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		group_throttle	Reduce the number of groups in multilevel BLUE using a throttle
Optional		pilot_samples	Initial set of samples for groups in the multilevel BLUE sampling method
Optional		solution_mode	Solution mode for multilevel/multifidelity methods
Optional (Choose One)	Optimization Solver	sqp	Use a sequential quadratic programming method for solving an optimization sub-problem
		nip	Use a nonlinear interior point method for solving an optimization sub-problem
		global_local	Use a hybrid global-local scheme for solving an optimization sub-problem
		competed_local	Use a competed local solver scheme for solving an optimization sub-problem
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		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		rng	Selection of a random number generator
Optional		model_pointer	Identifier for model block to be used by a method

Description

An adaptive multifidelity sampling method that improves performance relative to single-fidelity 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 lower-fidelity 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 group-based 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

\[\Psi q^B = y\]

where

\[\Psi = \sum_{k=1}^K m_k R_k^T C_k^{-1} R_k\]

\[y = \sum_{k=1}^K R_k^T C_k^{-1} S_k\]

\[S_k = \sum_{i=1}^{m_k} Y_k^{(i)}\]

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 ill-conditioning 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 group-based approaches will try to allocate the entire budget on the first iteration, the use of under-relaxation (see relaxation) 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 one-dimensional 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.