As noted in OP, this can lead to O(1/n) convergence of averages to underlying target values, rather than the expected O(1/sqrt(n)) convergence ("n" is the number of probes of the underlying system). This is shown, strikingly, in Fig. 2 of the OP.
There is a conference, tutorial papers etc. E.g.: http://mcqmc2018.inria.fr
In principle, the curse of dimensionality kicks in after a dozen or so dimensions, as convergence is O( log(N)^D / N). But in practice I have found that in a surprising number of applications and circumstances, quasirandom sequences can have offer a substantial improvement even when the number of dimensions are in the hundreds or even thousands. (Some authors have suggested this works because the solution space is of low dimensions but embedded in a much higher dimensional space...)
However, to get the full benefit I needed to find a low discrepancy quasirandom sequence that did not suffer the many of the parameter degeneracy problems that many of the conventional ones exhibit for very large D. For me, the new R-sequence nicely solves this parameter selection problem by not having any parameters to optimize!
BTW, in the past couple months I have directed several people who were interested in spiral phyllotaxis and the like at your page. I think your sequences could make for some interesting art projects.
Personally I plan to try them out for dithering, picking well-spaced colors for diagrams, picking points in a region when trying to optimize a map projection, etc.
I get that the uniform grid is not an open sequence, just want to understand which kind of convergence do you have in mind. For which class of functions?
Despite this, there are two major use cases for quasirandom sequences: The first is when n is not known upfront and so you can't create a lattice; and the second is in higher dimensions. The reason for this latter situation is because the number of points required to make a lattice in D dimensions scales exponentially in D, whereas the convergence rates for random or quasirandom sampling are (amazingly!) independent on the dimension.
I think that you would really like Owen's paper on this topic. It is very comprehensive and yet extremely readable [1]
To do anything of use, you have to draw samples and then simulate complex outcomes, very few of which reduce to simple averages over the drawn samples or functions of the drawn samples. And since you need to draw samples, you might as well use those samples for any parts that do happen to require integration, rather than a quasi mc calculation, except in extremely toy-problem situations where the convergence rate matters disproportionately for that smallset of outcomes.
I agree that for cases when you just want to evaluate an integral it could be useful.
I have never encountered a use case when anyone just wanted to calculate an integral, as opposed to also generating posterior uncertainty metrics, posterior test statistics for posterior predictive checking, posterior diagnostics like ordinal statistics among the posterior samples.
I’m sure outside of stats, there must be use cases. Just adding a counterpoint to the idea that quasi mc should always be interesting to practitioners. For a lot of people who work in mcmc methods, quasi mc is just not interesting and generally speaking could never be.
If you don’t mind: what’s the general application domain for your MCMC‘s? For me, it’s inference of (usually) spatial fields for various science data sets, like Earth or solar imagery or atmospheric composition.