# low_discrepancy

Uses low-discrepancy points to sample variables

Specification

• Alias: qmc

• Arguments: None

Child Keywords:

Required/Optional

Description of Group

Dakota Keyword

Dakota Keyword Description

Optional

rank_1_lattice

Uses rank-1 lattice points to sample variables

Optional

digital_net

Uses digital net points to sample variables

Description

The `low_discrepancy` keyword invokes low-discrepancy sampling as the means of drawing samples of uncertain variables according to their probability distributions. Low-discrepancy points have the desired property that they are more evenly distributed compared to random samples. We implement two flavors of low-discrepancy points: `rank_1_lattice` points [NC06] and `digital_net`s [JK08]. Digital net points are the default choice. The well-known Sobol sequence is a particular example of a digital net [Sobol67].

When low-discrepancy points are used for integration, the method is called the quasi-Monte Carlo method. Low-discrepancy points are also referred to as quasi-Monte Carlo points.

Note

Currently, only continuous variables are suppored for low discrepancy sampling.

Usage Tips

Low-discrepancy points easily scale up to hundreds or thousands of dimensions. They are generally more efficient than random samples (and sometimes more efficient than Latin Hypercube samples) for estimating the mean of a model response. Digital net sequences (such as the Sobol sequence) can even achieve higher-order convergence [DP10].

Quasi-Monte Carlo points work best if the random variables in the model are ordered according to their relative importance, i.e., the first variable in the model is the most important, the second one is less important, and so on, and if the importance of the random variables decays relatively rapidly. In modern quasi-Monte Carlo literature, this is understood as if the function belongs to a function class with a certain smoothness [JK08].

Just as Latin Hypercube samples, the low-discrepancy points are generated uniformly in the unit (hyper)cube and care must be taken when the points are transformed to other distributions. For example, when generating standard normally distributed samples, the points are mapped from the unit cube to an infinite domain using the inverse cumulative distribution function. This function becomes unbounded near the boundary, which makes it hard to integrate, and one may lose the high-order convergence observed when integrating in the (hyper)cube [DKS13].

Examples

```environment
tabular_data
tabular_data_file = 'samples.dat'
freeform

method
sampling
samples 32
sample_type
low_discrepancy

variables
uniform_uncertain = 2
lower_bounds 0.0 0.0
upper_bounds 1.0 1.0

interface
analysis_drivers = 'genz'
analysis_components = 'cp1'
direct

responses
response_functions = 1