[EM] Spatial models -- Polytopes vs Sampling

Forest Simmons forest.simmons21 at gmail.com
Tue Feb 1 20:26:39 PST 2022


Great idea ... and if you bevel/taper/dither the function near the boundary
of its support, the numerical quadrature will converge faster.

El mar., 1 de feb. de 2022 5:44 p. m., Daniel Carrera <dcarrera at gmail.com>
escribió:

>
> On Tue, Feb 1, 2022 at 5:43 AM Kristofer Munsterhjelm <
> km_elmet at t-online.de> wrote:
>
>> I'm hardly an expert on these matters, but I found a paper suggesting a
>> way to transform a simplex into a hypercube (by, I think, using extended
>> barycentric coordinates as a change of bases). If that's a workable
>> approach, then you could decompose the polytope into simplices,
>> transform these simplices into cubes, and do QMC on the cubes without
>> throwing any points out.
>>
>>
>> https://istam.iitkgp.ac.in/resources/2014/proceedings/59-istam-nm-fp-268.pdf
>>
>>
> So I've been thinking about this, and I think I've come to the conclusion
> that at least in this instance it is perfectly fine to throw out QMC
> points. Here's why: Suppose I want to integrate a function `f()` over some
> polytope `P`.
>
> Voters = Integral_P { f( x ) }
>
> Let `C` be some cube that contains the polytope `P`. I can always define a
> function `g()` that is defined on `C` but has compact support in `P`:
>
> g(x) = { f(x) if x in P; or 0 otherwise }
>
> There's no reason why I shouldn't be allowed to integrate `g(x)` over the
> cube `C` and technically I'm not throwing away any points. But that is
> exactly equivalent to the original plan of generating QMC points inside the
> `C`, selecting those in `P`, and evaluating `f()` on those.
>
> Wikipedia says that one of the key advantages of quasi-MC over regular MC
> is that a low-dispersion sequence gives you more even coverage of the space
> without making a lattice. Random sampling will naturally create clusters of
> points and voids with few points. It is the nature of randomness that
> clustering happens. Low-dispersion sequences avoid clustering. Now... I
> don't know why we're told to not throw away points. But if the reason is
> that in general throwing away points will create gaps, then... well...
> slicing the space would be an exception to the rule. If the points
> uniformly cover `C` then they uniformly cover any subset of `C`. You just
> need to be aware that you are effectively integrating over that subset of
> `C` and not all of `C`, but in our case that is exactly what we want.
>
> Cheers,
> --
> Dr. Daniel Carrera
> Postdoctoral Research Associate
> Iowa State University
> ----
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> info
>
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