ordering

Ordering of the points of this rank-1 lattice

Specification

• Alias: None

• Arguments: None

• Default: Radical inverse ordering

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional (Choose One)	Ordering	natural	Natural ordering of the points of this rank-1 lattice
		radical_inverse	Radical inverse ordering of the points of this rank-1 lattice

Description

Specify the order in which the points of this rank-1 lattice should be returned. Currently, Dakota supports natural and radical_inverse ordering. The default behavior is to return points using radical inverse ordering, i.e., the points will be generated according to

\[\boldsymbol{t}^{(i)} = \left\{ \phi_b(i) \boldsymbol{z} \right\}\]

where \(\phi_b(i)\) denotes the so-called radical inverse function in base \(b\) (usually, and also in Dakota, \(b = 2\)). This function transforms a number \(i = (\ldots i_2i_1)_b\) in its base-\(b\) representation to \(\phi_b(i) = (0.i_1i_2\ldots)_b\). Note that the radical inverse function agrees with the original formulation when \(N = b^m\) for any \(m \geq 0\). The advantage of the radical inverse ordering is that one can generate a good point set with an arbitrary number of points \(N\). Using the natural order, a good point set requires a number of points that is a power of 2. Semantically, the radical inverse ordering turns the lattice rule into a lattice sequence.

Usage Tips

When the natural ordering of the points is used, it is implicitly assumed that the user will only request a number of points that is a power of 2. Failing to do so will result in a set of points for which one or more dimensions are not uniformly covered, resulting in bad low-discrepancy properties, and ultimately bad performance of the method that uses these points. It is recommended to not use this option unless you know what you’re doing.

Examples

environment
  tabular_data
    tabular_data_file = 'samples.dat'
    freeform

method
  sampling
    samples 8
    sample_type
      low_discrepancy
        rank_1_lattice
          generating_vector inline 1 5 # this is a Fibonacci lattice
          m_max 3 # 8 points in total
          no_random_shift
          ordering natural

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