Enable low-discrepancy (quasi-random) numbers in addition to pseudorandom numbers

[KronosTheLate]

It has recently come to my attention that low-discrepancy sequences could be used in monte-carlo simulations for faster convergence. The benefit is that you require fewer points before the points truly represent the probability distribution. I quickly thought that this faster convergence should definitively be made use of in this package.

Suggestion: I would love the ability to chose a different samples via some option like set_sampler!(:sobol) (defaulting to set_sampler!(:prng). This would make it so that all current code could be used without modification. It also makes sense to make it a global setting, as I guess users would want to use the same sampler in all cases for a given script.

Alternatively, one could of course set a sampler as one constructs the particles (Particles(100, Uniform(0,2), :prng)), but that does not compose well with the nice unicode plus-minus constructor, which to me feels like incentivising users to not use non-default samplers, as it does not allow the same nice syntax. But if the global setting is a bad idea for some reason, this would be much better than nothing.

Below is some (python) code and a reference that I used personally, which might help to get started:

#From https://www.youtube.com/watch?v=DbFJdjpTPok
from scipy import stats
from scipy.stats import qmc

def sobol(m, d=1):
    sampler = qmc.Sobol(d, scramble=True)
    #return sampler.random_base2(m)  #! This would ensure balance properties
    return sampler.random(m)

def soboln(m):
    return stats.norm.ppf(sobol(m))

def sobol_balanced(m, d=1):
    sampler = qmc.Sobol(d, scramble=True)
    return sampler.random_base2(m)

def soboln_balanced(m):
    return stats.norm.ppf(sobol_balanced(m))

[baggepinnen]

Already used for scalar quantities
https://baggepinnen.github.io/MonteCarloMeasurements.jl/stable/examples/#Monte-Carlo-sampling-properties-1

[KronosTheLate]

Oh, that is really nice! My bad then. I guess I just assumed that is this package did this, it would be advertised more and at an earlier place in the documentation/readme. But I can hardly complain when there is a dedicated page explaining both how the samples are drawn, and the benefits ^_^