multifidelity_polynomial_chaos

Multifidelity uncertainty quantification using polynomial chaos expansions

Specification

Alias: None
Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional		p_refinement	Automatic polynomial order refinement
Optional		max_refinement_iterations	Maximum number of expansion refinement iterations
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		statistics_mode	type of statistical metric roll-up for multifidelity UQ methods
Optional		allocation_control	Sample allocation approach for multifidelity expansions
Optional		discrepancy_emulation	Formulation for emulation of model discrepancies.
Required (Choose One)	Chaos Coefficient Estimation Approach	quadrature_order_sequence	Sequence of quadrature orders used in a multi-stage expansion
		sparse_grid_level_sequence	Sequence of sparse grid levels used in a multi-stage expansion
		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.
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 and an additive discrepancy approach, 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.

The same specification options are available as described in polynomial_chaos with one key difference: many of the coefficient estimation inputs change from a scalar input for a single expansion to a <i>sequence</i> specification for a low-fidelity expansion followed by multiple discrepancy expansions.

To obtain the coefficients \(\alpha_i\) and \(\delta_i\) for each of the expansions, the following options are provided:

multidimensional integration by a tensor-product of Gaussian quadrature rules (specified with quadrature_order_sequence, and, optionally, dimension_preference).
multidimensional integration by the Smolyak sparse grid method (specified with sparse_grid_level_sequence and, optionally, dimension_preference)
multidimensional integration by Latin hypercube sampling (specified with expansion_order_sequence and expansion_samples_sequence).
linear regression (specified with expansion_order_sequence and either collocation_points_sequence or collocation_ratio), using either over-determined (least squares) or under-determined (compressed sensing) approaches.
orthogonal least interpolation (specified with orthogonal_least_interpolation and collocation_points_sequence)

It is important to note that, while quadrature_order_sequence, sparse_grid_level_sequence, expansion_order_sequence, expansion_samples_sequence, and collocation_points_sequence are array inputs, only one scalar from these arrays is active at a time for a particular expansion estimation. In order to specify anisotropy in resolution across the random variable set, a dimension_preference specification can be used to augment scalar specifications for quadrature order, sparse grid level, and expansion order.

Multifidelity UQ using PCE requires that the model selected for iteration by the method specification is an ensemble surrogate model (see ensemble), which defines an ordered sequence of model fidelities or resolutions. Two types of hierarchies are supported: (i) a hierarchy of model forms composed from more than one model within the ordered_model_fidelities specification, or (ii) a hierarchy of discretization levels comprised from a single model (either from a truth_model_pointer specification or a single entry within an ordered_model_fidelities specification) which in turn specifies a solution_level_control (see solution_level_control).

In both cases, an expansion will first be formed for the low fidelity model or coarse discretization, using the first value within the coefficient estimation sequence, 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 coefficient estimation sequence (if the sequence does not provide a new value, then the previous value is reused) along with any specified refinement strategy. The number of discrepancy expansions is determined by the number of model forms or discretization levels in the hierarchy.

After formation and refinement of the constituent expansions, each of the expansions is 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.

Additional Resources

Dakota provides access to multifidelity PCE methods through the NonDMultilevelPolynomialChaos class. Refer to the Stochastic Expansion Methods chapter of the Theory Manual [ABD+23] for additional information on the Multifidelity 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,
 multifidelity_polynomial_chaos
   model_pointer = 'HIERARCH'
   sparse_grid_level_sequence = 4 3 2

model,
 id_model = 'HIERARCH'
 surrogate ensemble
   ordered_model_fidelities = 'LF' 'MF' 'HF'
   correction additive zeroth_order

multifidelity_polynomial_chaos

Exceptional service in the national interest