Dakota Reference Manual
Version 6.2
LargeScale Engineering Optimization and Uncertainty Analysis

Uncertainty quantification using polynomial chaos expansions
Alias: nond_polynomial_chaos
Argument(s): none
Required/Optional  Description of Group  Dakota Keyword  Dakota Keyword Description  

Optional  p_refinement  Automatic polynomial order refinement  
Optional (Choose One)  Basis polynomial family (Group 1)  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.  
Required (Choose One)  Coefficient estimation approach (Group 2)  quadrature_order  Cubature using tensorproducts of Gaussian quadrature rules  
sparse_grid_level  Set the sparse grid level to be used when peforming sparse grid integration or sparse grid interpolation  
cubature_integrand  Cubature using Stroud rules and their extensions  
expansion_order  The (initial) order of a polynomial expansion  
orthogonal_least_interpolation  Build a polynomial chaos expansion from simulation samples using orthogonal least interpolation.  
import_expansion_file  Build a Polynomial Chaos Expansion (PCE) by import coefficients and a multiindex from a file  
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  normalized  The normalized specification requests output of PCE coefficients that correspond to normalized orthogonal basis polynomials  
Optional  sample_type  Selection of sampling strategy  
Optional  probability_refinement  Allow refinement of probability and generalized reliability results using importance sampling  
Optional  export_points_file  Output file for evaluations of a surrogate model  
Optional  export_expansion_file  Export the coefficients and multiindex of a Polynomial Chaos Expansion (PCE) to a file  
Optional  fixed_seed  Reuses the same seed value for multiple random sampling sets  
Optional  reliability_levels  Specify reliability levels at which the response values will be estimated  
Optional  response_levels  Values at which to estimate desired statistics for each response  
Optional  distribution  Selection of cumulative or complementary cumulative functions  
Optional  probability_levels  Specify probability levels at which to estimate the corresponding response value  
Optional  gen_reliability_levels  Specify generalized relability levels at which to estimate the corresponding response value  
Optional  rng  Selection of a random number generator  
Optional  samples  Number of samples for samplingbased methods  
Optional  seed  Seed of the random number generator  
Optional  model_pointer  Identifier for model block to be used by a method 
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 is a deterministic coefficient, is a multidimensional orthogonal polynomial and 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, , of the expansion. Three approaches may be employed:
For supporting correlated random variables, certain fallbacks must be implemented.
Refer to variable_support for additional information on supported variable types, with and without correlation.
Coefficient estimation approach (Group 2)
To obtain the coefficients of the expansion, seven options are provided:
quadrature_order
, and, optionally, dimension_preference
). sparse_grid_level
and, optionally, dimension_preference
) cubature_integrand
. expansion_order
and expansion_samples
). expansion_order
and either collocation_points
or collocation_ratio
), using either overdetermined (least squares) or underdetermined (compressed sensing) approaches. orthogonal_least_interpolation
and collocation_points
) 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, while quadrature_order
, sparse_grid_level
, and expansion_order
are array inputs, only one scalar from these arrays is active at a time for a particular expansion estimation. These scalars can be augmented with a dimension_preference
to support anisotropy across the random dimension set. The array inputs are present to support advanced use cases such as multifidelity UQ, where multiple grid resolutions can be employed.
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.
Multifidelity UQ
The advanced use case of multifidelity UQ automatically becomes active if the model selected for iteration by the method specification is a multifidelity surrogate model (see hierarchical). In this case, an expansion will first be formed for the model discrepancy (the difference between response results if additive
correction
or the ratio of results if multiplicative
correction
), using the first quadrature_order
, sparse_grid_level
, or expansion_order
value along with any specified refinement strategy. Second, an expansion will be formed for the low fidelity surrogate model, using the second quadrature_order
, sparse_grid_level
, or expansion_order
value (if present; the first is reused if not present) along with any specified refinement strategy. Then the two expansions are 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.
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 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 . 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 ( ) or an order m for a tensor grid ( ). 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 the Uncertainty Quantification Capabilities chapter of the Users Manual[5] and the Stochastic Expansion Methods chapter of the Theory Manual[4] for additional information on the PCE algorithm.
method, polynomial_chaos sparse_grid_level = 2 samples = 10000 seed = 12347 rng rnum2 response_levels = .1 1. 50. 100. 500. 1000. variance_based_decomp
These keywords may also be of interest: