multifidelity_stoch_collocation

Multifidelity uncertainty quantification using stochastic collocation

Specification

  • Alias: None

  • Arguments: None

Child Keywords:

Required/Optional

Description of Group

Dakota Keyword

Dakota Keyword Description

Optional (Choose One)

Automated Refinement Type

p_refinement

Automatic polynomial order refinement

h_refinement

Employ h-refinement to refine the grid

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)

Interpolation Grid Type

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

Optional (Choose One)

Basis Polynomial Family

piecewise

Use piecewise local basis functions

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

use_derivatives

Use derivative data to construct surrogate models

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 stoch_collocation, stochastic collocation is a general framework for approximate representation of random response functions in terms of finite-dimensional interpolation bases, using interpolation polynomials that may be either local or global and either value-based or gradient-enhanced.

In the multifidelity case, we decompose this interpolant 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:

\[R = \sum_{i=1}^{N_p^{LF}} r^{LF}_i L_i(\xi) + \sum_{i=1}^{N_p^{HF}} \delta_i L_i(\xi)\]

where \(\delta_i\) is a coefficient for the discrepancy expansion.

The same specification options are available as described in stoch_collocation with one key difference: 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 \(r_i\) and \(\delta_i\) for each of the expansions, the following options are provided:

  1. multidimensional integration by a tensor-product of Gaussian quadrature rules (specified with quadrature_order_sequence, and, optionally, dimension_preference).

  2. multidimensional integration by the Smolyak sparse grid method (specified with sparse_grid_level_sequence and, optionally, dimension_preference)

It is important to note that, while quadrature_order_sequence and sparse_grid_level_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 these scalar specifications.

Multifidelity UQ using SC requires that the model selected for iteration by the method specification is an ensemble surrogate model (see ensemble), which defines an 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.

Additional Resources

Dakota provides access to multifidelity SC methods through the NonDMultilevelStochCollocation class. Refer to the Stochastic Expansion Methods chapter of the Theory Manual [ABD+23] for additional information on the Multifidelity SC 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,
 multifidelity_stoch_collocation
   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