multifidelity_sampling

Multifidelity sampling methods for UQ

Specification

• Alias: multifidelity_mc mfmc

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
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		pilot_samples	Initial set of samples for multilevel/multifidelity sampling methods.
Optional		solution_mode	Solution mode for multilevel/multifidelity methods
Optional		numerical_solve	Specify the situations where numerical optimization is used for MFMC sample allocation
Optional		sample_type	Selection of sampling strategy
Optional		export_sample_sequence	Enable export of multilevel/multifidelity sample sequences to individual files
Optional		convergence_tolerance	Stopping criterion based on relative error reduction
Optional		max_iterations	Number of iterations allowed for optimizers and adaptive UQ methods
Optional		max_function_evaluations	Stopping criterion based on maximum function evaluations
Optional		final_statistics	Indicate the type of final statistics to be returned by a UQ method
Optional		rng	Selection of a random number generator
Optional		model_pointer	Identifier for model block to be used by a method

Description

An adaptive sampling method that utilizes multifidelity relationships in order to improve efficiency through variance reduction.

Two variants are currently supported, with the former now deprecated and to be replaced by the latter:

• In the case of a hierarchical surrogate model, the two-model approach of Ng and Willcox (2014) is supported and the two most extreme model fidelities or resolutions are employed as the truth model and approximation model.

• In the case of a non-hierarchical surrogate model, the multi-model approach of Peherstorfer et al. (2016) is supported for which all model instances can be integrated into the scheme. Both methods can be used with either a model form sequence or a resolution level sequence.

Control Variate Monte Carlo

In the case of two model fidelities (low fidelity denoted as LF and high fidelity denoted as HF), we employ a control variate approach as described in Ng and Willcox (2014):

\[\hat{Q}_{HF}^{CV} = \hat{Q}_{HF}^{MC} - \beta (\hat{Q}_{LF}^{MC} - \mathbb{E}[Q_{LF}])\]

As opposed to the traditional control variate approach, we do not know \(\mathbb{E}[Q_{LF}]\) precisely, but rather we estimate it more accurately than \(\hat{Q}_{LF}^{MC}\) based on a sampling increment applied to the LF model. This sampling increment is based again on a total cost minimization procedure that incorporates the relative LF and HF costs and the observed Pearson correlation coefficient \(\rho_{LH}\) between \(Q_{LF}\) and \(Q_{HF}\) . The coefficient \(\beta\) is then determined from the observed LF-HF covariance and LF variance.

Multifidelity Monte Carlo

This approach can be extended to a sequence of low-fidelity approximations using a recusive sampling approach as in Peherstorfer et al. (2016).

\[\hat{Q}_{HF}^{CV} = \hat{Q}_{HF}^{MC} - \sum_{i=1}^M \beta_i (\hat{Q}_{LF_i}^{MC} - \mathbb{E}[Q_{LF_i}])\]

In this case, the variance in the estimate of the \(i^{th}\) control mean is reduced by the \((i+1)^{th}\) control variate, such that the variance reduction is limited by the case of an exact estimate of the first control mean (referred to as OCV-1 in Gorodetsky et al., 2020).

Default Behavior

The multifidelity_sampling method employs Monte Carlo sample sets by default, but this default can be overridden to use Latin hypercube sample sets using sample_type lhs.

Expected Output

The multifidelity_sampling method reports estimates of the first four moments and a summary of the evaluations performed for each model fidelity and discretization level. The method does not support any level mappings (response, probability, reliability, generalized reliability) at this time.

Expected HDF5 Output

If Dakota was built with HDF5 support and run with the environment-results_output-hdf5 keyword, this method writes the following results to HDF5:

• hdf5_results-sampling_moments (moments only, not confidence intervals)

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

Usage Tips

The multifidelity_sampling method can be used in combination with either a hierarchical or non-hierarchical model specification for either a model form sequence or a discretization level sequence. For a model form sequence, each model must provide a scalar solution_level_cost. For a discretization level sequence, it is necessary to identify the variable string descriptor that controls the resolution levels using solution_level_control as well as the associated array of relative costs using solution_level_cost.

The hierarchical two-model approach is a special case of the non-hierarchical multi-model approach. The latter gives identical results to the former when restricted to one approximation model; as such, the hierarchical two-model approach is deprecated.

Examples

We provide an example of a multifidelity Monte Carlo study using a non-hierarchical model specification employing multiple approximations.

The following method block:

method,
 model_pointer = 'NONHIER'
 multifidelity_sampling
   pilot_samples = 20 seed = 1237
   max_iterations = 10
   convergence_tolerance = .001

specifies MFMC in combination with the model identified by the NONHIER pointer.

This NONHIER model specification provides a one-dimensional sequence, here defined by a single truth model and a set of unordered approximation models, each with a single (or default) discretization level:

model,
 id_model = 'NONHIER'
 surrogate non_hierarchical
   truth_model = 'HF'
   unordered_model_fidelities = 'LF1' 'LF2'

model,
 id_model = 'LF1'
 interface_pointer = 'LF1_INT'
 simulation
   solution_level_cost = 0.01

model,
 id_model = 'LF2'
 interface_pointer = 'LF2_INT'
 simulation
   solution_level_cost = 0.1

model,
 id_model = 'HF'
 interface_pointer = 'HF_INT'
 simulation
   solution_level_cost = 1.

Refer to dakota/test/dakota_uq_*_cvmc.in and dakota/test/dakota_uq_*_mfmc.in in the source distribution for additional examples.