Dakota Reference Manual: multilevel_multifidelity_sampling

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multilevel_multifidelity_sampling

Multilevel-Multifidelity sampling methods for UQ

## Specification

Alias: multilevel_multifidelity_mc mlmfmc

Argument(s): 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 | 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_moments | Output moments of the specified type and include them within the set of final statistics.
| Optional | distribution | Selection of cumulative or complementary cumulative functions
| 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 both multilevel and multifidelity relationships within a hierarchical surrogate model in order to improve efficiency through variance reduction.

In the case of a multilevel relationship, multilevel Monte Carlo methods are used to compute an optimal sample allocation per level, and in the case of a multifidelity relationship, control variate Monte Carlo methods are used to compute an optimal sample allocation per fidelity. These two approaches can also be combined, resulting in the multilevel-multifidelity sampling approach below.

Multilevel Control Variate Monte Carlo

If both multilevel and multifidelity structure are included within a hierarchical model specification, then an inner control variate can be applied across two model fidelities for each level within an outer multilevel approach.

On each level, a control variate is active for the discrepancy based on

where .

The optimal parameter is computed from the correlation properties between model forms and discretization levels (see the theory manual for further details) and the optimal allocation (per level) is finally obtained from it.

Default Behavior

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

Expected Output

The `multilevel_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 `multilevel_multifidelity_sampling` method must be used in combination with a hierarchical model specification. The highest and lowest fidelity model must provide multiple discretization levels, for which 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

The following method block

    method,
        model_pointer = 'HIERARCH'
        multilevel_multifidelity_sampling
          pilot_samples = 20 seed = 1237        #s0,#s1,#s2,#s3,#p0,#p1
          convergence_tolerance = .01           #s0,#s2,#s3,#p0,#p1

specifies a multilevel-multifidelity Monte Carlo study in combination with the model identified by the HIERARCH pointer. This model specification provides a two-dimensional hierarchy, comprised of two model forms each with four discretization levels:

    model,
        id_model = 'HIERARCH'
        surrogate hierarchical
          ordered_model_fidelities = 'LF' 'HF'

    model,
        id_model = 'LF'
        simulation
          solution_level_control = 'N_x'
          solution_level_cost = 375. 10125. 81000. 648000.

    model,
        id_model = 'HF'
        simulation
          solution_level_control = 'N_x'
          solution_level_cost = 5.67e+5 4.536e+6 2.1e+7 1.68e+8

Refer to `dakota/test/dakota_uq_heat_eq_mlcvmc.in` in the source distribution for this case as well as additional examples.

## See Also

These keywords may also be of interest:

- multilevel_sampling
- multifidelity_sampling