Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
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polynomial_chaos


Uncertainty quantification using polynomial chaos expansions

Specification

Alias: nond_polynomial_chaos

Argument(s): none

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Optional samples_on_emulator

Number of samples at which to evaluate an emulator (surrogate)

Optional seed

Seed of the random number generator

Optional fixed_seed

Reuses the same seed value for multiple random sampling sets

Optional max_iterations

Stopping criterion based on number of iterations

Optional convergence_tolerance

Stopping criterion based on convergence of the objective function or statistics

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 user-specified 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_sequence Cubature using tensor-products of Gaussian quadrature rules
sparse_grid_level_sequence

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_sequence

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 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 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 export_expansion_file

Export the coefficients and multi-index of a Polynomial Chaos Expansion (PCE) to a file

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 model_pointer

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 finite-dimensional series expansions in standardized random variables

\[R = \sum_{i=0}^P \alpha_i \Psi_i(\xi) \]

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 numerically-generated orthogonal polynomials for non-Askey probability density functions. Extended polynomial selections replace each of the sub-optimal Askey basis selections for bounded normal, lognormal, bounded lognormal, loguniform, triangular, gumbel, frechet, weibull, and bin-based 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.

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:

  1. multidimensional integration by a tensor-product of Gaussian quadrature rules (specified with quadrature_order, and, optionally, dimension_preference).
  2. multidimensional integration by the Smolyak sparse grid method (specified with sparse_grid_level and, optionally, dimension_preference)
  3. multidimensional integration by Stroud cubature rules and extensions as specified with cubature_integrand.
  4. multidimensional integration by Latin hypercube sampling (specified with expansion_order and expansion_samples).
  5. linear regression (specified with expansion_order and either collocation_points or collocation_ratio), using either over-determined (least squares) or under-determined (compressed sensing) approaches.
  6. orthogonal least interpolation (specified with orthogonal_least_interpolation and collocation_points)
  7. 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 multi-index 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 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.

Multifidelity PCE

The advanced use case of multifidelity UQ using PCE 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 low fidelity surrogate model, using the first value within the quadrature_order_sequence, sparse_grid_level_sequence, or expansion_order_sequence (if multiple values are present; the first is reused if not present) along with any specified refinement strategy. Second, expansions are formed for one or more model discrepancies (the difference between response results if additive correction or the ratio of results if multiplicative correction), using all subsequent values in the quadrature_order_sequence, sparse_grid_level_sequence, or expansion_order_sequence along with any specified refinement strategy. The number of discrepancy expansions is determined by the length of the ordered_model_sequence within the hierarchical model specification (see hierarchical). Then each of these 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.

Multilevel PCE

Experimental: For the case of regression-based PCE (either least squares or compressed sensing), 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 PCE-based estimator of the statistics, requiring approximation of the variance of these estimators.

Initial prototypes for multilevel PCE can be explored using dakota/test/dakota_uq_diffusion_mlpce.in, and will be stabilized in future releases.

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 $m=2l+1$ 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 $i=2m-1=4l+1$. 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 one-dimensional precision requirement implied by either a level l for a sparse grid ( $i=4l+1$) or an order m for a tensor grid ( $i=2m-1$). 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.

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

See Also

These keywords may also be of interest: