1. Fast Multipole Method
For Solving N-Body Problems
Amir Masoud Abdol
University of Amsterdam

2. N-Body Problem
▪ The n-body problem is the problem of predicting the motion of a group of particles
that interacts with each other gravitationally or by other forces.
▪ First introduced by Newton to understand the motion of Sun, planets and other visible stars.
▪ Electrostatics
▪ Magneto statics
▪ Acoustic Scattering
▪ Even Diffusion
▪ They are computationally intensive. O(n2)
▪ They can be huge because of large number of particles.

3. N-Body Problem
▪ Problem definition
▪ mi is the mass, and Φ is potential function
▪ and we are interested in equilibrium configuration of a set of particles.
▪ Consider potentials of the form
▪ Φ = Φ near+ Φ external+ Φ far
▪ Φ near is not a problem. O(n)
▪ Φexternal is independent of the number of particles. O(n)
▪ Φ far is a BIG problem. O(n2)

4. Different Solutions
▪ Solutions:
▪ Simple Numerical Integration: Euler Method, O(n2)
▪ Truncation Based Methods*
▪ Barnes-Hut Method, and Appel
▪ Consider a pseudo-particle for a group of particles in distant.
▪ Then, perform integration or solving method.
▪ O(n log(n))
▪ Fast Multipole Method (Multipole Expansion + Local Expansion)
▪ O(n)

5. Fast Multipole Method
Idea
▪ Recursively divide the computational domain.
▪ The algorithm group particles together and compute the centralized potential. (Multipole
Expansion)
▪ Distribute information to upper level in grid decomposition.
▪ Use local information (interaction list) in coarser mesh level to transfer information from
long-range to near-range. (Local Expansion)
▪ Redistribute coarser mesh level to finer mesh level and again use local information of finer
mesh to localize the information even more. Up to finest mesh level!
▪ Compute long-range potential using local expansion
▪ Combine long-range and near-range potential and build the potential function.

6. Fast Multipole Method
Multipole Expansion
▪ Suppose that m charges of strengths qi are located at points zi with |zi|< r. Then for any
z∈C with |z|>r, the potential Φ(z) is given by
There are another Lemmas that
allows us to shift this function. For
instance shifting the center of circle
into origin.
[1]

7. Fast Multipole Method
Local Expansion
▪ Local expansion about the center of box i at level l, describing the potential field due to
all particles outside i’s parent box and the parent box’s nearest neighbors.
▪ Interaction list:
For box i at level l, it’s the set of boxes which
are children of the nearest neighbors of I’s
parent and which are well separated from box
i. (Red Boxes)

8. Algorithm
Computing Multipole Expansion
1. Form multipole expansion of potential
field due to particles in each box about
the box center at the finest mesh level.

9. Algorithm
Computing Multipole Expansion
2. Form Multipole expansions about the
centers of all boxes at all coarser mesh
levels, each expansion representing the
potential field due to all particles
contained in one box.
(Continuing this procedure to coarser
mesh level)

10. Algorithm
Computing Local Expansion
3. Form a local expansion about the
center of each box at each mesh level
l<=n-1.
This local expansion describes the field
due to all particles in the system that are
not contained in the current box or its
nearest neighbors.

11. Algorithm
Computing Local Expansion
4. Shifting the local expansion for a given
box to the center of the box’s
children, forming the initial expansion
for the boxes at the next level.

12. Algorithm
Computing Local Expansion
5. Continue 3 and 4 to shift local
expansion to finest mesh level.
Now we have the local expansion in
finest mesh level. And we can compute
the local expansion at particle position
by Φ(z) for z inside each box.

13. Algorithm
Evaluate Local Expansion
6. Evaluate Local Expansions at particle
positions.
• For every particle pj located at the
point zj in box, evaluate the Φbox(zj)

14. Algorithm
Computing the Φ near
6. For every particle pj in box, compute
interactions with all other particles
within the box and its nearest neighbors
as Φ near .
Now adding up Φ near and Φ far , we can
have all forces in one equation. Putting
this function into Newton formulation
will form complete n-body model.

15. Conclusion
1. The complexity is O(n)
2. The error estimation is easy because it based on polynomial expansion (Multipole
and Taylor Expansion). Actually, error bound is available.
3. There is no need for numerical differentiation because Φ are polynomials.
4. It also support boundary conditions.

16. References
1. Greengard, L. and Rokhlin, V. (1987). A fast algorithm for particle simulations. Journal
of Computational Physics, 73(2):325–348.