Dakota Reference Manual  Version 6.12
Explore and Predict with Confidence
 All Pages
multilevel_polynomial_chaos


Multilevel uncertainty quantification using polynomial chaos expansions

Specification

Alias: none

Argument(s): none

Child Keywords:

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

Number of iterations allowed for optimizers and adaptive UQ methods

Optional allocation_control

Sample allocation approach for multilevel expansions

Optional discrepancy_emulation

Formulation for emulation of model discrepancies.

Required
(Choose One)
Coefficient Computation Approach (Group 1) expansion_order_sequence

Sequence of expansion orders used in a multi-stage expansion

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

Stopping criterion based on objective function or statistics convergence

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 seed_sequence

Sequence of seed values for a multi-stage random sampling

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

Description

As described in polynomial_chaos, the polynomial chaos expansion (PCE) is a general framework for the approximate representation of random response functions in terms of 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.

In the multilevel and multifidelity cases, we decompose this expansion into several constituent expansions, one per model form or solution control level. In a bi-fidelity case with low-fidelity (LF) and high-fidelity (HF) models, we have:

\[R = \sum_{i=0}^{P^{LF}} \alpha^{LF}_i \Psi_i(\xi) + \sum_{i=0}^{P^{HF}} \delta_i \Psi_i(\xi) \]

where $\delta_i$ is a coefficient for the discrepancy expansion.

For the case of regression-based PCE (least squares, compressed sensing, or orthogonal least interpolation), 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/share/dakota/test/dakota_uq_diffusion_mlpce.in, and will be stabilized in future releases.

Additional Resources

Dakota provides access to multilevel PCE methods through the NonDMultilevelPolynomialChaos class. Refer to the Stochastic Expansion Methods chapter of the Theory Manual[4] for additional information on the Multilevel 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:

In addition, the execution group has the attribute equiv_hf_evals, which records the equivalent number of high-fidelity evaluations.

Examples

method,
    multilevel_polynomial_chaos
      model_pointer = 'HIERARCH'
      pilot_samples = 10
      expansion_order_sequence = 2
      collocation_ratio = .9
      seed = 1237
      orthogonal_matching_pursuit
      convergence_tolerance = .01
    output silent

model,
    id_model = 'HIERARCH'
    surrogate hierarchical
      ordered_model_fidelities = 'SIM1'
      correction additive zeroth_order

model,
    id_model = 'SIM1'
    simulation
      solution_level_control = 'mesh_size'
      solution_level_cost = 1. 8. 64. 512. 4096.

See Also

These keywords may also be of interest: