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	Approximate control variate (ACV) algorithm that employs independent samples per model
		acv_multifidelity	Approximate control variate (ACV) algorithm that mimics MFMC by employing a nested pyramid sample pattern This ACV variant uses sample set definitions that are similar to multifidelity Monte Carlo (MFMC), in that sample sets are nested with each new level adding an increment on top of the previous.
		acv_recursive_diff	Weighted recursive difference option for approximate control variate sampling (ACV-RD)
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	Uses a sequential quadratic programming method for underlying optimization
		nip	Uses a nonlinear interior point method for underlying optimization
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		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

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 relaxes the nested sampling of a recursive emulator, instead targeting the truth model’s variance with each control variate pair. 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 Gorodetsky et al. (2020), Figure 2. Two ACV variants are currently implemented, ACV-MF and ACV-IS, with ACV-KL to follow.

Default Behavior

The approximate_control_variate 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 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 defines 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, 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 = '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 in the source distribution for this case as well as additional examples.

Refer to [Gorodetsky et al., JCP (408), 2020] for more detailed algorithm descriptions, theoretical considerations, and a helpful sample set diagram.