[EM] Exact spatial model probabilities?

[Forest Simmons]

Thanks Ted and Daniel. Very interesting!

In the early 70's we did our minuteman missile simulations on a room size
mainframe IBM 360/65 computer with FORTRAN code, double precision
arithmetic ... punched cards interface.... and all. We got one turnaround
per night.... night because the Top Secret runs had to be totally isolated
from the daytime use of the computer. In 1974 our group got ahold of a
couple of the mini-computers that were just coming out ... TTY "ticker
tape" interface at first then (unreliable, but more convenient) floppy
discs. Very slow, single precision, but interactive BASIC for the spine of
the simulation. We employed pseudo-double precision for the numerical
integration, and modified BASIC so we could call on assembled bottle-neck
subroutines, etc.

The Top Secret, computationally high cost, high precision, spherical
harmonics gravity model "GRAV 74" (mylar tape in a secure safe) became the
main bottle- neck. Our project leader came up with the idea that cut the
run time from several hours to real time; our simulation had to be a
complete virtual reality operation with real time speed in order to fool
the D37 guidance and control computer that had been extracted from a MM2
missile to test the target tapes.

The idea was this  ... start with a quick (fraction of a second)
approximate trajectory, and in another fraction of a second, fit a simpler
model "GRAVCHICO" to the precise GRAV 74 values at half a dozen points near
the approximate trajectory. So near that approximate trajectory GRAVCHICO
was accurate to seven significant figures. Then we transformed the
spherical coordinates to the Kustanheimo coordinates from Stiefel and
Scheifele's 1971 book, Linear and Regular Celestial Mechanics.

After that our simulation on the little dedicated, stand alone minicomputer
(32K bytes) system was good for real time use ... no trouble keeping up
with the D37 guidance and control computer.

Now here are some thoughts on our current problem. We need an
"electo-potential" scalar function V defined at every point in voter space,
such that the (opposite of) grad V gives rise to a force field that points
towards the (generalized) median voter.

It seems to me that a point partical located at point p0, should contribute
an amount ||p-p0||=distance(p, p0) to the potential V(p) at point p.

A test particle at p would experience a force of (p0-p)/||p0-p||, a force
of unit magnitude towards p0.

If the particles (voters) are distributed according to some probability
density P, and dP(r) is the probability in an infinitesimal volume centered
at r, then a test particle at p will experience a force of

F=Integral(over r)of (r-p)/|r-p|||dP(r)

or something like that.

Visualize the potential function V as a mountain range. The force field F
is represented by tiny arrows pointing in the direction of steepest descent
to (wet or dry) lake bottoms, local minima of V.

If there is only one drainage basin, the global minimum is the sink of that
basin, and is the generalized median of the probability distribution.

To find a local minimum start at a random point p0 and evolve the initial
value problem dp/dt=-gradV until two consecutive points in the numerical
solution are within some preset tolerance.

Which points would make good starting points?

How about the vertices of each of the Voronoi cells centered on the
candidate positions?

What do we do with the local minima?

For each of them, calculate the candidate preference order. Those will be
the stable orders that the voters of the respective drainage systems
gravitate towards.

How do we find the total probability of each drainage system?

Go back and find which Voronoi vertices end up in which sinks. If
necessary, triangulate the Voronoi cells with progressively finer mesh, and
continue following vertices to sinks.

I hope this stimulates some ideas.

We have all of the tools of force fields and numerical solutions of ODE's
at our disposal!

-Forest

El lun., 24 de ene. de 2022 3:19 p. m., Daniel Carrera <dcarrera at gmail.com>
escribió:

>
> On Mon, Jan 24, 2022 at 4:46 PM Ted Stern <dodecatheon at gmail.com> wrote:
>
>> Hi Forest,
>>
>> To digress from the topic a bit, there is an O(N) method for N-body
>> problems which has been around since the 1980s. It is called the Fast
>> Multiple Method.
>>
>> The essence of the method is that the effect of a group of objects can be
>> approximated (outside a certain radius) as the effect of a single composite
>> object. These composite objects can further be combined into larger scale
>> composite objects. This is similar to Isaac Newton's approximation of
>> gravitational effect being due to a point mass.
>>
>
> Not too similar though. Planets and stars are really well approximated by
> spheres, and the Shell theorem tells you that the gravitational field from
> a sphere is exactly identical to that of a point mass. The quadrupole
> moments for a planet are not usually relevant except for problems that
> require extremely high precision (most notably, studies of Saturnian
> moons). Tree codes and the Shell theorem are more or less at opposite ends
> in terms of levels of approximation. You wouldn't use a tree code for the
> solar system, for example.
>
>
> If you just stop there, this reduces the order of operations to O(N
>> logN).  That first stage of approximation is the effect of collections of
>> masses on everything else in the Universe.  N-body simulations using this
>> idea are called Barnes-Hut methods.
>>
>> The key to FMM is that you can do the same thing in the other direction,
>> finding the effect of everything *outside* a given radius on the objects
>> inside. That can also be described as a composite object -- a multipole.
>> Then there is a similar set of outer-inner nested collections to convey
>> information from the rest of the Universe to each individual particle.
>> This brings the number of operations to O(N).
>>
>
>
> Fun fact, FMM only entered astronomy relatively recently, while it has
> been dominant in particle physics for a very long time. The difference is
> that the FMM methods from condensed physics are only faster than BH trees
> if the N-body system is largely homogeneous. That's great for condensed
> matter physics, but in astronomy we have ridiculously hierarchical systems
> and the FMM that you'll find if you do a Google search will work poorly in
> astronomy. It was only 8 years ago that someone developed a FMM that beats
> BH for hierarchical systems (Dehnen 2014).
>
> Cheers,
> --
> Dr. Daniel Carrera
> Postdoctoral Research Associate
> Iowa State University
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20220124/83c9c74e/attachment-0001.html>