Dakota Reference Manual
Version 6.16
Explore and Predict with Confidence

Multifidelity sampling methods for UQ
Alias: multifidelity_mc mfmc
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/multifidelity sampling methods.  
Optional  solution_mode  Solution mode for multilevel/multifidelity methods  
Optional  numerical_solve  Specify the situations where numerical optimization is used for MFMC sample allocation  
Optional  sample_type  Selection of sampling strategy  
Optional  export_sample_sequence  Enable export of multilevel/multifidelity sample sequences to individual files  
Optional  convergence_tolerance  Stopping criterion based on relative error reduction  
Optional  max_iterations  Number of iterations allowed for optimizers and adaptive UQ methods  
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 multifidelity relationships in order to improve efficiency through variance reduction.
Two variants are currently supported, with the former now deprecated and to be replaced by the latter:
Control Variate Monte Carlo
In the case of two model fidelities (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 precisely, but rather we estimate it more accurately than 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 between and . The coefficient is then determined from the observed LFHF covariance and LF variance.
Multifidelity Monte Carlo
This approach can be extended to a sequence of lowfidelity approximations using a recusive sampling approach as in Peherstorfer et al. (2016).
In this case, the variance in the estimate of the control mean is reduced by the control variate, such that the variance reduction is limited by the case of an exact estimate of the first control mean (referred to as OCV1 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:
In addition, the execution group has the attribute equiv_hf_evals
, which records the equivalent number of highfidelity evaluations.
Usage Tips
The multifidelity_sampling
method can be used in combination with either a hierarchical or nonhierarchical 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
.
The hierarchical twomodel approach is a special case of the nonhierarchical multimodel approach. The latter gives identical results to the former when restricted to one approximation model; as such, the hierarchical twomodel approach is deprecated.
We provide an example of a multifidelity Monte Carlo study using a nonhierarchical 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 onedimensional sequence, here defined by a single truth model and a set of unordered approximation models, each with a single (or default) discretization level:
model, id_model = 'NONHIER' surrogate non_hierarchical 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.
These keywords may also be of interest: