Generate low discrepancy sequencesΒΆ

In this examples we are going to expose the available low discrepancy sequences in order to approximate some integrals.

The following low-discrepancy sequences are available:

  • Sobol

  • Faure

  • Halton

  • reverse Halton

  • Haselgrove

To illustrate these sequences we generate their first 1024 points and compare with the sequence obtained from the pseudo random generator (Merse Twister) as the latter has a higher discrepancy.

[1]:

from __future__ import print_function
import openturns as ot
import math as m

[2]:

# 1. Sobol sequence
dimension = 2
size = 1024
sequence = ot.SobolSequence(2)
sample = sequence.generate(size)
graph = ot.Graph("Sobol", "", "", True, "")
cloud = ot.Cloud(sample)
graph.add(cloud)
graph

[2]:
../../_images/examples_reliability_sensitivity_low_discrepancy_sequence_3_0.svg
[3]:

# 2. Halton sequence
dimension = 2
sequence = ot.HaltonSequence(2)
sample = sequence.generate(size)
graph = ot.Graph("Halton", "", "", True, "")
cloud = ot.Cloud(sample)
graph.add(cloud)
graph

[3]:
../../_images/examples_reliability_sensitivity_low_discrepancy_sequence_4_0.svg
[4]:

# 3. Reverse Halton sequence
sequence = ot.ReverseHaltonSequence(dimension)
sample = sequence.generate(size)
print('discrepancy=', ot.LowDiscrepancySequenceImplementation.ComputeStarDiscrepancy(sample))
graph = ot.Graph("Reverse Halton", "", "", True, "")
cloud = ot.Cloud(sample)
graph.add(cloud)
graph

discrepancy= 0.0035074981424325635

[4]:
../../_images/examples_reliability_sensitivity_low_discrepancy_sequence_5_1.svg
[5]:

# 4. Haselgrove sequence
sequence = ot.HaselgroveSequence(dimension)
sample = sequence.generate(size)
graph = ot.Graph("Haselgrove", "", "", True, "")
cloud = ot.Cloud(sample)
graph.add(cloud)
graph

[5]:
../../_images/examples_reliability_sensitivity_low_discrepancy_sequence_6_0.svg
[6]:

# Compare with uniform random sequence
distribution = ot.ComposedDistribution([ot.Uniform(0.0, 1.0)]*2)
sample = distribution.getSample(size)
print('discrepancy=', ot.LowDiscrepancySequenceImplementation.ComputeStarDiscrepancy(sample))
graph = ot.Graph("Mersenne Twister", "", "", True, "")
cloud = ot.Cloud(sample)
graph.add(cloud)
graph

discrepancy= 0.03685516518292309

[6]:
../../_images/examples_reliability_sensitivity_low_discrepancy_sequence_7_1.svg