multilevel_sampling

Multilevel Monte Carlo (MLMC) sampling method 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		weighted	Add control variate weights to the recursive differences using in multilevel Monte Carlo
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		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 an ensemble 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 \(Y_\ell\) .

Weighting and Model Selection

Similar to MFMC (see multifidelity_sampling), weighted MLMC is a special case of generalized ACV using the ACV-RD sampling scheme ([BLWL22]) in combination with a hierarchical DAG (each approximation node points to the next approximation of higher fidelity, ending with the truth model at the root node). As such, the selection of a weighted MLMC approach is promoted to the generalized ACV solver in order to gain access to both weighting and optional model selection capabilities by activating the weighted and model_selection options, respectively. This results in one or more numerical solutions for a fixed hierarchical DAG.

Default Behavior

The multilevel_sampling method employs a number of important default settings:

• By default, MLMC is unweighted and provides a convenient analytic solution for the optimal sample allocations. The option of weighted MLMC is also available as a numerical solution and can be combined with model selection as described above.

• Solution mode will be online_pilot, an approach which iterates toward a set of shared samples whose size is consistent with the optimal allocation.

• Monte Carlo sample sets are used by default and are most consistent with the underlying theory, but this default can be overridden to use Latin hypercube sample sets using sample_type lhs. Allocations remain governed by Monte Carlo variance for all cases.

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:

• 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 an ordered ensemble surrogate 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. An alternative to prescribing the cost profile is estimating it on the fly using cost metadata that is returned from the different simulation instances.

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 ensemble
   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.