Dakota Reference Manual  Version 6.16
Explore and Predict with Confidence
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Multilevel sampling methods for UQ


Alias: multilevel_mc mlmc

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 sampling methods.

Optional solution_mode

Solution mode for multilevel/multifidelity methods

Optional sample_type

Selection of sampling strategy

Optional export_sample_sequence

Enable export of multilevel/multifidelity sample sequences to individual files

Optional allocation_target

Allocation statistics/target for the MLMC sample allocation.

Optional qoi_aggregation

Aggregation strategy for the QoIs statistics for problems with multiple responses in the MLMC algorithm

Optional convergence_tolerance

Stopping criterion based on relative error

Optional convergence_tolerance_type

Sets the type of the convergence tolerance. Can be absolute or relative.

Default Behavior "relative"

Optional convergence_tolerance_target

Sets the target of the MLMC sample allocation, i.e. the constraint of the MLMC sample allocation problem

Default Behavior "variance_constraint"

Optional max_iterations

Stopping criterion based on number of refinement iterations within the multilevel sample allocation

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


An adaptive sampling method that utilizes multilevel 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.

Multilevel Monte Carlo

The Monte Carlo estimator for the mean is defined as

\[ \mathbb{E}[Q] \equiv \hat{Q}^{MC} = \frac{1}{N} \sum_{i=1}^N Q^{(i)} \]

In a multilevel method with $L$ levels, we replace this estimator with a telescoping sum:

\[ \mathbb{E}[Q] \equiv \hat{Q}^{ML} = \sum_{l=0}^L \frac{1}{N_l} \sum_{i=1}^{N_l} (Q_l^{(i)} - Q_{l-1}^{(i)}) \equiv \sum_{l=0}^L \hat{Y}^{MC}_l \]

This decomposition forms discrepancies for each level greater than 0, seeking reduction in the variance of the discrepancy $Y$ relative to the variance of the original response $Q$. The number of samples allocated for each level ( $N_l$) is based on a total cost minimization procedure that incorporates the relative cost and observed variance for each of the $Y_\ell$.

Default Behavior

The multilevel_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 multilevel_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:

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

Usage Tips

The multilevel sampling method must be used in combination with a hierarchical model specification, and supports either a sequence of model forms or a sequence of discretization levels. For the former, each model form must provide a scalar solution_level_cost and for the latter, 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 following method block

    model_pointer = 'HIERARCH'
      pilot_samples = 20 seed = 1237
      max_iterations = 10
      convergence_tolerance = .001

specifies a multilevel Monte Carlo study in combination with the model identified by the HIERARCH pointer. This model specification provides a one-dimensional hierarchy, typically defined by a single model fidelity with multiple discretization levels, but may also be provided as multiple ordered model fidelities, each with a single (or default) discretization level. An example of the former (single model fidelity with multiple discretization levels) follows:

    id_model = 'HIERARCH'
    surrogate hierarchical
      ordered_model_fidelities = 'SIM1'
      correction additive zeroth_order

    id_model = 'SIM1'
      solution_level_control = 'N_x'
      solution_level_cost = 630. 1260. 2100. 4200.

Refer to dakota/test/dakota_uq_*_mlmc.in in the source distribution for additional examples.

See Also

These keywords may also be of interest: