# 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 |
Uses rank-1 lattice points to sample variables |
||

Optional |
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
no_gradients
no_hessians
```