An implementation of the Fast Multipole Method without multipoles

Transcription

1 An implementation of the Fast Multipole Method without multipoles Robin Schoemaker Scientific Computing Group Seminar The Fast Multipole Method Theory and Applications Spring 2002

2 Overview Fast Multipole Algorithm Anderson s method Poisson s formula instead of multipole expansions Application

3 N-body interactions (1) Particle simulations in physical systems require the evaluation of a potential function φ( r), (yielding a force field); These pair wise interactions are Coulombic/gravitational in nature: φ ( ) N r = q log r r, j = 1,, N j i j i i= 1 for a system of N charged particles in 2 dimensions with strengths q i.

4 N-body interactions (2) If N is the total number of particles involved, the temporal complexity of such a system is ( 2 O N ).

5 { z, i = 1,, m} i Multipole expansion {, = 1,, } For m charges of strengths q i m located at points, with z < r, the potential φ() z is approximated by the multipole expansion i i k a qz i i φ() z " log() z q ith a, k m p m k + w = i k k i= 1 k= 1 z i= 1 for any z # with z > r.

6 Error bound m = i i= 1 For A q and any p 1, m p a k A 1 φ() z log() z q i k i= 1 k= 1 z z/ r 1 z/ r p 1 A 2 p, for z / r 2.

7 Example of speed-up (1) Two sets of charges are well-separated : x x < r for i = 1,, m i 0 y y < r for j = 1,, n j 0 x y > 3r 0 0.,, x 1 r y 1 r r x 0 x m y 0 y n

8 Example of speed-up (2) The potential at the points {y i } due to charges at {x j } is computed directly via m i= 1 φ x Onm i ( ) ( y ) n points). j for all j = 1,, n and requires computations (evaluating m fields at x 1 r x 0 x m r y 1 y 0 r y n

9 Example of speed-up (3) A p-term multipole expansion due to charges q 1,, q m about x 0 requires O(mp) operations; Evaluating this expansion at all {y i } : O(np) operations; total : O(mp + np) The O(p) operations can be neglected for large m and n, resulting in O(m) + O(n) << O(mn).

10 Translation operators Shifting the centre of a multipole expansion: M2M; Convert a shifted expansion into a local (Taylor) expansion: M2L; Shifting the centre of a local (Taylor) expansion: L2L.

11 M2M φ() z " b l a log( z z ) + φ() z " a log( z) + p l= 1 b z l l p ( z l l 1 l k az = a z k 0 k= 1 k 1 l k= 1 0 l 0 0 a k z w ith ) k into D z 0 R φ() z R+ z 0 D 1

12 M2L z > ( c + 1) R with c > 1 0 φ() z " b z with p l= 0 l a b a z b l p 0 k " k k= 1 z0 0 0 l D 2 φ() z R k ( 1) + log( ) p 1 a l k 1 + k k a0 " ( 1) l 1 l k l z0 k= 1 z k 1 for l z 0 0 cr D 1 z 0 R

13 L2L Shifting a Taylor expansion via an exact translation operation with a finite number of terms: { a } For any complex z, z, a nd, k = 0,, n, 0 k a ( z z ) = a ( z ) l n n n k k l k 0 k 0 k= 0 l= 0 k= l k z l.

14 well separated cells Hierarchy: FMA (1) parent cell A M2L parent cell B M2M L2L children of A children of B

15 Levels of refinement Hierarchy: FMA (2)

16 Anderson s method FMM: φ() z due to large numbers of particles; Not forced to use a multipole expansion as the computational element. Instead use Poisson s formula

17 Poisson s formula (1) Solving Laplace s equation for the potential outside a disk with radius a containing sources : φ( r, θ) = κlog( r) + 2π 2 1 a 1 ( ) ( ( ) ( )) r φ as, κlog a ds a 2 a 2 2π 1 2( ) co s( ) ( ) 0 r θ s + r for r > a.

18 Poisson s formula (2) and for the potential inside a hole with radius a from sources outside: 2π 2 1 ( r 1 ) a φ( r, θ) = φ( a, s) ds ( r) cos( ) ( r π ) 0 a θ s + a for r < a.

19 Outer ring approximation (1) Numerical representation : With M an integer and K = 2M + 1, we set ( ( ) ( )) h = 2 π / K and s = acos ih, asin ih, i = 1,, K : i φ( r, θ) " κlog( r) + K 2 1 ( a 1 ) ( (, ) log ) r φ as κ ( a) h i ( a) cos( ) ( a π i= r θ s + r) i

20 Outer ring approximation (2) K φ( r, θ)

21 Construction via Poisson s formula: Parent outer rings

22 Inner ring approximation Similar as the outer ring construction: With M an integer and K = 2M + 1, we set ( ) ( ) ( ) h = 2 π / K and s = acos ih, asin ih, i = 1,, K : i φ( r, θ) " K 1 1 φ( a, s ) 2π i= cos( θ ( ) i a 2 a 2 ( r) ( r) i a r 2 s ) + h φ( r, θ) K for r < a.

23 Children inner rings K K

24 Levels of refinement Hierarchical tree

25 Hierarchy in action

26 Temporal complexity (1) Seven steps in the hierarchy : 1. Finest level direct interactions N 2 ; 2. Finest level outer ring construction KN; 3. Finest level contribution by outer rings in well-separated area KN; 4. Outer ring approximations K 2 ; 5. Inner ring approx. by outer rings K 2 ; 6. Inner ring approx. by inner ring parent K 2 ; 7. Finest level contribution by inner ring parent KN.

27 Temporal complexity (2) Total operation count depends on l f, N, and K ; with K << N and l f = f (N); For large l f this leads to : Complexity l α4 f N + βkn + γ4 f K 2 l 2 with αβγ,,, and K of order O(10) uniform distribution resulting in of particles, for a

28 Temporal complexity (3) an operation count linear in N Uniform distribution Non-uniform distribution

29 Numerical method for simulating : Contour dynamics (1) 2D, inviscid, incompressible vortex flows; Patches of uniform vorticity; Velocity field determined by the evolution of the contour bounding the patch: ur (,) t = % ln r r dr. C() t

30 Contour dynamics (2)

31 An O(N 2 ) method Contour dynamics (3)

32 Simple case : N = O(10 2 ) : Contour dynamics (4) Hard case : N = O(10 4 ) :

33 Instead of point sources, use source distributions like vorticity : Apply Anderson s method

34 Accuracy of Poisson kernel (1) Kernel is very inaccurate for r Ø a : K 2 1 a 1 ( ) (, ) (, ) r φ r θ " φ a s h i a 2 a 2 2π i= 1 1 2( r) cos( θ s ) + ( r) i Anderson solves for this: C.R. Anderson, SIAM J.Sci.Stat.Comput. Vol 13, No. 4, 1992; Vosbeek solves for this HEM (Hierarchical-Element Method): P.W.C. Vosbeek et al., J.Comput.Phys., Vol. 161, 2000.

35 Accuracy of Poisson kernel (2) From: P.W.C. Vosbeek et al., J.Comput.Phys., Vol. 161, 2000.

36 Not only point sources! Advantages of AM/HEM Operations for constructing and combining elements are easy to formulate: only function evaluation; These operations are almost identical in 2 and 3 dimensions (not discussed); HEM more accurate than AM; O(N).