[EM] High Order Numerical Cubature2.0

[Daniel Carrera]

On Sat, Feb 5, 2022 at 3:55 PM Daniel Carrera <dcarrera at gmail.com> wrote:

> https://postimg.cc/zV34pv0F
>
> Unfortunately, it doesn't seem to be space-filling. It looks like you have
> reinvented Sierpiński's gasket.
>

Incidentally, my own idea of using multiple barycenters doesn't work
either. It gives points very clustered in the center of the simplex.

I did find a solution that works great for a 2D simplex, but I don't see an
obvious way to generalize it. We begin by generating a pair of points (u,v)
from the 2D Sobol sequence. We want to map these to (x,y) inside the
simplex in a way that retains at least some of the uniformity guarantees of
the Sobol sequence. The simplex is defined by the boundary:

x + y < 1

That means for x = 0 there are more possible values of y than for x = 0.9,
so we should produce more values close to x=0. This is a familiar problem
in the context of drawing random numbers and it has a known solution.

Let `y = f(x) = 1 - x` be (not normalized) the probability of generating
the value `x`. The cumulative distribution is:

F(x) = x - x^2/2

Normalize it to `F(x) = 2x - x^2` so that we have the property that `F(0) =
0` and `F(1) = 1`. So we can let F(x) be equal to the first Sobol number:

F(x) = u

=> x = 1 - sqrt(1 - u)

This gives an `x` with the correct probability distribution and we are free
to draw `y` randomly between 0 and (1-x). We accomplish that with the
second Sobol number:

=> y = (1 - x)*v

The result is a nicely uniform distribution of points without the
clustering that you'd expect from random points:

https://postimg.cc/XBbnD7fQ

There's probably a way to generalize these ideas to higher dimensions.

Cheers,
-- 
Dr. Daniel Carrera
Postdoctoral Research Associate
Iowa State University
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