[EM] Exact spatial model probabilities?

[Ted Stern]

Cool, thanks! I'll look up Dehnen.

I was in grad school when Greengard and Rokhlin was first published. It was
a big deal at the time in the applied math world. I knew the astro
community was continuing to use BH, but didn't know why.

On Mon, Jan 24, 2022 at 3:19 PM Daniel Carrera <dcarrera at gmail.com> wrote:

>
> 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
>
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