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
Required (Choose One)	Coefficient Computation Approach	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.
Optional (Choose One)	Basis Polynomial Family	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
Optional (Choose One)	Covariance Type	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:

Integration and Expansion Moments (expansion moments only)
Probability Density
Level Mappings

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.

multilevel_polynomial_chaos

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