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# 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 | search_model_graphs | Perform a search over admissible model relationships for a given model ensemble
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.

Multifidelity sampling is a recursive sampling scheme for which model ordering is important. In the case of an ensemble surrogate model with more than two model instances (either in terms of model forms or resolutions), the multi-model approach of Peherstorfer et al. (2016) is supported for which all model instances can be integrated into the scheme. In the special case of two model instances, this collapses to the approach of Ng and Willcox (2014). The approach can be used with either a model form sequence or a resolution level sequence, but not both.

Control Variate Monte Carlo

In the case of two model instances (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 `hdf5` keyword, this method writes the following results to HDF5:

- 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 an ensemble 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`.

Examples

We provide an example of a multifidelity Monte Carlo study using an ensemble 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 truth model and a set of unordered approximation models, each with a single (or default) discretization level:

    model,
     id_model = 'NONHIER'
     surrogate ensemble
       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.

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