Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
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multilevel_sampling


Multilevel methods for sampling-based UQ

Specification

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

Description

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

\[ \mathbf{E}[Q] \equiv \hat{Q}^{MC} = \frac{1}{N} \sum_{i=1}^N Q^{(i)} \]

In a multilevel method with $L$ levels, we replace this estimator with a telescoping sum:

\[ \mathbf{E}[Q] \equiv \hat{Q}^{ML} = \sum_{l=0}^L \frac{1}{N_l} \sum_{i=1}^{N_l} (Q_l^{(i)} - Q_{l-1}^{(i)}) \equiv \sum_{l=0}^L \hat{Y}^{MC}_l \]

This decomposition forms discrepancies for each level greater than 0, seeking reduction in the variance of the discrepancy $Y$ relative to the variance of the original response $Q$. The number of samples allocated for each level ( $N_l$) is based on a total cost minimization procedure that incorporates the relative cost and observed variance for each of the $Y_l$.

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:

\[ \hat{Q}_{HF}^{CV} = \hat{Q}_{HF}^{MC} - \beta (\hat{Q}_{LF}^{MC} - \mathbf{E}[Q_{LF}]) \]

As opposed to the traditional control variate approach, we do not know $\mathbf{E}[Q_{LF}]$ precisely, but rather etimate it more accurately than $\hat{Q}_{LF}^{MC}$ 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 $\rho_{LH}$ between $Q_{LF}$ and $Q_{HF}$. The coefficient $\beta$ is then determined from the observed LF-HF 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.

Examples

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.

See Also

These keywords may also be of interest: