A brief overview of techniques for n-body simulations, ending with the Fast Multiple Method, developed by Leslie Greengard

1. (Fast) Methods for N-
Body Simulations
Speaker: Leslie Greengard
Presenters: Ryan Lee, Jeffrey Wang

2. Problem Statement

3. World’s Largest Singles Meetup
N-(human) Body Simulations

4. Astrophysical Systems

5. Molecular Dynamics

6. Body Deformation and Medical
Imaging

7. Robotics Control and Simulations (IoT)

8. In the Beginning
LOGY

9. Lolwut!?

10. They always said hardware was
better...
Holmberg 1941

11. Direct Method
Can we hit the O(n) ideal?

12. Mean-Field Theory

13. Hartree-Fock Method

14. Tree codes: Barnes-Hut Algo
Ventimiglia & Wayne, 2003

15. Populating the Tree
Ventimiglia & Wayne, 2003

16. Populating the Tree
Ventimiglia & Wayne, 2003

17. Populating the Tree
Ventimiglia & Wayne, 2003

18. Computing Forces
Ventimiglia & Wayne, 2003

19. Computing Forces
Ventimiglia & Wayne, 2003

20. Computing Forces
Ventimiglia & Wayne, 2003
O(n log(n))

21. The Top Ten Algorithms of the 20th Century
Jack Dongarra and Francis Sullivan editors of Computing in Science & Engineering published a list of
"The Top Ten Algorithms of the Century."
1. the Monte Carlo method or Metropolis algorithm, devised by John von Neumann, Stanislaw
Ulam, and Nicholas Metropolis;
2. the simplex method of linear programming, developed by George Dantzig;
3. the Krylov Subspace Iteration method, developed by Magnus Hestenes, Eduard Stiefel, and
Cornelius Lanczos;
4. the Householder matrix decomposition, developed by Alston Householder;
5. the Fortran compiler, developed by a team lead by John Backus;
6. the QR algorithm for eigenvalue calculation, developed by J Francis;
7. the Quicksort algorithm, developed by Anthony Hoare;
8. the Fast Fourier Transform, developed by James Cooley and John Tukey;
9. the Integer Relation Detection Algorithm, developed by Helaman Ferguson and Rodney
Forcade;
10. the fast Multipole algorithm, developed by Leslie Greengard and Vladimir Rokhlin;
● Invented in 1987
● O(N^2) to O(N)

22. FMM Innovations
● Duality Principle
○ Greengard approximated relationship between children and parents
● Merging and Shifting Expansions
○ If multipole has been already calculated for parent can avoid
calculating for children (and vice versa)
● Requires More Memory
http://www.physics.buffalo.edu/phy411-506-2004
http://courses.cs.washington.edu/courses/cse548/06wi/files/benchmarks/fmm.pdf
X
V
W
U

23. Simple Overview of Steps: all O(N)
X
V
W
U
1. Build Tree
2. Construct interaction lists
3. Upward Pass (bottom up)
4. Compute List Interactions
a. C+U
b. C+V (expansions localized)
c. C+W
d. C+X no need to calculate!
since X is dual of W
5. Downward Pass
6. Calculate at local level

24. How it works in brief
http://www.bu.edu/pasi/courses/12-steps-to-having-a-fast-
multipole-method-on-gpus/

25. Leslie Greengard
● Son of Paul Greengard
○ Nobel Laureate (Medicine)
○ Neuroscience + Signal Transduction
● Born in London (56 yo)
● Educated in the US
○ B.A. in mathematics from the Wesleyan
University (1979)
○ MD/PhD from the Yale School of Medicine
(1987)
■ Developed FMM here!
○ Ph.D. in computer science from Yale
University (1987)
● Director of the Courant Institute of Mathematical
Sciences (NYU)
● Current
○ Professor of Mathematics and Computer
Science at Courant Institute of Mathematical
Sciences (NYU)

26. Current Research and Future Directions