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

Design of Computer Experiments - Quasi-Monte Carlo sampling

## Topics

This keyword is related to the topics:

## Specification

Alias: none

Argument(s): none

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Required
(Choose One)
sequence type (Group 1) halton Generate samples from a Halton sequence
hammersley Use Hammersley sequences
Optional latinize Adjust samples to improve the discrepancy of the marginal distributions
Optional quality_metrics Calculate metrics to assess the quality of quasi-Monte Carlo samples
Optional variance_based_decomp

Activates global sensitivity analysis based on decomposition of response variance into contributions from variables

Optional samples

Number of samples for sampling-based methods

Optional fixed_sequence Reuse the same sequence and samples for multiple sampling sets
Optional sequence_start Choose where to start sampling the sequence
Optional sequence_leap Specify how often the sequence is sampled
Optional prime_base The prime numbers used to generate the sequence
Optional max_iterations

Stopping criterion based on number of iterations

Optional model_pointer

Identifier for model block to be used by a method

## Description

Quasi-Monte Carlo methods produce low discrepancy sequences, especially if one is interested in the uniformity of projections of the point sets onto lower dimensional faces of the hypercube (usually 1-D: how well do the marginal distributions approximate a uniform?)

This method generates sets of uniform random variables on the interval [0,1]. If the user specifies lower and upper bounds for a variable, the [0,1] samples are mapped to the [lower, upper] interval.

The user must first choose the sequence type:

• `halton` or
• `hammersley`

Then three keywords are used to define the sequence and how it is sampled:

• `prime_base`
• `sequence_start`
• `sequence_leap`

Each of these has defaults, so specification is optional.

## Theory

The quasi-Monte Carlo sequences of Halton and Hammersley are deterministic sequences determined by a set of prime bases. Generally, we recommend that the user leave the default setting for the bases, which are the lowest primes. Thus, if one wants to generate a sample set for 3 random variables, the default bases used are 2, 3, and 5 in the Halton sequence. To give an example of how these sequences look, the Halton sequence in base 2 starts with points 0.5, 0.25, 0.75, 0.125, 0.625, etc. The first few points in a Halton base 3 sequence are 0.33333, 0.66667, 0.11111, 0.44444, 0.77777, etc. Notice that the Halton sequence tends to alternate back and forth, generating a point closer to zero then a point closer to one. An individual sequence is based on a radix inverse function defined on a prime base. The prime base determines how quickly the [0,1] interval is filled in. Generally, the lowest primes are recommended.

The Hammersley sequence is the same as the Halton sequence, except the values for the first random variable are equal to 1/N, where N is the number of samples. Thus, if one wants to generate a sample set of 100 samples for 3 random variables, the first random variable has values 1/100, 2/100, 3/100, etc. and the second and third variables are generated according to a Halton sequence with bases 2 and 3, respectively.