CS 170: Computing for the Sciences and Mathematics

1. CS 170:Computing for the Sciences and Mathematics N-Body I

2. Administrivia • Today • N-Body Simulations • HW #10 assigned • Ongoing • Project! • A Final Exam time solution

3. N-Body Simulations I Modeling the Interactions of Lots of Things

4. N Bodies

5. N-Body Problems • An N-body problem is a problem involving N “bodies” – that is, particles (stars, atoms) – each of which applies some force to all of the others. • For example, if you have N stars, then each of the N stars exerts a force (gravity) on all of the other N–1 stars. • Likewise, if you have N atoms, then every atom exerts a force (nuclear) on all of the other N–1 atoms.

6. 1-Body Problem • When N is 1, you have a simple 1-Body Problem: a single particle, with no forces acting on it. • Given the particle’s position P and velocity V at some time t0, you can trivially calculate the particle’s position at time t0+Δt: P(t0+Δt) = P(t0) + VΔt V(t0+Δt) = V(t0)

7. 2-Body Problem • When N is 2, you have – surprise! – a 2-Body Problem: exactly 2 particles, each exerting a force that acts on the other. • The relationship between the 2 particles can be expressed as a differential equation that can be solved analytically, producing a closed-form solution. • So, given the particles’ initial positions and velocities, you can trivially calculate their positions and velocities at any later time.

8. 3-Body Problem • When N is 3, you have – surprise! – a 3-Body Problem: exactly 3 particles, each exerting a force that acts on the other. • The relationship between the 3 particles can be expressed as a differential equation that can be solved using an infinite series, producing a closed-form solution, due to Karl Fritiof Sundman in 1912. • However, in practice, the number of terms of the infinite series that you need to calculate to get a reasonable solution is so large that the infinite series is impractical, so you’re stuck with the generalized formulation.

9. N-Body Problems (N > 3) • For N greater than 3 (and for N of 3 in practice), no one knows how to solve the equations to get a closed form solution. • So, numerical simulation is pretty much the only way to study groups of 3 or more bodies. • Popular applications of N-body codes include: • astronomy (that is, galaxy formation, cosmology); • chemistry (that is, protein folding, molecular dynamics). • Note that, for N bodies, there are on the order of N2 forces, denoted O(N2).

10. N Bodies

11. Force #1 A

12. Force #2 A

13. Force #3 A

14. Force #4 A

15. Force #5 A

16. Force #6 A

17. Force #N-1 A

18. N-Body Problems • Given N bodies, each body exerts a force on all of the other N – 1 bodies. • Therefore, there are N • (N – 1) forces in total. • You can also think of this as (N • (N – 1)) / 2 forces, in the sense that the force from particle A to particle B is the same (except in the opposite direction) as the force from particle B to particle A.

19. Aside: Big-O Notation • Let’s say that you have some task to perform on a certain number of things, and that the task takes a certain amount of time to complete. • Let’s say that the amount of time can be expressed as a polynomial on the number of things to perform the task on. • For example, the amount of time it takes to read a book might be proportional to the number of words, plus the amount of time it takes to settle into your favorite easy chair. C1•N + C2

20. Big-O: Dropping the Low Term C1•N + C2 • When N is very large, the time spent settling into your easy chair becomes such a small proportion of the total time that it’s virtually zero. • So from a practical perspective, for large N, the polynomial reduces to: C1•N • In fact, for any polynomial, if N is large, then all of the terms except the highest-order term are irrelevant.

21. Big-O: Dropping the Constant C1•N • Computers get faster and faster all the time. And there are many different flavors of computers, having many different speeds. • So, computer scientists don’t care about the constant, only about the order of the highest-order term of the polynomial • Separates the cost of the fundamental algorithm from computer-specifics • They indicate this with Big-O notation: O(N) • This is often said as: “of order N.”

