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

Optional 
Number of iterations allowed for optimizers and adaptive UQ methods 

Optional 
Sample allocation approach for multilevel expansions 

Optional 
Stopping criterion based on objective function or statistics convergence 

Optional 
define scaling of statistical metrics when adapting UQ surrogates 

Optional 
Formulation for emulation of model discrepancies. 

Required (Choose One) 
Coefficient Computation Approach 
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, we have:
where \(\delta_i\) 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 [ABD+23] 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:
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,
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 ensemble
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.