polynomial_chaos
Uncertainty quantification using polynomial chaos expansions
Specification
Alias: nond_polynomial_chaos
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 

Required (Choose One) 
Chaos coefficient estimation approach 
Order for tensorproducts of Gaussian quadrature rules 

Level to use in sparse grid integration or interpolation 

Cubature using Stroud rules and their extensions 

The (initial) order of a polynomial expansion 

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

Build a Polynomial Chaos Expansion (PCE) by importing expansion coefficients and a corresponding multiindex from a file 

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 
Seed of the random number generator 

Optional 
Reuses the same seed value for multiple random sampling sets 

Optional 
Identifier for model block to be used by a method 
Description
The polynomial chaos expansion (PCE) is a general framework for the approximate representation of random response functions in terms of finitedimensional 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. An important distinguishing feature of the methodology is that the functional relationship between random inputs and outputs is captured, not merely the output statistics as in the case of many nondeterministic methodologies.
Basis polynomial family (Group 1)
Group 1 keywords are used to select the type of basis, \(\Psi_i\) , of the expansion. Three approaches may be employed:
Wiener: employs standard normal random variables in a transformed probability space, corresponding to Hermite orthogonal basis polynomials (see
wiener
).Askey: employs standard normal, standard uniform, standard exponential, standard beta, and standard gamma random variables in a transformed probability space, corresponding to Hermite, Legendre, Laguerre, Jacobi, and generalized Laguerre orthogonal basis polynomials, respectively (see
askey
).Extended (default if no option is selected): The Extended option avoids the use of any nonlinear variable transformations by augmenting the Askey approach with numericallygenerated orthogonal polynomials for nonAskey probability density functions. Extended polynomial selections with numericallygenerated polynomials that are orthogonal to the prescribed probability density functions replace each of the suboptimal Askey basis selections for bounded normal, lognormal, bounded lognormal, loguniform, triangular, gumbel, frechet, weibull, and binbased histogram.
For supporting correlated random variables, certain fallbacks must be implemented.
The Extended option is the default and supports only Gaussian correlations.
If needed to support prescribed correlations (not under user control), the Extended and Askey options will fall back to the Wiener option on a per variable basis. If the prescribed correlations are also unsupported by Wiener expansions, then Dakota will exit with an error.
These defaults can be overridden by the user by supplying the keyword
askey
to request restriction to the use of Askey bases only or by
supplying the keyword wiener
to request restriction to the use of
exclusively Hermite bases.
Refer to Variable Support for additional information on supported variable types, with and without correlation.
Coefficient estimation approach (Group 2)
To obtain the coefficients \(\alpha_i\) of the expansion, seven options are provided:
multidimensional integration by a tensorproduct of Gaussian quadrature rules (specified with
quadrature_order
, and, optionally,dimension_preference
).multidimensional integration by the Smolyak sparse grid method (specified with
sparse_grid_level
and, optionally,dimension_preference
)multidimensional integration by Stroud cubature rules and extensions as specified with
cubature_integrand
.multidimensional integration by Latin hypercube sampling (specified with
expansion_order
andexpansion_samples
).linear regression (specified with
expansion_order
and eithercollocation_points
orcollocation_ratio
), using either overdetermined (least squares) or underdetermined (compressed sensing) approaches.orthogonal least interpolation (specified with
orthogonal_least_interpolation
andcollocation_points
)coefficient import from a file (specified with
import_expansion_file
). The expansion can be comprised of a general set of expansion terms, as indicated by the multiindex annotation within the file.
It is important to note that, for polynomial chaos using a single
model fidelity, quadrature_order
, sparse_grid_level
, and
expansion_order
are scalar inputs used for a single expansion
estimation. These scalars can be augmented with a
dimension_preference
to support anisotropy across the random dimension
set. This differs from the use of sequence arrays in advanced use
cases such as multilevel and multifidelity polynomial chaos, where
multiple grid resolutions can be employed across a model hierarchy.
Active Variables
The default behavior is to form expansions over aleatory
uncertain continuous variables. To form expansions
over a broader set of variables, one needs to specify
active
followed by state
, epistemic
, design
, or all
in the variables specification block.
For continuous design, continuous state, and continuous epistemic uncertain variables included in the expansion, Legendre chaos bases are used to model the bounded intervals for these variables. However, these variables are not assumed to have any particular probability distribution, only that they are independent variables. Moreover, when probability integrals are evaluated, only the aleatory random variable domain is integrated, leaving behind a polynomial relationship between the statistics and the remaining design/state/epistemic variables.
Covariance type (Group 3)
These two keywords are used to specify how this method computes, stores, and outputs the covariance of the responses. In particular, the diagonal covariance option is provided for reducing postprocessing overhead and output volume in high dimensional applications.
Optional Keywords regarding method outputs
Each of these sampling specifications refer to sampling on the PCE approximation for the purposes of generating approximate statistics.
sample_type
samples
seed
fixed_seed
rng
probability_refinement
distribution
reliability_levels
response_levels
probability_levels
gen_reliability_levels
which should be distinguished from simulation sampling for generating
the PCE coefficients as described in options 4, 5, and 6 above
(although these options will share the sample_type
, seed
, and
rng
settings, if provided).
When using the probability_refinement
control, the number of
refinement samples is not under the user’s control (these evaluations
are approximationbased, so management of this expense is less
critical). This option allows for refinement of probability and
generalized reliability results using importance sampling.
Usage Tips
If n is small (e.g., two or three), then tensorproduct Gaussian quadrature is quite effective and can be the preferred choice. For moderate to large n (e.g., five or more), tensorproduct quadrature quickly becomes too expensive and the sparse grid and regression approaches are preferred. Random sampling for coefficient estimation is generally not recommended due to its slow convergence rate.
For incremental studies, approaches 4 and 5 support reuse of previous
samples through the incremental_lhs
and reuse_points
specifications, respectively.
In the quadrature and sparse grid cases, growth rates for nested and
nonnested rules can be synchronized for consistency. For a
nonnested Gauss rule used within a sparse grid, linear
onedimensional growth rules of \(m=2l+1\) are used to enforce odd
quadrature orders, where l is the grid level and m is the number
of points in the rule. The precision of this Gauss rule is then
\(i=2m1=4l+1\) . For nested rules, order growth with level is
typically exponential; however, the default behavior is to restrict
the number of points to be the lowest order rule that is available
that meets the onedimensional precision requirement implied by either
a level l for a sparse grid ( \(i=4l+1\) ) or an order m for a
tensor grid ( \(i=2m1\) ). This behavior is known as “restricted
growth” or “delayed sequences.” To override this default behavior in
the case of sparse grids, the unrestricted
keyword can be used; it
cannot be overridden for tensor grids using nested rules since it also
provides a mapping to the available nested rule quadrature orders. An
exception to the default usage of restricted growth is the
dimension_adaptive
p_refinement
generalized
sparse grid case
described previously, since the ability to evolve the index sets of a
sparse grid in an unstructured manner eliminates the motivation for
restricting the exponential growth of nested rules.
Additional Resources
Dakota provides access to PCE methods through the NonDPolynomialChaos class. Refer to Uncertainty Quantification and Stochastic Expansion Methods for additional information on the 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:
Examples
method,
polynomial_chaos
sparse_grid_level = 2
samples = 10000 seed = 12347 rng rnum2
response_levels = .1 1. 50. 100. 500. 1000.
variance_based_decomp