approximate_control_variate

Approximate control variate (ACV) sampling methods for UQ

Specification

• Alias: acv_sampling

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required (Choose One)	Solution Approach	acv_independent_sampling	Sampling scheme within the approximate control variate (ACV) algorithm that employs independent samples (IS) across model pairings
		acv_multifidelity	Approximate control variate (ACV) algorithm that employs a nested pyramid sample pattern, similar to MFMC
		acv_recursive_diff	Sampling scheme within the approximate control variate (ACV) algorithm that employs weighted recursive differences (RD) across model pairings
Optional		search_model_graphs	Perform a recursion of admissible DAGs for a given model ensemble
Optional		pilot_samples	Initial set of samples for multilevel/multifidelity sampling methods.
Optional		solution_mode	Solution mode for multilevel/multifidelity methods
Optional		truth_fixed_by_pilot	Option to suppress any increment to the number of truth samples
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 sampling method that utilizes multifidelity relationships in order to improve efficiency through variance reduction. It employs an ensemble model to manage an unordered set of lower-fidelity approximations to a single truth model.

Compared to multifidelity Monte Carlo (MFMC), ACV is not a recursive emulator based on hierarchical control variate pairings. Rather, the truth model’s variance is targeted using each control variate pairing. These pairings are codified with a directed acyclic graph (DAG), such that ACV defines a “peer” DAG where all approximation nodes point to the root node and the root node identifies the reference “truth” model. MFMC, on the other hand, employs a “hierarchical” DAG, where each approximation node points to the next approximation of higher fidelity, ending with the truth model at the root node.

While the ensemble of control variates appears identical to MFMC:

\[\hat{Q}_{HF}^{CV} = \hat{Q}_{HF}^{MC} - \sum_{i=1}^M \beta_i (\hat{Q}_{LF_i}^{MC} - \mathbb{E}[Q_{LF_i}])\]

the sample patterns used for the constituent estimators differ as depicted in [GGEJ20], Figure 2, with the \(z^1\) and \(z^2\) definitions reflective of the control variate pairings.

In the generalized ACV case ([BLWL22]) available from the search_model_graphs specification, three different ACV sampling schemes are supported: ACV-MF (multifidelity), ACV-IS (independent sampling), and ACV-RD (recursive difference), where each scheme supports an enumerative search over admissible DAGs and model subset selections. Each of these three schemes is also supported in non-generalized form for specific DAG definitions, as follows:

• ACV-MF: peer DAG in approximate_control_variate and hierarchical DAG in multifidelity_sampling

• ACV-IS: peer DAG in approximate_control_variate.

• ACV-RD: hierarchical DAG in multilevel_sampling using the weighted option.

Finally, the ACV-KL variant ([GGEJ20]) is not another sampling scheme, but rather a parametrically defined estimator that enumerates a relatively small subset of admissible DAGs. It is available under the kl_recursion option, using the approaches described in [BLWL22].

Default Behavior

The approximate_control_variate method employs a number of important default settings:

• The DAG defining control variate pairings defaults to a “peer” DAG where all approximation nodes point to the root node and the root node identifies the reference “truth” model. This means that every approximation participates in a sample set that is shared with the truth model (the root allocation), and each approximation (each of the nodes connecting to the root) then has its own augmentation to this shared set. For ACV-IS and ACV-RD, these augmentations are independent. To explore other admissible DAGs, search_model_graphs can be used.

• 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 model ensembles, a purely local approach is 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.

• 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 approximate_control_variate 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:

• Sampling Moments (moments only, not confidence intervals)

In addition, the execution group has the attribute equiv_hf_evals, which records the equivalent number of high-fidelity evaluations.

Usage Tips

The approximate_control_variate method must be used in combination with an ensemble model specification that enumerates a truth model and a set of model approximations using either a model form sequence, a discretization level sequence, or a combination of model forms and discretization levels. For each model form and discretization, each instance must either provide a solution_level_cost or return metadata for estimating cost. 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. An alternative to prescribing the cost profile is estimating it on the fly using cost metadata that is returned from the different simulation instances.

Examples

The following method block:

method,
 model_pointer = 'NONHIER'
 approximate_control_variate
   acv_mf nip
   pilot_samples = 20 seed = 1237
   max_iterations = 10
   convergence_tolerance = .001

specifies ACV-MF using the nonlinear interior point (NIP) solver in combination with the model identified by the NONHIER pointer.

This NONHIER model specification provides a one-dimensional 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 ensemble
   truth_model = 'HF'
   unordered_model_fidelities = 'LF1' 'LF2'

model,
 id_model = 'LF1'
 interface_pointer = 'LF1_INT'
 simulation
   solution_level_cost = 1

model,
 id_model = 'LF2'
 interface_pointer = 'LF2_INT'
 simulation
   solution_level_cost = 16

model,
 id_model = 'HF'
 interface_pointer = 'HF_INT'
 simulation
   solution_level_cost = 256.

Refer to dakota/test/dakota_uq_diffusion_acv3_cost4.in and dakota/test/dakota_uq_tunable_acv.in within the source distribution for this case as well as additional examples.

Theory

Refer to [GGEJ20] for more detailed algorithm descriptions, theoretical considerations, and helpful sample set diagrams.

Refer to [BLWL22] for understanding ACV generalizations for the different control variate pairings that are possible when codified into a more comprehensive set of potential DAGs.