides access to multilevel PCE methods through the NonDMultilevelPolynomialChaos class. Refer to the Stochastic Expansion Methods chapter of the Theory Manual[4] for additional information on the Multilevel 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, multilevel_polynomial_chaos model_pointer = 'HIERARCH' pilot_samples = 10 expansion_order_sequence = 2 collocation_ratio = .9 seed = 1237 orthogonal_matching_pursuit convergence_tolerance = .01 output silent model, id_model = 'HIERARCH' surrogate hierarchical ordered_model_fidelities = 'SIM1' correction additive zeroth_order model, id_model = 'SIM1' simulation solution_level_control = 'mesh_size' solution_level_cost = 1. 8. 64. 512. 4096.
These keywords may also be of interest: