The Fast Multipole Method (FMM)

Transcription

1 The Fast Multipole Method (FMM)

2 Motivation for FMM Computational Physics Problems involving mutual interactions of N particles Gravitational or Electrostatic forces Collective (but weak) long-range forces cannot be ignored Example Application DNA, proteins, biomollecular processes 200k to 1M atoms History 1977 Pincus, Scheraga Group of distant particles replaced with a Pseudoparticle 1987 Greengard, Rokhlin Rigorous numerical analysis: how many terms are needed Local expansion: Groups interact with groups 1992 Rokhlin: Applied to Helmholz Operator (MoM) FMM Slide 2

3 Need for Fast Algorithms in EM Basic Problem in EM Everything is coupled to everything Problem complexity tends to scale as N 2, where N is the number of unknowns Large EM Problems Radar, Remote Sensing (Large scatterers) Wireless Communications (Environment) Antenna/environment studies (body, vehicles) Complete RF system simulation FMM Slide 3

4 Example of a Fast Algorithm in SP Example: DFT Computation: Complexity: O(N 2 ) Alternative: FFT Partition (odd/even) Recursion (Reuse information) Complexity: O(NlogN) FMM Slide 4

5 Fast Multipole Algorithm Method of Moments (MoM) Get system of the form v = Z i solution i = Z -1 v Coupling matrix Z is a dense NxN matrix Computing inverse O(N 2 ) Fast Multipole Method Exploit structure of Z fast algorithm to compute Z i Combined with conjugate gradient method for inverse Obtain O(N 3/2 ) algorithm Can also do multistage: O(NlogN) algorithm Follow development in Coifman, et. al., The Fast Multipole Method for the Wave Equation: A Pedestrian Prescription, IEEE Antennas and Propagation Magazine FMM Slide 5

6 MoM Problem Scalar wave equation Boundary condition S Application of MoM yields FMM Slide 6

7 Computation of Inverse Small systems (N 100) Larger systems Compute inverse iteratively (conjugate gradient) Repeatedly solve Problem: Matrix vector multiply is O(N 2 ) Can we accelerate this? FMM Slide 7

8 Airports Problem: Connect all airports with flights Option 1: Connect all cities to all cities Problem: Inefficient Bridgeport New York Newark Atlantic City Philadelphia Wilmington Washington Richmond Virgina Beach FMM Slide 8

9 Airports Problem: Connect all airports with flights Option 1: Connect all cities to all cities Problem: Inefficient Bridgeport New York Newark Atlantic City Philadelphia Wilmington Washington Richmond Virgina Beach FMM Slide 9

10 Airports Problem: Connect all airports with flights Option 1: Connect all cities to all cities Problem: Inefficient Option 2: Use hubs All cities served Far fewer connections to deal with! Newark New York Atlantic City Richmond Bridgeport Philadelphia Wilmington Washington Virgina Beach FMM Slide 10

11 Airports Problem: Connect all airports with flights Option 1: Connect all cities to all cities Problem: Inefficient Option 2: Use hubs All cities served Far fewer connections to deal with! Newark New York Atlantic City Richmond Bridgeport Philadelphia Wilmington Washington Virgina Beach FMM Slide 11

12 FFT Analogy Revisited Transformation Matrix Can generate sparse factorization Exploits algebraic properties (exact) More efficient to compute y = W x with FMM Slide 12

13 Sparse Factorization of Z Z can be decomposed as All sparse matrices Exploits analytic properties of Z Approximate, but can control level of approximation (down to precision of computer) Result: O(N 3/2 ) method to compute product of Z with an arbitrary vector Can repeat decomposition (approach O(NlogN)) Not really necessary FMM Slide 13

14 Factoring Z: Identity 1 Consider the Green s function Spherical Bessel Function Legendre Polynomial Spherical Hankel Function Transformed spherical wave from x to x Sum of product of spherical waves Separate long interaction (X) from near interactions (d) FMM Slide 14

15 Factoring Z: Identity 2 Expand spherical factor in d as Substituting, Write j(.) as sum of plane waves (all directions on unit sphere) FMM Slide 15

16 Meaning of Factorization Superimpose all plane waves Phase shift required due to small offset d Complex amplitude of plane wave traveling in k direction from x to x +X Fields at x +X are now a superposition of plane waves Fields at a small distance d from x +X found by shifting the phase of the plane waves Note: L > kd for good accuracy (d should be kept small) FMM Slide 16

