Additional low-discrepancy sequences for (hyper)cubic integration

[Samuel-Vangu]

Following #241 (lattice rules) -- since that's already moving forward in #293 --
I wanted to propose a few complementary low-discrepancy constructions that
aren't covered by lattice rules alone:

• Sobol' sequences -- digital net construction, widely used, extensible in
  n (unlike lattice rules, which need n fixed in advance for a given
  generating vector).
• Halton sequences -- radical-inverse construction, also extensible,
  simpler to construct than Sobol' but with somewhat worse discrepancy in
  higher dimensions.
• (Randomized) Latin Hypercube Sampling -- not itself a low-discrepancy
  sequence in the classical sense, but a well-established variance-reduction
  alternative to plain Monte Carlo (Stein, 1987; McKay, Beckman & Conover,
  1979), often used alongside QMC methods for the same integration problems.
• Korobov lattice rules -- as far as I can tell, Add lattice rules for (hyper)cubic grid integration #293 implements the
  general CBC-constructed lattice rules (via the Kuo/UNSW tabulated vectors).
  Korobov lattices are a restricted special case (generating vector
  z(a) = (1, a, a^2, ..., a^{d-1}) mod n, a single integer parameter a),
  which trades a smaller search space for near-instant table lookup and much
  smaller storage -- a lighter-weight complement to the general CBC lattice
  rather than a duplicate of it.

I've implemented all four of these from scratch already: Halton, Korobov
lattice rules, and randomized LHS for QMCPy
(merged), and Halton and rank-1 lattice sequences for
torchquad (open PR).

To keep review manageable, I'd plan to open a separate PR per sequence
rather than one large PR.

Would this be of interest? Happy to start whenever convenient, and to adjust
scope/order based on what's most useful for the library.

[Zohaibcode740]

Hi Samuel, I’d be interested in contributing to this. I agree that keeping each sequence in a separate PR would make the implementation and review much easier. I’d be happy to start with the Halton sequence as the first contribution.

Before I begin, could you confirm if there are any preferred API/design conventions or existing abstractions in the project that I should follow?

[Samuel-Vangu]

Hi @Zohaibcode740 , thanks for the interest! Just to clarify -- I'm not a maintainer here,
just proposing this and hoping to work on it myself, same as you. The design/
convention questions would be best answered by @PaulWAyers or whoever from the
core team weighs in.

That said, if the maintainers are on board and open to splitting the work, I'd
be happy to divide it up -- you could take Halton as you suggested, and I'd
take the rest (Sobol', randomized LHS, Korobov). Let's see what they say.

[Zohaibcode740]

Thanks for clarifying, Samuel. That works for me — I’ll wait for the maintainers’ feedback on the proposed split and the preferred API/design before starting the Halton implementation.

[PaulWAyers]

Hi @Samuel-Vangu and @Zohaibcode740 . Our primary use case for lattice rules for hypercubes, which we typically map to real space with a nonlinear transformation to integrate (very) inhomogeneous functions. In those applications, the integrand is usually smooth (almost everywhere) so expansion in a Fourier basis is sensible, which was the motivation for the use of lattice rules. (We, traditionally, also support simpler 1D-product quadatures, but I haven't been following this issue recently.)

I think of quasirandom (minimum discrepancy) approaches as optimal for functions that are continuous but have limited differentiability, as opposed to more structured grids that integrate all trigonometric/polynomial functions up to a specified degree, which tend to be more useful in the chemical/molecular applications grid was originally intended to support. That said, if you feel like this is useful for your work, I am not averse to including it in grid. It would be important to provide a tutorial notebook with a relevant use case for the work, which I'm happy to (try to) help with.

[Samuel-Vangu]

Hi @PaulWAyers

Thanks for the detailed context . I'd still like to implement these in grid -- I'll put together a tutorial notebook as you
suggested, and try to find a relevant use case from your domain (chemistry/
molecular integration) to make the motivation concrete . Will follow up once I have a draft.

[Zohaibcode740]

I’ll proceed with the Halton sequence as a separate contribution.

I’ll follow the existing API and implementation conventions in grid, add the necessary tests, and include a tutorial notebook with a relevant chemistry/molecular integration use case as requested.

I’ll prepare the implementation and tutorial, then open a PR for review.

[PaulWAyers]

I'm assigning the issue to the two of you. When a PR is made, please look at our contributing guidelines and:

1. Review each other's code first.
2. I will then do a copilot review.
3. @marco-2023 and @FarnazH will make a final review.

[Zohaibcode740]

Hi @Samuel-Vangu ,

I’ve opened a separate PR for the Halton sequence implementation, following the scope we discussed.

The implementation includes:

• a multidimensional Halton sequence based on radical inverses,
• uniform integration weights,
• tests covering points, weights, shape, and invalid arguments.

The full test suite passes (602 passed, 1 skipped).

I’ll also review your implementation once it’s available, as requested by the maintainers.

[Samuel-Vangu]

Hi @Zohaibcode740,

I've also opened a PR on the LatinHypercube sampler. Please take a look when
you get a chance -- I'll review yours as well.

[Zohaibcode740]

Thanks, @Samuel-Vangu . I’ll take a look at PR #324 and review the implementation and tests. I’ll leave any comments directly on the PR if I find anything that needs discussion or changes.