# 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 |
Sequence of seed values for multi-stage random sampling |
||

Optional |
Reuses the same seed value for multiple random sampling sets |
||

Optional |
Initial set of samples for multilevel/multifidelity sampling methods. |
||

Optional |
Solution mode for multilevel/multifidelity methods |
||

Optional |
Specify the situations where numerical optimization is used for MFMC sample allocation |
||

Optional |
Perform a search over admissible model relationships for a given model ensemble |
||

Optional |
Selection of sampling strategy |
||

Optional |
Enable export of multilevel/multifidelity sample sequences to individual files |
||

Optional |
Stopping criterion based on relative error reduction |
||

Optional |
Number of iterations allowed for optimizers and adaptive UQ methods |
||

Optional |
Stopping criterion based on maximum function evaluations |
||

Optional |
Indicate the type of final statistics to be returned by a UQ method |
||

Optional |
Selection of a random number generator |
||

Optional |
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):

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).

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.