multilevel_polynomial_chaos

Multilevel uncertainty quantification using polynomial chaos expansions

Specification

  • Alias: None

  • Arguments: 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

convergence_tolerance

Stopping criterion based on objective function or statistics convergence

Optional

metric_scale

define scaling of statistical metrics when adapting UQ surrogates

Optional

discrepancy_emulation

Formulation for emulation of model discrepancies.

Required (Choose One)

Coefficient Computation Approach

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

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

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_sequence

Sequence of seed values for 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 [ABD+23] 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 ensemble
   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.