Dakota Reference Manual
Version 6.16
Explore and Predict with Confidence

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 multistage 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
In a multilevel method with levels, we replace this estimator with a telescoping sum:
This decomposition forms discrepancies for each level greater than 0, seeking reduction in the variance of the discrepancy relative to the variance of the original response . The number of samples allocated for each level ( ) is based on a total cost minimization procedure that incorporates the relative cost and observed variance for each of the .
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 highfidelity 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
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 onedimensional 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.
These keywords may also be of interest: