Dakota Reference Manual
Version 6.4
LargeScale Engineering Optimization and Uncertainty Analysis

Multilevel methods for samplingbased UQ
Alias: multilevel_mc
Argument(s): none
Required/Optional  Description of Group  Dakota Keyword  Dakota Keyword Description  

Optional  seed  Seed of the random number generator  
Optional  pilot_samples  Initial set of samples for multilevel sampling methods.  
Optional  sample_type  Selection of sampling strategy  
Optional  max_iterations  Stopping criterion based on number of refinement iterations within the multilevel sample allocation  
Optional  convergence_tolerance  Stopping criterion based on relative error  
Optional  distribution  Selection of cumulative or complementary cumulative functions  
Optional  probability_levels  Specify probability levels at which to estimate the corresponding response value  
Optional  gen_reliability_levels  Specify generalized relability levels at which to estimate the corresponding response value  
Optional  rng  Selection of a random number generator  
Optional  model_pointer  Identifier for model block to be used by a method 
A nascent sampling method that utilizes both multifidelity and multilevel relationships within a hierarchical surrogate model in order to improve convergence behavior in sampling methods.
In the case of a multilevel relationship, multilevel Monte Carlo methods are used to compute an optimal sample allocation per level, and in the case of a multifidelity relationship, control variate Monte Carlo methods are used to compute an optimal sample allocation per fidelity. These two approaches can also be combined, resulting in the three approaches below.
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 .
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 opposed to the traditional control variate approach, we do not know precisely, but rather etimate 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.
Multilevel Control Variate Monte Carlo
Experimental: if both multifidelity and multilevel structure are included within the hierarchical model specification, then control variates can be applied across fidelities for each level within an outer multilevel approach.
An initial prototype for multilevel control variate MC can be explored using dakota/test/dakota_uq_heat_eq_mlcvmc.in
, and will be stabilized in future releases.
Default Behavior
The multilevel sampling method employs Monte Carlo sampling be default, but this default can be overriden to use Latin hypercube sampling 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.
Usage Tips
The multilevel sampling method must be used in combination with a hierarchical model specification. When exploiting multiple discretization levels, it is necessary to identify the variable string identifier that controls these levels using solution_level_control
. Associated relative costs also need to be supplied 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
results in multilevel Monte Carlo when the HIERARCH model specification contains a single model fidelity with multiple discretization levels, in control variate Monte Carlo when the HIERARCH model specification has multiple ordered model fidelities each with a single discretization level, and multilevel control variate Monte Carlo when the HIERARCH model specification contains multiple model fidelities each with multiple discretization levels.
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_heat_eq_{mlmc
,cvmc,mlcvmc}.in for additional examples.
These keywords may also be of interest: