regularized fast multipole method (rfmm) for geometric...

FMM and Velocity Verlet scheme

Regularized Fast Multipole Method (RFMM)for Geometric Numerical Integrations

Eric DARRIGRAND

Universite de Rennes 1 – INRIA - [email protected]

http://perso.univ-rennes1.fr/eric.darrigrand-lacarrieu

joint work with Philippe CHARTIER and Erwan FAOU

Eric Darrigrand Seminaire IECN – 15 juin 2010


Outline

• Hamiltonian systems

? symplectic integrators

? motivation of fast methods

• a classical FMM

? derivation of the FMM

? FMM and symplectic integrators

? some improvements of the FMM

• a regularized FMM (RFMM)

? regularization of the classical FMM

? numerical application to the Solar System



Hamiltonian systems

ODE system: q = M−1p ∈ R3N

p = −∇U(q) ∈ R3N

where M = diag(m1IR3 , · · · ,mNIR3)

Hamiltonian of the system:

H(p, q) = T (p) + U(q)

? T (p) =12pTM−1p is the kinetic energy

? U(q) is the potential function

−→ For invariance energy: Use of symplectic integrators.



Symplectic integrators

Velocity Verlet scheme: ([Hairer, Lubich, Wanner - 06])qn+ 1

2= qn + h

2 vn

vn+1 = vn − h∇U(qn+ 12)

qn+1 = qn+ 12

+ h2 vn+1

where qn ≈ q(nh) and vn ≈ v(nh) with v = q = M−1p

−→ explicit, symplectic and symmetric

Calculation of the potential:

astronomy / molecular dynamics =⇒ evaluation of∇U of order N2.

For instance, for the Outer Solar System,

U(q) = −γ5∑i=1

i−1∑j=0

mimj

‖qi − qj‖



The Fast Multipole Method (FMM): Basic idea

• Compute: ∀i = 1, N , Yi =N∑j=1

Mij Xj −→ Complexity = O(N2)

• Suppose: ∃ (ai)i , (bj)j /Mij = ai bj

Algorithm:Step 1: F =

N∑j=1

bj Xj

Step 2: ∀i, Yi = aiF

−→ Complexity = O(N)

• Suppose: ∃ (ali)il , (bl

j)jl /Mij =L∑

l=1

ali bl

j , L << N

Algorithm:

Step 1: ∀l , F l =N∑j=1

blj Xj

Step 2: ∀i , Yi =L∑l=1

aliFl

−→ Complexity = O(N L)



FMM: a 1D simple example

Speed up of matrix-vector productsMX with given X and

Mi j =

1

xi − xjif i 6= j

1 if i = j

Suppose the configuration

B1 B2 B3 B4

0 1

•xi xj

•

For xj ∈ B3 ∪B4

1xi − xj

=1

C1 − xj − (C1 − xi)=

1C1 − xj

∞∑l=0

(C1 − xiC1 − xj

)l.



Hence∑

j/xj∈B3∪B4

Mi j .Xj =Lε∑l=0

(C1 − xi)l∑

j/xj∈B3∪B4

Xj

(C1 − xj)l+1.

•Ckxi

◦xj1◦

xj2◦

xj3◦

Complexity of a matrix-vector product: O(KN ln N + N2/K)with K = number of boxes and N = number of points xj)

Optimal complexity: O(N3/2 ln N) obtained with K ∼ N1/2.



FMM for the Outer Solar System

Coulomb forces of the Outer Solar System

∇iU(q) = −γ∑j 6=i

mimj∇1G(qi, qj) = −γ∑j

Mi,j

with

G(x, y) =1

‖x− y‖Mj,j = 0

Mi,j = mimj∇1G(qi, qj) for i 6= j

−→ common matrix-vector product calculable with FMM(V. Rokhlin - L. Greengard)

First step: FMM expansion for∑j 6=i

wj‖xi − yj‖

for given {wj}j .



18 interactions

BT1

BS2

BS3

x11

x12

x13

y21y22

y23

y31y32

y33 11 interactions

BT1

BS2

BS3

C2◦

C3◦

C1

◦x11

x12

x13

y21y22

y23

y31y32

y33



Source boxes – multipole expansion:(yj , wj) source points in a box Bsrc of center Csrc and xi a target point far away.

