Fast multipole method

Transcription

1 203/0/ page Chapter 4 Fast multipole method Originally designed for particle simulations One of the most well nown way to tae advantage of the low-ran properties Applications include Simulation of physical systems (a) Gravitational potential due to a distribution of masses (b) Electrostatic potential due to a distribution of charges Electromagnetics of complex systems (c) Stellar clusters (d) Protein folding (e) Acoustics (f) Turbulence 2 Grand challenge problems in large numbers of variables 3 Learning theory (a) Kernel methods (b) Support Vector Machines 4 Graphics and Vision, Light scattering Introduced by Rohlin & Greengard in 987 One of the 0 most significant advances in computing of the 20th century Approximation result with O(m + n) cost Use analytical manipulation of series to achieve faster summation Approximate result with arbitrary accuracy This section is mainly based on [4]

2 203/0/ page 2 2 Chapter 4 Fast multipole method Figure 4 (Well separated) source and target points 40 A basic idea m source particle i with charge q i at position s i By Coulomb s law, the potential at target j located at t j for n targets: ϕ(t j ) = q i K(s i, t j ), j = : n, where, for example, K(s, t) = s t Write Then K = (K(s i, t j )), q = (q i ), Φ = (ϕ(t j )) Φ = Kq This usually costs O(mn) However, this cost can be reduced if K is separable Assume r K(x, y) = f (x)g (y) = ( f (x) f p (x) ) Then K = = f (s ) f r (s ) f (s 2 ) f r (s 2 ) } f (s m ) {{ f r (s m ) } m r r f g T, (f = (f (x i )) i, g = (f (x j )) j ) Consider a special K as an example g (y) g p (y) g (t ) g (t 2 ) g (t n ) = F G T g r (t ) g r (t 2 ) g r (t n ) }{{} Theorem 4 δ (0, ), [0, ) For any x, y [0, ) satisfying have r K(x, y) = (x y) 2 = f (x)g (y) + R r, f (x) = (x ), g (y) = R r (r + )δ r r n (y ) +, x y δ we

3 203/0/ page 3 3 x { y { 0 Figure 42 Well separated points Proof Then use (x y) 2 = ((y ) (x )) 2 = (y ) 2 ( ) 2 x r ( t) 2 = t + R r, t δ < Note that, to reach a given tolerance τ, (r + )δ r < τ, y we need r = O( log τ ) Definition 42 Two sets {x i }, {y j } are well separated if there exist points x 0, y 0 and d > 0 st where α > 2 is a constant x i x 0 < d, y i y 0 < d, x 0 y 0 > αd, Corollary 43 If K is formed for all the subsets of a total of n points, then any off-diag bloc of K has numerical ran O(log n) Proof The interval can be repeatedly partitioned 402 Math preliminaries Multipole expansion Consider K(, 0 ) = log( 0 )

4 203/0/ page 4 4 Chapter 4 Fast multipole method Lemma 44 A point change is located at 0 Then for any st > 0 log( 0 ) = log ( 0 ) Proof log( 0 ) log = log( t) Taylor Theorem 45 (Multipole expansion (with center 0)) q i, i = : m at points i, i = : m with i < d Then for any C with > d, the potential is Furthermore, with truncation ϕ() = q log + a q = q i, a = ϕ() = M p (, 0 ) + R r r a M p (, 0 ) = q log + q i i ( ) r+ ( A R r α c c c =, Q m = q i, α = d ) ( ) r c A /c Proof The previous lemma gives the expansion: ϕ() = q i K( i, ) = q i [log ( m ) = q i log = Q log + a =r+ ( m q =r+ q i q i i ) d q ( i ) =r+ ] ( i ) ( ) c In particular, if c 2 or 2d, R r q ( ) r 2

