QuasiMonteCarlo.jl

This is a lightweight package for generating Quasi-Monte Carlo (QMC) samples using various different methods.

Tutorials and Documentation

For information on using the package, see the stable documentation. Use the in-development documentation for the version of the documentation, which contains the unreleased features.

Example

using QuasiMonteCarlo, Distributions
lb = [0.1,-0.5]
ub = [1.0,20.0]
n = 5
d = 2

s = QuasiMonteCarlo.sample(n,lb,ub,GridSample([0.1,0.5]))
s = QuasiMonteCarlo.sample(n,lb,ub,UniformSample())
s = QuasiMonteCarlo.sample(n,lb,ub,SobolSample())
s = QuasiMonteCarlo.sample(n,lb,ub,LatinHypercubeSample())
s = QuasiMonteCarlo.sample(n,lb,ub,LatticeRuleSample())
s = QuasiMonteCarlo.sample(n,lb,ub,LowDiscrepancySample([10,3], false))

The output s is a matrix, so one can use things like @uview from UnsafeArrays.jl for a stack-allocated view of the ith point:

using UnsafeArrays
@uview s[:,i]

API

Everything has the same interface:

A = QuasiMonteCarlo.sample(n,lb,ub,sample_method)

where:

• n is the number of points to sample.
• lb is the lower bound for each variable. The length determines the dimensionality.
• ub is the upper bound.
• sample_method is the quasi-Monte Carlo sampling strategy.

Additionally, there is a helper function for generating design matrices:

k=2
As = QuasiMonteCarlo.generate_design_matrices(n,lb,ub,sample_method,k)

which returns As which is an array of k design matrices A[i] that are all sampled from the same low-discrepancy sequence.

Available Sampling Methods

• GridSample(dx) where the grid is given by lb:dx[i]:ub in the ith direction.
• UniformSample for uniformly distributed random numbers.
• SobolSample for the Sobol sequence.
• LatinHypercubeSample for a Latin Hypercube.
• LatticeRuleSample for a randomly-shifted rank-1 lattice rule.
• LowDiscrepancySample(base) where base[i] is the base in the ith direction.
• GoldenSample for a Golden Ratio sequence.
• KroneckerSample(alpha, s0) for a Kronecker sequence, where alpha is an length-d vector of irrational numbers (often sqrt(d)) and s0 is a length-d seed vector (often 0).
• SectionSample(x0, sampler) where sampler is any sampler above and x0 is a vector of either NaN for a free dimension or some scalar for a constrained dimension.
• Additionally, any Distribution can be used, and it will be sampled from.

Adding a new sampling method

Adding a new sampling method is a two-step process:

1. Add a new SamplingAlgorithm type.
2. Overload the sample function with the new type.

All sampling methods are expected to return a matrix with dimension d by n, where d is the dimension of the sample space and n is the number of samples.

Example

struct NewAmazingSamplingAlgorithm{OPTIONAL} <: SamplingAlgorithm end

function sample(n,lb,ub,::NewAmazingSamplingAlgorithm)
    if lb isa Number
        ...
        return x
    else
        ...
        return reduce(hcat, x)
    end
end