Φ(xi) =∑j

wj‖xi − yj‖

=∑j

wj‖(xi − Csrc)− (yj − Csrc)‖

Consider (xi − Csrc)↔ (r′, θ′, ϕ′) and (yj − Csrc)↔ (ρj , αj , βj). Then

Φ(xi) =∞∑n=0

n∑m=−n

Mmn

r′n+1Y mn (θ′, ϕ′)

Mmn =

∑j

wj ρnj Y

−mn (αj , βj)

Choice of the truncation, with a = radius of the boxes:∣∣∣∣∣Φ(xi)−L∑n=0

n∑m=−n

Mmn

r′n+1Y mn (θ′, ϕ′)

∣∣∣∣∣ ≤∑j |wj |r′ − a

( ar′

)L+1



Target boxes – local expansion:xi in a box Btrg of center Ctrg ; (xi − Ctrg)↔ (ρi, αi, βi)If (Csrc − Ctrg)↔ (r, θ, ϕ) with r > (c+ 1)a, c > 1 and ρi ≤ a.

Then Φ(xi) =∞∑ν=0

ν∑µ=−ν

Lµν ρνi Y

µν (αi, βi)

Lµν =∞∑n=0

n∑m=−n

ı|µ−m|−|µ|−|m| Amn Aµν

(−1)n Am−µν+n

Y m−µν+n (θ, ϕ)rν+n+1

Mmn

with

Amn =(−1)n√

(n−m)!(n+m)!

Error estimates:∣∣∣∣∣Φ(xi)−L∑ν=0

ν∑µ=−ν

Lµν ρνi Y

µν (αi, βi)

∣∣∣∣∣ ≤∑j |wj |

ca− a

(1c

)L+1



Algorithm:

• Step 0: w-independent quantities:

? Translation operator: multipole exp. around Csrc→ local exp. around Ctrg.

? Far moments fn,mj and local moments gν,µi .

• Step 1: Far fields: ∀Bsrc, ∀yj ∈ Bsrc, Fn,mBsrc← wj · fn,mj .

• Step 2: Local fields: ∀Btrg, ∀Bsrc far from Btrg, (Gν,µBtrg)ν,µ ← (Fn,mBsrc

)n,m.

• Step 3: Far interactions: ∀Btrg, ∀xi ∈ Btrg, ∀(ν, µ), Φfar(xi)← Gν,µBtrg· gν,µi .

• Step 4: Close interactions: ∀Btrg, ∀xi ∈ Btrg,Φclose(xi)← neighbor-source-points contribution.

• Step 5: The matrix-vector product: ∀Btrg, ∀xi ∈ Btrg,Φ(xi) ≈ Φclose(xi) + Φfar(xi).



Complexity:With N = number of degrees of freedom.

K = number of boxes.

T = number of translations between boxes.

L = truncature parameter.

• Translations between boxes (step 2): T × L4.

• Local translations inside the boxes (steps 1 and 3): N × L2.

• Close interactions (step 4): N2/K.

• One-level FMM (SL-FMM): total cost ∼N2/K + K2 L4 + N L2.

• Multilevel FMM (ML-FMM): total cost ∼N2/K + K L4 + N L2.−→ Optimal choice: K ∼ N ; complexity N L4.



FMM discontinuities:

•CBT

CBS1• •CBS2

xi1◦

yj1◦ ◦

yj2

CBT1• •CBT2

xi1◦ ◦xi2

Far interactions of BT1

Far interactions of BT2

Close interactions of xi1 ∈ BT1

Close interactions of xi2 ∈ BT2

−→ loss of regularity and preservation of the Hamiltonian



Some algorithm improvements:

• ML-FMM using multipole-multipole and local-local translations.

• Optimization of the translation costs:

? vFMM: Number of multipoles L adapted to the level in ML-FMM.Petersen et al., 1994;

? Rotation of the system: translations are more efficient along the axes.Greengard and Rokhlin, 1997; −→ reduces L4 to L2.

? Convolution ((n− ν), (m− µ)) and FFT for translations Mmn −→ Lµν .

Elliott and Board, 1996; −→ reduces L4 to L2 ln L.

• Alternatives:

? FFTM: Convolution (Csrc − Ctrg) and FFT for the translations Csrc to Ctrg.Ong, Lim, and Lee, 2003. A modified SL-FMM.

? Generalization of the FMM: e.g., using the SVD concept.



RFMM – a regularized FMM: Overlapping boxes and partition of unity.

