(Fast) Methods for N-Body Simulations

Speaker: Leslie GreengardPresenters: Ryan Lee, Jeffrey Wang

Problem Statement

World’s Largest Singles MeetupN-(human) Body Simulations

Astrophysical Systems

Molecular Dynamics

Body Deformation and Medical Imaging

Robotics Control and Simulations (IoT)

In the Beginning

LOGY

Lolwut!?

They always said hardware was better...

Holmberg 1941

Direct Method

Can we hit the O(n) ideal?

Mean-Field Theory

Hartree-Fock Method

Tree codes: Barnes-Hut Algo

Ventimiglia & Wayne, 2003

Populating the Tree

Ventimiglia & Wayne, 2003

Populating the Tree

Ventimiglia & Wayne, 2003

Populating the Tree

Ventimiglia & Wayne, 2003

Computing Forces

Ventimiglia & Wayne, 2003

Computing Forces

Ventimiglia & Wayne, 2003

Computing Forces

Ventimiglia & Wayne, 2003

O(n log(n))

The Top Ten Algorithms of the 20th CenturyJack 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)

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/phy411-506-2004http://courses.cs.washington.edu/courses/cse548/06wi/files/benchmarks/fmm.pdf

X

V

W

U

Simple Overview of Steps: all O(N)

X

V

W

U

1. Build Tree2. Construct interaction lists3. 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 W5. Downward Pass6. Calculate at local level

How it works in brief

http://www.bu.edu/pasi/courses/12-steps-to-having-a-fast-multipole-method-on-gpus/

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)

Current Research and Future Directions