22. N-Body Problems • Given N bodies, each body exerts a force on all of the other N – 1 bodies. • Therefore, there are N • (N – 1) forces total. • In Big-O notation, that’s O(N2) forces. • So, calculating the forces takes O(N2) time to execute. • But, there are only N particles, each taking up the same amount of memory, so we say that N-body codes are of: • O(N) spatial complexity (memory) • O(N2) time complexity

23. O(N2) Forces A Note that this picture shows only the forces between A and everyone else.

24. How to Calculate? • Whatever your physics is, you have some function, F(A,B), that expresses the force between two bodies A and B. • For example, for stars and galaxies: F(A,B) = G·mA·mB / (dist(A,B)2) where G is the gravitational constant and m is the mass of the body in question. • If you have all of the forces for every pair of particles, then you can calculate their sum, obtaining the force on every particle. • From that, you can calculate every particle’s new velocity and position.

25. Algorithm Set up initial positions and velocities of all particles FOR time steps from 1 to T FOR each particle p from 1 to N Initialize force on p to 0. FOR each other particle q from 1 to N calculate force on p from q add to p’s forces Calculate the velocity of p based on forces Calculate the position of p based on velocity

26. Example: GalaxSee http://www.shodor.org/galaxsee-simsurface/galaxsee.html

27. HOMEWORK! • HW #10 is posted • Last one of the semester! • 1 Monte Carlo problem • Class in here on Thursday and Monday

28. Algorithm – Parallel Version Set up initial positions and velocities of all particles FOR time steps from 1 to T FOR some sub-set of particlesp from 1 to N Initialize force on p to 0. FOR each particle q from 1 to N (excluding p) calculate force on p from q add to p’s forces Calculate the velocity of p based on forces Calculate the position of p based on velocity Send position information of my subset to other CPUs

29. Parallelization of the Direct Force Algorithm • The steps needed for each calculation: • Single set of Instructions over Multiple Data • Each process calculates some of the accelerations (calculate) • Each process calculates some of the new positions (calculate) • Each process shares its position information (communicate!) • The point of diminishing returns: • The more you split up the problem, the less work each processor does, thus the ratio of concurrent work to communication reduces.

30. Amdahl's Law and optimal efficiency • General Law • Best case simulation time = • Speedup approaches a limit • N-body is worse: • time = (aN2 / P) + (cN) + (dP) • Speedup falls off as 1/P for large P (time increases linearly) • Large N, less communication can increase the value of P before speedup falls off.

31. Particle-MeshandParticle-Particle Particle-Mesh

32. Particle Mesh Algorithm • Replace solution of force through particle-particle interaction with solution of force as the gradient of a density function satisfying Poisson’s equation.

33. Particle Mesh Algorithm • Step 1: Generate Density Distribution Function from point sources • Step 2: Take FFT of density distribution function • Step 3: Solve Poisson’s equation for gravitational potential in Fourier space • Step 4: Transform back to Euclidean space • Step 5: Compute force from potential

34. Step 1: Generate Density Distribution Function • For each body, determine which grid sites are near the body, and determine how to apply a density distribution to those nearby grid points • Simplest approach, assume point mass fills some radius R and volume V, and any grid point within that radius has its density increased by M/V

35. Translate N bodies onto grid

36. Overlay grid onto space

37. Soften particles

38. Map density distribution onto grid

39. Solve for potential of density

40. Step 2/3/4: Solving Poisson’s equation using Fourier Transform

41. Step 5: Solve for the force using the potential

42. Win with PM • Now we only have to communicate based on the number of grid nodes, instead of the number of bodies. • There’s very little point-to-point force calculations. • Very fast! • What’s the bad news?

43. PM Issues • Because we’re “softening” the particles to a grid, we have no calculations reflecting “close” (local/short range) forces. • Those are the biggest forces! • Recall that gravitational force diminishes with r2

44. Other Concerns when using PM • Limitations on size of grid • Memory requirements, particularly in 3-D • NG*NG*NG • Time requirements to map points to grid • NP*NG*NG*NG • Time requirements to solve FFT • NG*log(NG)

45. Improvements for nearest neighbors • Use PM method for long range forces only, and calculate short range forces using direct calculation of nearest neighbors. • Particle-Particle Particle-Mesh or P3M • (Or use tree-based hybrid methods) • i.e. Barnes-Hut

46. Timing