fast n-body methods
DESCRIPTION
A brief overview of techniques for n-body simulations, ending with the Fast Multiple Method, developed by Leslie GreengardTRANSCRIPT
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(Fast) Methods for N-Body Simulations
Speaker: Leslie GreengardPresenters: Ryan Lee, Jeffrey Wang
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Problem Statement
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World’s Largest Singles MeetupN-(human) Body Simulations
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Body Deformation and Medical Imaging
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In the Beginning
LOGY
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Lolwut!?
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They always said hardware was better...
Holmberg 1941
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Direct Method
Can we hit the O(n) ideal?
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Mean-Field Theory
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Hartree-Fock Method
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Tree codes: Barnes-Hut Algo
Ventimiglia & Wayne, 2003
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Populating the Tree
Ventimiglia & Wayne, 2003
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Populating the Tree
Ventimiglia & Wayne, 2003
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Populating the Tree
Ventimiglia & Wayne, 2003
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Computing Forces
Ventimiglia & Wayne, 2003
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Computing Forces
Ventimiglia & Wayne, 2003
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Computing Forces
Ventimiglia & Wayne, 2003
O(n log(n))
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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)
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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
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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
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How it works in brief
http://www.bu.edu/pasi/courses/12-steps-to-having-a-fast-multipole-method-on-gpus/
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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)
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Current Research and Future Directions