Virtual box B1

Virtual box B2

Virtual box B3

Box 1 Box 2 Box 3x1• x2•x3•

Φ(x1) ≈ Φclose(x1 ∈ B2) + Φfar(x1 ∈ B2)

Φ(x2) ≈ (1− χ(x2))[Φclose(x2 ∈ B2) + Φfar(x2 ∈ B2)

]+ χ(x2)

[Φclose(x2 ∈ B3) + Φfar(x2 ∈ B3)

]Φ(x3) ≈ (1− χ(x3))


]+ χ(x3)


]Eric Darrigrand Seminaire IECN – 15 juin 2010


Algorithm consequencies:

? The overlapping of the boxes may affect the speed of the convergence of thelocal/multipole expansions

=⇒ larger order of neighborhood.

? A target point may belong to several boxes.

? A low increase of the number of points in each target box.=⇒ A low increase of the cost of the step involving the local moments.

? No change of the global algorithm complexity.

=⇒ a regularized FMM for a comparable computational cost



1D illustration: x1, · · · , x800, uniformly distributed on [0, 1] ;

Compute ∀i = 1, · · · , 800: Si =400∑

j 6=i,j=250

1‖xi − xj‖

.

0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10 x 10−3

x

erro

r on

Sum

y( G(x

,y) )

Error on Sumy( G(x,y) )

classical FMMsmooth FMM

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−2

0

2

4

6

8

10 x 10−3

x

erro

r on

Sum

y( G(x

,y) )

Error on Sumy( G(x,y) )

classical FMMsmooth FMM



RFMM and the Outer Solar System

Problem: The solar system (Sun, Jupiter, Saturn, Uranus, Neptune, Pluto).Initial data: positions/velocities on sept. 4, 1994.FMM for the computation of

∇iU(q) = −γ∑j 6=i

mimj∇1G(qi, qj) , G(x, y) =1

‖x− y‖

Notations:

? L: number of multipoles. Typical value ≈ 6, very accurate with L = 15 or even 20.

? No: order of neighborhood (defines close and far interactions).

? NL: number of levels of the oc-tree. Here, a good tradeoff is NL = 7.

? Rreg: ratio (regularization zone on each side of a group / length of the group).




L = 3, NL = 7, No = 1, Rreg = 0.25

0 0.5 1 1.5 2 2.5x 106

−7

−6

−5

−4

−3

−2

−1

0

1

time (unit = 1 day) −− Time−step = 10 days

Log 10

(rela

tive

erro

r on

Ham

ilton

ian)

Log10 of relative error on Hamiltonian versus time in days − 3p7b

regular FMMclassical FMMwithout FMM

0 0.5 1 1.5 2 2.5x 106

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2


Rel

ativ

e er

ror o

n H

amilt

onia

n

Relative error on Hamiltonian versus time in days − 3p7b

regular FMMclassical FMM




L = 3, NL = 7, No = 1, Rreg = 0.25




L = 5 or 6, NL = 7, No = 1, Rreg = 0.25

0 1 2 3 4 5 6 7 8x 105

−8

−7

−6

−5

−4

−3

−2

−1


Log 10

(rela

tive

erro

r on

Ham

ilton

ian)



0 0.5 1 1.5 2x 106

−8

−7

−6

−5

−4

−3

−2

−1


Log 10

(rela

tive

erro

r on

Ham

ilton

ian)






L = 6, NL = 7, No = 1, Rreg = 0.25




L = 10, NL = 7, No = 1, Rreg = 0.25

0 2 4 6 8 10x 105

−10

−9

−8

−7

−6

−5

−4

−3

−2


Log 10

(rela

tive

erro

r on

Ham

ilton

ian)



0 0.5 1 1.5 2x 105

−11

−10

−9

−8

−7

−6

−5

−4

−3

−2


Log 10

(rela

tive

erro

r on

Ham

ilton

ian)






L = 3, NL = 7, No = 2, Rreg = 0.45

0 0.5 1 1.5 2x 106

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5


Log 10

(rela

tive

erro

r on

Ham

ilton

ian)

Log10 of relative error on Hamiltonian versus time in days − 3p7b 2Nei Tr=0.45


0 0.5 1 1.5 2x 106

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03


Rel

ativ

e er

ror o

n H

amilt

onia

n

Relative error on Hamiltonian versus time in days − 3p7b 2Nei Tr=0.45

regular FMMclassical FMM




L = 3, NL = 7, No = 2, Rreg = 0.45



A first conclusion

The regularized FMM

• recovers the invariance of energy of an Hamiltonian system,

• has the same algorithm complexity as the usual FMM.

Coming work

• Application of improvements regarding the source-to-target translations.

• Application of the regular FMM to molecular dynamics.


regularized fast multipole method (rfmm) for geometric...

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