regularized fast multipole method (rfmm) for geometric...
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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
FMM and Velocity Verlet scheme
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
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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.
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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‖
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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)
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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.
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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.
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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 .
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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).
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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.
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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.
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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))
[Φclose(x3 ∈ B1) + Φfar(x3 ∈ B1)
]+ χ(x3)
[Φclose(x3 ∈ B2) + Φfar(x3 ∈ B2)
]Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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).
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
RFMM and the Outer Solar System
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
time (unit = 1 day) −− Time−step = 10 days
Rel
ativ
e er
ror o
n H
amilt
onia
n
Relative error on Hamiltonian versus time in days − 3p7b
regular FMMclassical FMM
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
RFMM and the Outer Solar System
L = 3, NL = 7, No = 1, Rreg = 0.25
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
RFMM and the Outer Solar System
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
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 − 5p7b
regular FMMclassical FMMwithout FMM
0 0.5 1 1.5 2x 106
−8
−7
−6
−5
−4
−3
−2
−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 − 6p7b
regular FMMclassical FMMwithout FMM
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
RFMM and the Outer Solar System
L = 6, NL = 7, No = 1, Rreg = 0.25
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
RFMM and the Outer Solar System
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
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 − 10p7b
regular FMMclassical FMMwithout FMM
0 0.5 1 1.5 2x 105
−11
−10
−9
−8
−7
−6
−5
−4
−3
−2
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 − 10p7b
regular FMMclassical FMMwithout FMM
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
RFMM and the Outer Solar System
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
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 2Nei Tr=0.45
regular FMMclassical FMMwithout FMM
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
time (unit = 1 day) −− Time−step = 10 days
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
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
RFMM and the Outer Solar System
L = 3, NL = 7, No = 2, Rreg = 0.45
Eric Darrigrand Seminaire IECN – 15 juin 2010
FMM and Velocity Verlet scheme
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.
Eric Darrigrand Seminaire IECN – 15 juin 2010