5 203/0/ page 5 5 By the theorem ϕ xi (y j ) M p (y j, x 0 ) q ( ) r 2 The coefs of in a multipole expansion costs mp, once for all y i To compute the potential (force) at {y j } due to the charges at {x i }, we need m ϕ x i (y j ) for all j The evaluation of the expansion at {y j } is nr Total cost mr + nr Translation/shift Theorem 46 (Shifting center to 0) Suppose ϕ() = a 0 log( 0 ) + a ( 0 ) (4) is a ME of the potential due to q i, i = : m located inside the circle D of radius R with center 0 Then for outside the circle D of radius R + 0 and centered at the origin, ϕ() = a 0 log + b l = l a l 0 l= b l l ( l With truncation, the error is bounded by A 0 + R 0 +R ) a 0 l 0 l r+ Proof log( 0 ) log = log( t), etc Taylor expansion at 0 This means we can shift the center without loss of accuracy Theorem 47 (Multipole to local expansion (for the purpose of later evaluation)) q i, i = : m located inside the circle D with radius R and centered at 0, 0 > (c + )R, c > Then (4) converges inside the circle D 2 of radius R about the origin Inside D 2, ϕ() = b 0 = b l = b l l l= 0 a a +l 0 ( ) + a 0 log( 0 ) ( l + ) ( ) a 0 l0, l

6 203/0/ page 6 6 Chapter 4 Fast multipole method D R 0 q q i 0 For r max(2, 2c/(c )), error bound A(4e(r + c)(c + ) + c 2 ) c(c ) ( ) r+ c D R 0 q q i cr D 2 R 0 C Figure 43 Theorem 48 (Shifting center within a region of analyticity) For any 0,, a, = 0 : n, [ n n n ( ] a ( 0 ) = a )( 0 ) l l =0 l=0 =l l 403 FMM See slides-xuepdf Patition the domain into multiple levels of boxes Φ li : r -term multipole expansion about the box center of the potential field due to all the particles in box i at level l

7 203/0/ page 7 7 Figure 44 Single-level FMM and multi-level FMM Ψ li : r -term local expansion about the box center of the potential field due to all the particles outside box i and its nearest neighbors Ψ li : r -term local expansion about the box center of the potential field due to all the particles outside box i s parent and its parent nearest neighbors I i : Interaction list: children of the nearest neighbors of i s parent which are well separated from i Main steps: Upward: Figure 45 Computational boxes at different levels (a) for i = : 4 l max form ME of Φ lmax,i coefficients of ME only and NO evaluation! (b) for l = l max : 0 for i = : 4 l form ME of Φ l,i by shifting the center of each child coefficients of ME only and NO evaluation!

8 203/0/ page 8 8 Chapter 4 Fast multipole method 2 Downward: for l = : l max 3 Sum: for i = : 4 l for j I i, Ψ li = Ψ li + Φ lj with center of Φ lj shifted to that of i if l < l max for i = : 4 l for j I ch(i), form Ψ l+,i by shifting the center of Ψ li to j for i = : 4 l max for j box i, evaluate Ψ lmax,i( j ), ˆΨ lmax,ij of j with i s other nodes and i s neighbors i j Ψ l max,i( j ) + ˆΨ lmax,ij

9 203/0/ page 9 Bibliography [] S Börm, L Grasedyc, and W Hacbusch, Introduction to hierarchical matrices with applications, Eng Anal Bound Elem, 27 (2003), pp [2] S Chandrasearan, P Dewilde, M Gu, and N Somasunderam, On the numerical ran of the off-diagonal blocs of Schur complements of discretied elliptic PDEs, SIAM J Matrix Anal Appl, 3 (200), pp [3] S Chandrasearan, M Gu, X Sun, J Xia, and J Zhu, A superfast algorithm for Toeplit systems of linear equations, SIAM J Matrix Anal Appl, 29 (2007), pp [4] L Greengard and V Rohlin, A Fast Algorithm for Particle Simulations, J Comput Phys, 73 (987), pp [5] J Carrier, L Greengard, and V Rohlin, A fast adaptive multipole algorithm for particle simulations, SIAM J Sci Stat Comput, 9 (998), pp [6] P G Martinsson, V Rohlin, and M Tygert, A fast algorithm for the inversion of general Toeplit matrices, Comput Math Appl 50 (2005), pp [7] P Starr, On the Numerical Solution of One-Dimensional Integral and Differential Equations, Thesis advisor: V Rohlin, Research Report YALEU/DCS/RR-888, Department of Computer Science, Yale University, New Haven, CT, 99 [8] J Xia, On the complexity of some hierarchical structured matrix algorithms, SIAM J Matrix Anal Appl, 33 (202), pp [9] J Xia, S Chandrasearan, M Gu, and X S Li, Fast algorithms for hierarchically semiseparable matrices, Numer Linear Algebra Appl, 7 (200), pp