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Dakota Reference Manual
Version 6.15
Explore and Predict with Confidence
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Uncertainty quantification using polynomial chaos expansions
Alias: nond_polynomial_chaos
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 | convergence_tolerance | Stopping criterion based on objective function or statistics convergence | ||
Optional | metric_scale | define scaling of statistical metrics when adapting UQ surrogates | ||
Required (Choose One) | Chaos coefficient estimation approach (Group 1) | quadrature_order | Order for tensor-products of Gaussian quadrature rules | |
sparse_grid_level | Level to use in sparse grid integration or 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 multi-index from a file | |||
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 user-specified 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 multi-index 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 | import_approx_points_file | Filename for points at which to evaluate the PCE/SC surrogate | ||
Optional | export_approx_points_file | Output file for surrogate model value evaluations | ||
Optional | seed | Seed of the random number generator | ||
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 |
The polynomial chaos expansion (PCE) is a general framework for the approximate representation of random response functions in terms of finite-dimensional 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 over-determined (least squares) or under-determined (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 multi-index 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 post-processing 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 approximation-based, 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 tensor-product Gaussian quadrature is quite effective and can be the preferred choice. For moderate to large n (e.g., five or more), tensor-product 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 non-nested rules can be synchronized for consistency. For a non-nested Gauss rule used within a sparse grid, linear one-dimensional 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 one-dimensional 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[4] and the Stochastic Expansion Methods chapter of the Theory Manual[15] 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:
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: