multilevel_blue
The multilevel best linear unbiased estimator (ML BLUE) sampling method for UQ
Specification
Alias: None
Arguments: None
Child Keywords:
Required/Optional 
Description of Group 
Dakota Keyword 
Dakota Keyword Description 

Optional 
Reduce the number of groups in multilevel BLUE using a throttle 

Optional 
Initial set of samples for groups in the multilevel BLUE sampling method 

Optional 
Solution mode for multilevel/multifidelity methods 

Optional (Choose One) 
Optimization Solver 
Use a sequential quadratic programming method for solving an optimization subproblem 

Use a nonlinear interior point method for solving an optimization subproblem 

Use a hybrid globallocal scheme for solving an optimization subproblem 

Use a competed local solver scheme for solving an optimization subproblem 

Optional 
Sequence of seed values for multistage random sampling 

Optional 
Reuses the same seed value for multiple random sampling sets 

Optional 
Selection of sampling strategy 

Optional 
Enable export of multilevel/multifidelity sample sequences to individual files 

Optional 
Stopping criterion based on relative error reduction 

Optional 
Number of iterations allowed for optimizers and adaptive UQ methods 

Optional 
Stopping criterion based on maximum function evaluations 

Optional 
Selection of a random number generator 

Optional 
Identifier for model block to be used by a method 
Description
An adaptive multifidelity sampling method that improves performance relative to singlefidelity Monte Carlo sampling, either of terms of greater accuracy (reduced variance in the estimated statistics) for a prescribed budget or reduced cost for specified accuracy. It employs an ensemble model to manage a set of lowerfidelity approximations to a single truth model.
Compared to other estimators (MLMC, MLCV MC, MFMC, ACV, generalized ACV), ML BLUE is distinguished in that it employs a groupbased approach, where independent samples are allocated for each group and each group is composed of unordered combinations of models. The number of groups grows combinatorially with the total number of models, and a few throttle options are provided to prevent the number of groups from growing to excess.
As described in [SU20], the ML BLUE estimator for QoI expected values is
where
given group sample count \(m_k\), group restriction operator \(R_k\), group covariance estimate \(C_k\), and group QoI sum \(S_k\) for each group \(k\). For a linear combination of the model means \(q_{\beta}^B = \beta^T q^B\), the variance of ML BLUE is \(\beta^T \Psi^{1} \beta\). The numerical solution for \(m\) minimizes this variance subject to a prescribed budget (or minimizes cost for specified accuracy), where the common choice of \(\beta = [1, 0, 0, \dots]^T\) targets the HF mean.
Status
This method is currently under active development. It exhibits competitive performance for smaller numbers of models (roughly 5 or less), whereas larger model ensembles lead to illconditioning in the matrix solutions and inaccurate results. Throttling can delay these problems in some cases, but more advanced numerical treatments are needed (work in progress).
Default Behavior
The multilevel_blue
method employs a number of important default settings:
The pilot sampling strategy involves shared samples for initial estimation of group covariances (refer to
pilot_samples
).The number of groups is not throttled by default (refer to
group_throttle
).The default solver is
global_local
, starting with the DIRECT global solver and proceeding to available local solvers (SQP and NIP) in competition. For larger group counts, a purely local approach can be more scalable.Solution mode will be
online_pilot
, an approach which iterates toward a set of shared samples whose size is consistent with the optimal allocation. Since groupbased approaches will try to allocate the entire budget on the first iteration, the use of underrelaxation (seerelaxation
) can be especially beneficial.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_blue
method reports estimates of the first four
moments and a summary of the evaluations performed for each group and
for each model instance. The method does not support any level
mappings (response, probability, reliability, generalized reliability)
at this time.
Usage Tips
The multilevel_blue
method must be used in combination with an
ensemble model specification that enumerates a truth model and
approximation models using either a model form ensemble, a set of
resolution levels, or some combination. By default, all model forms
and resolution levels will be enumerated within the model ensemble.
For each model instance, cost data must by provided either using
solution_level_cost
or metadata that is returned from simulations
for estimating cost on the fly. For a sequence of discretization
levels, solution_level_control
must identify the variable string
descriptor that controls the resolution levels and the associated
array of relative costs must be provided using
solution_level_cost
.
Examples
The following method block:
method,
model_pointer = 'HIERARCH'
multilevel_blue
solution_mode online_pilot
relaxation factor_sequence = .5 .75 1.
pilot_samples = 25 #independent
seed = 8674132
max_function_evaluations = 500
specifies ML BLUE using an iterated online pilot in combination with the default optimization solver strategy (global_local), the default shared pilot estimation of group covariances, and an ensemble model identified by the HIERARCH pointer.
This HIERARCH model specification provides a onedimensional sequence, here defined by a set of 3 discretization levels:
model,
id_model = 'HIERARCH'
variables_pointer = 'HF_VARS'
surrogate ensemble
truth_model = 'HF'
model,
id_model = 'HF'
variables_pointer = 'HF_VARS'
interface_pointer = 'HF_INT'
simulation
solution_level_control = 'mesh_size'
solution_level_cost = 1 16 256
Refer to dakota/test/dakota_uq_diffusion_mlblue_cost4
.in within
the source distribution for this case as well as additional examples.
Theory
Refer to [SU20] for more detailed algorithm descriptions, theoretical considerations, and illustrative examples.