An implementation of the Fast Multipole Method

without multipoles

Robin Schoemaker

Scientific Computing Group Seminar

The Fast Multipole Method – Theory and ApplicationsSpring 2002

Overview

• Fast Multipole Algorithm

• Anderson’s method– Poisson’s formula instead of multipole

expansions

• Application

N-body interactions (1)

• Particle simulations in physical systems require the evaluation of a potential function

, (yielding a force field);• These pair wise interactions are

Coulombic/gravitational in nature:

for a system of N charged particles in 2dimensions with strengths qi .

( )φ r

( )1

log 1, ,N

j i j ii

q j Nφ=

= − =∑r r r …,

N-body interactions (2)

If N is the total number of particles involved,

the temporal complexity of such a system is

.( )2O N

Multipole expansion

{ }{ }

1 1 1

, 1, ,

, 1

( ) log( )

, ,

( )

kpm mk i

i

i i

iki k

i k i

m q i m

z i m

a q zz z q a

z

z r

k

zφ

φ= = =

=

=

+

<

−=∑ ∑ ∑

…

…

"

For charges of strengths located

at points , with , the potential

is approximated by the

w

multipole expansion

ith ,

for z z r∈ >#any with .

Error bound

1

1 ,m

ii

A q p=

= ≥∑For and any

1 1

/ 2.

1( ) log( )/ 1 /

12

ppm

kki

i k

p

a Az z qz z r z

z r

r

A

φ= =

− − ≤ −

≤ ≥

∑ ∑

, for

Example of speed-up (1)

Two sets of charges are well-separated :

0

0

0 0

1, ,

1, ,

3

i

j

x x r i m

y y r j n

x y r

− < =

− < =

− >

…

…

for

for

,

,

.

y0

y1

yn

r

x0

x1

xmr

r

Example of speed-up (2)

The potential at the points {yi} due to charges

at {xj} is computed directly via

( )( )

1

1, ,i

m

x ji

O nm

n

y n

m

jφ=

=∑ …for all and requires

computations (evaluating fields at

p

oint

s).

y0x0

x1

xm

y1

yn

r

r

r

Example of speed-up (3)

• A p-term multipole expansion due to charges q1 ,…, qm about x0 requires O(mp) operations;

• Evaluating this expansion at all {yi} : 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) .

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.

M2M

0 0

01

0 00

1

1 0

log( )( )

( ) lo

(

1

1

)

g( )p

ll

l

ll

pk

kk

l kl k

k

az

bz a z

zl a z

b a zk

zz z

l

a zφ

φ=

−

=

=

+

− = − −

−

− +

∑

∑

∑"

"

into

w ith

R

R+|z0|

z0

D1

D

( )zφ

M2L

( ) ( )

( )

0

0 0 01 0

0

10 0 0

( )

1 log

11 1 11

pl

ll

pkk

kk

pkk

l k llk

z b z

ab a z

z

l ka ab l

kz z l z

φ=

=

=

⋅

− + −

+ − − − ≥ − ⋅

∑

∑

∑

"

"

"

with

for

RR z0

D1

D2 ( )zφ

cR

( )0 1 1z c R c> + > h wit

L2L

Shifting a Taylor expansion via an exact translation operation with a finite number of terms:

{ }

( ) ( )0 00 0

0, , , 0, , ,k

n n nk k l lk k

k l k l

ka z z a z

z a

l

k n

z

z

−

= = =

− = −

=

∑ ∑ ∑

…For any complex nd

a

.

Hierarchy: FMA (1)

well separated cells

parent cell Bparent cell AM2L

M2M L2L

children of A children of B

Hierarchy: FMA (2)

Levels of refinement

Anderson’s method

• FMM: due to large numbers of particles;

• Not forced to use a multipole expansion as the ‘computational element’.

Instead use Poisson’s formula

( )zφ

Poisson’s formula (1)

Solving Laplace’s equation for the potential outside a disk with radius a containing ‘sources’ :

( ) ( )

( ) ( )( ) ( )( ) ( )

2 2

2 20

, log

1 1, log2 1 2 co )

.

s(

ar

a ar r

r r

a s a dss

r a

π

φ θ κ

φ κπ θ

= +

− − − − +

>

∫

for

Poisson’s formula (2)

… and for the potential inside a hole with radius a from ‘sources’ outside:

( ) ( )( ) ( )

2 2

2 20

1 1, ( , )2 1 2 cos( )

.

ra

r ra a

r

r a dss

a

sπ

φ θ φπ θ

− = − − +

<

∫

for

Outer ring approximation (1)

Numerical representation :

( ) ( )( )

( ) ( )

( ) ( )( ) ( )( ) ( )

2

2 21

, log

1 1, log2 1 2 cos( )

2 1,

2 / cos , sin , 1, ,

K ar

i a ar ri i

i

r r

a s a hs

M K M

h K s a ih a ih i K

φ θ κ

φ κπ θ

π

=

+

− − − − +

= +

= = =

∑

"

…

With an integer and we set

and :

Outer ring approximation (2)

K

( ),rφ θ

Parent outer rings

Construction via Poisson’s formula:

Inner ring approximation

Similar as the outer ring construction:

( ) ( )( )

( )

( ) ( )( ) ( )

2

2 21

,

1 1,2 1 2 cos(

.

)

2 1,

2 / cos , sin , 1, ,

K ar

i a ar ri i

i

r

a

r

ss

a

h

M K M

h K s a ih a ih i K

φ θ

φπ θ

π

=

− −

<

− +

= +

= = =

∑

"

…

f

or

With an integer and we set

and :

( ),rφ θ

K

Children inner rings

K

K

Hierarchical tree

Levels of refinement

Hierarchy in action

Temporal complexity (1)

Seven steps in the hierarchy :1. Finest level direct interactions ∝ N2 ;

2. Finest level outer ring construction ∝ KN;

3. Finest level contribution by outer rings in well-separated area ∝ KN;

4. Outer ring approximations ∝ K2;

5. Inner ring approx. by outer rings ∝K2;

6. Inner ring approx. by inner ring parent ∝ K2;

7. Finest level contribution by inner ring parent ∝ KN.

Temporal complexity (2)

• Total operation count depends on lf , N, and K ; with K << N and lf = f (N);

• For large lf this leads to :

2 244 f fl lN N KKα β γ− + +∼Complexity

, , , (10)K Oα β γwith and of order

uniform distributi

for a

of particles,

resulting

on

in

Temporal complexity (3)

an operation count linear in NUniform distribution Non-uniform distribution

Contour dynamics (1)

Numerical method for simulating :

• 2D, inviscid, incompressible vortex flows;

• Patches of uniform vorticity;

• Velocity field determined by the evolution of the contour bounding the patch:

( )

( , ) ln | | .C t

t d′ ′= −∫u r r r r%

Contour dynamics (2)

Contour dynamics (3)

An O(N2) method

Contour dynamics (4)

• Simple case : N = O(102) :

• Hard case : N = O(104) :

Apply Anderson’s method

Instead of point sources, use

source distributions like vorticity :

Accuracy of Poisson kernel (1)

( ) ( ) ( )( ) ( )

2

2 21

1 1, ,2 1 2 cos( )

K ar

i a ar ri i

r a s hs

φ θ φπ θ=

− − − +

∑"

Kernel is very inaccurate for r Ø a :

• 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.

Accuracy of Poisson kernel (2)

From: P.W.C. Vosbeek et al., J.Comput.Phys., Vol.161, 2000.

Advantages of AM/HEM

• Not only point sources!

• 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).