17 Back to MoM Result so far Note: Substitute into coupling matrix Interpretation #1 FMM Slide 17

18 Back to MoM Result so far Note: Substitute into coupling matrix Interpretation #1 Source Basis Function (a source with local support) FMM Slide 18

19 Back to MoM Result so far Note: Substitute into coupling matrix Spatial Fourier Transform in x => Transform source to plane waves (k is pw dir) Interpretation #1 FMM Slide 19

20 Back to MoM Result so far Note: Substitute into coupling matrix Interpretation #1 Complex Transfer Coefficient: Attenuation/ phase shift of pw in k dir from x to x +X FMM Slide 20

21 Back to MoM Result so far Note: Substitute into coupling matrix Interpretation #1 Inverse Fourier Transform: pw back to fields in observation space FMM Slide 21

22 Back to MoM Result so far Note: Substitute into coupling matrix Project onto testing functions (Galerkin) Interpretation #1 FMM Slide 22

23 Back to MoM Interpretation #2 Rearrange Let X=X m -X m : X m is coordinate near local support of f n (x) FT of nth weighting function (plane waves received by f n (x) One-to-one coupling of pw s from source to observation FT of n th basis function (plane waves radiated by f n (x) Fields due to current i FMM Slide 23

24 Hub Interpretation Old MoM (far interactions) Apply current to each source basis function Observe field on each weighting function N 2 operations f n (x) V = Z i f n (x ) FMM Slide 24

25 Hub Interpretation Using Factorization Use pw basis functions: Currents to plane waves For single hub, plane waves superimpose 1-to-1 connection of K plane waves from source to dest. At destination, use pw weighting functions Transform from pw to observed fields (integrate over k) V n (k) V n (k) X m X m FMM Slide 25

26 Algorithm Setup 1/4 Divide N basis functions into M groups: M N 1/2 α=1 α=3 α=2 α=4 α=1 α=3 α=2 α=1 α=2 X 1 X 2 X 3 α=4 α=3 α=4 α=5 α=6 α=5 α=6 α=5 α=6 m=1 m=2 m=3 m = Group number α = Local element number n(m, α) = Global element number (like with normal MoM) X m = coordinate at center of mth group FMM Slide 26

27 Algorithm Setup 2/4 Close Interactions Require many terms L in expansion Relatively few close elements for large-scale problem Just use normal dense computation of Z for these For group pairs (m, m ) that are close Close means min{dist[n(m,α),n(m,α )]} = 2π/k Z mαm α is computed using Here we have to handle singularities of Green s function Requires O(N 3/2 ) computations FMM Slide 27

28 Algorithm Setup 3/4 Discretize Directions Form set of K directions Accuracy requires K L 2 > (kd) 2 D = max(d) Compute excitation vectors Fourier transform of basis/weighting functions From partitioning, kd on order of (N/M) 1/2 Requies O(N 2 /M) computations FMM Slide 28

29 Algorithm Setup 4/4 Regions that are not nearby i.e. Z mαm α = 0 Compute the matrix elements FMM Slide 29

30 Fast Matrix-Vector Multiply 1/3 Need to compute Compute KM quantities FT current in mα => plane waves Superimpose => plane waves from mth element Computation: O(N 2 /M) FMM Slide 30

31 Fast Matrix-Vector Multiply 2/3 Compute KM quantities FMM Slide 31

32 Fast Matrix-Vector Multiply 2/3 Compute KM quantities Coupling of plane waves from group m to group m Note: 1-to-1 coupling in k! FMM Slide 32

33 Fast Matrix-Vector Multiply 2/3 Compute KM quantities Observed plane-waves in group m Coupling of plane waves from group m to group m Note: 1-to-1 coupling in k! FMM Slide 33

34 Fast Matrix-Vector Multiply 2/3 Compute KM quantities Observed plane-waves in group m Coupling of plane waves from group m to group m Note: 1-to-1 coupling in k! Note: Just for distant terms Requires O(MN) operations FMM Slide 34

35 Fast Matrix-Vector Multiply 3/3 Put it all together Dense terms Transform pw s in group m back to fields Requires O(N 2 /M) operations FMM Slide 35

36 Conclusion Large-scale Problems in Physics/Engineering Problems involving mutual interactions of N particles Classical methods often lead to O(N 2 ) algorithms Fast Multipole Method Separate near and far interactions (hub analogy) Near interactions captured with exact (dense) operator Far interactions Collect particle interactions into groups Relate groups with each other (not individual particles) Application to MoM in EM Reduced an O(N 2 ) problem to O(N 3/2 ) Significant savings for large scale problems FMM Slide 36