multilevel_sampling

Multilevel sampling methods for UQ

Specification

• Alias: multilevel_mc mlmc

• 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 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	Select absolute or relative convergence tolerance
Optional		convergence_tolerance_target	Select target for MLMC sample allocation
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

Description

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 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 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.

Examples

The following method block

method,
 model_pointer = 'HIERARCH'
 multilevel_sampling
   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:

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

model,
 id_model = 'SIM1'
 simulation
   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.