Dakota Reference Manual  Version 6.1
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 p_refinement Automatic polynomial order refinement
Optional
(Choose One)
Group 1 askey Select the Askey basis polynomials that match the basis random variable
wiener Use Hermite polynomial basis functions
Required
(Choose One)
Group 2 quadrature_order Cubature using tensor-products of Gaussian quadrature rules
sparse_grid_level Set the maximum sparse grid level to be used when peforming sparse grid integration Cubature using spare grids
cubature_integrand Cubature using Stroud rules and their extensions
expansion_order The maximum 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 contributions from variables

Optional
(Choose One)
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 multi-index 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 sampling-based methods

Optional seed

Seed of the 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. The following provides details on the various polynomial chaos method options in Dakota.

Groups 1 and 2, plus the optional keywords p_refinement and fixed_seed relate to the specification of a PCE method. In addition, this method treats variables that are not aleatoric-uncertain different, despite the active keyword.

Group 3, and the remainder of the optional keywords relate to the output of the method.

polynomial_chaos Group 1

Group 1 keywords are used to select the type of basis, $\Psi_i$, of the expansion. Three approaches may be employed:

  • Extended (default if no option is selected)
  • Wiener
  • Askey

The selection of Wiener versus Askey versus Extended is partially automated and partially under the user's control.

  • 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.
  • 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 with numerically-generated polynomials that are orthogonal to the prescribed probability density functions (for bounded normal, lognormal, bounded lognormal, loguniform, triangular, gumbel, frechet, weibull, and bin-based histogram).

polynomial_chaos Group 2

To obtain the coefficients $\alpha_i$ of the expansion, six options are provided:

  • multidimensional integration by a tensor-product 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_samples).
  • linear regression (specified with either collocation_points or collocation_ratio).
  • coefficient import from a file (specified with import_expansion_file). A total-order expansion is assumed and must be specified using expansion_order.

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.

polynomial_chaos Group 3

These two keywords are used to specify how this method outputs the covariance of the responses.

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
  • 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 and 5 above (although options 4 and 5 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.

Multi-fidelity 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 or sparse_grid_level value along with any specified refinement strategy. Second, an expansion will be formed for the low fidelity surrogate model, using the second quadrature_order or sparse_grid_level 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.

Theory

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. Dakota provides access to PCE methods through the NonDPolynomialChaos class. Refer to the Uncertainty Quantification Capabilities chapter of the Users Manual [[4] "Adams et al., 2010"] for additional information on the PCE algorithm.

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.

See Also

These keywords may also be of interest: