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OSRC-Preconditioning and FMM
OSRC-Preconditioning and Fast Multipole Method:Eigenvalues investigation
Eric DARRIGRAND †
joint work with Marion DARBAS ‡ and Yvon LAFRANCHE †
† IRMAR – Universite de Rennes 1‡ LAMFA – Universite de Picardie
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Outline
• Initial motivation: 3-D Helmholtz equation
• Integral equation strategy: CFIE
• Analytical preconditioning strategy: OSRC
• Efficient calculations: FMM
• Numerical resolution results
? Validation on the unit sphere
? A trapping domain
• Eigenvalues numerical investigation
? The unit sphere
? A trapping domain
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
3-D exterior domain Helmholtz equation
Acoustic scattering in R3:
Ω−
Γu +
n
Ω+ u inc
Find u+ solution of the problem
∆u+ + k2u+ = 0, in Ω+,
u+|Γ = −uinc|Γ or ∂nu+|Γ = −∂nuinc|Γ, on Γ,
lim|x|→+∞
|x|(∇u+ · x
|x|− iku+
)= 0,
with uinc(x) = e−ikθinc·x, θinc the incidence direction, k the wavenumber.
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
CFIE integral formulation
Find ϕ = −γ+0 (u+ − uinc) ∈ H1/2(Γ) such that(
I
2+M + ηD
)ϕ = −γ+
0 uinc − ηγ+
1 uinc, on Γ,
with η ∈ C∗ the coupling parameter and with the integral operators
Mϕ(x) := −∫
Γ
∂n(y)G(x,y)ϕ(y)dΓ(y), ∀x ∈ Γ
Dϕ(x) := −∂n(x)
∫Γ
∂n(y)G(x,y)ϕ(y)dΓ(y), ∀x ∈ Γ
with G(x, y) = eik|x−y|/(4π|x− y|).
Two essential difficulties:
• implication of dense operators −→ high resolution cost
• bad spectral properties −→ low or non convergence of iterative solvers
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
OSRC-preconditioned integral formulation
Find ϕ = −γ+0 (u+ − uinc) ∈ H1/2(Γ) such that(
I
2+M − V D
)ϕ = −γ+
0 uinc + V γ+
1 uinc, on Γ.
where V is an approximation of the NtD operator V ex, derived from On-SurfaceRadiation Condition methods:
V =1
ik
(1 +
∆Γ
k2ε
)−1/2
, with kε = k + iε.
Essential remarks:• V evaluated using a Pade approximation.
• V involves only differential operators like ∆Γ.
• spectral properties strongly improved thanks to the relation I/2 +M − V exD = I
• the equation still involves the integral operators M and D.
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Use of FMM
Aim: Fast calculation of matrix-vector products with the matrix
[L]i j =
∫Γ
∫Γ
G(x,y)ϕj(y)ϕi(x)dΓ(y)dΓ(x), i, j = 1, · · · , N,
Principle: For i far from j
[L]i j ≈P∑p=1
cp∑
B/B∩suppϕi 6=∅
g(p)i,B
∑B/B∩suppϕj 6=∅
T (p)
B,Bf
(p)
j,B,
with g(p)i,B local moment in box B of target d.o.f. ϕi,
f(p)
j,Bfar moment in box B of source d.o.f. ϕj ,
T (p)
B,Btranslation operator from B to B.
Tools: Gegenbauer series and Funk-Hecke formula.
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Precisely:
cp =ik
(4π)2wp , f
(p)
j,B=
∫B∩suppϕj
e−ik<sp,y−CB>ϕj(y)dΓ(y) ,
g(p)i,B =
∫B∩suppϕi
eik<sp,x−CB>ϕi(x)dΓ(x) ,
T (p)
B,B=
L∑`=1
(−i)`(2`+ 1)h(1)` (k|CB − CB |)P`(cos(sp, CB − CB)),
CB center of B ; h(1)` spherical Hankel function ; P` Legendre polynomial;
(wp, sp)p ←→ quadrature rule on the unit sphere;∑Pp=1 ←→ integration on the unit sphere (P = (L+ 1)(2L+ 1));∑L`=1 ←→ Gegenbauer series (L = kd+ C(kd)3).
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
18 interactions
BT1
BS2
BS3
x11
x12
x13
y21
y22
y23
y31
y32y33 11 interactions
BT1
BS2
BS3
C2
C3
C1
x11
x12
x13
y21
y22
y23
y31
y32y33
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Numerical resolution (using Gmsh, MUMPS, and F90-personal codes)
Case of the unit sphere
• For numerical validation (with Mie series)
• Observations:
? Radar Cross Section
? GMRES convergence
Case of a cube with cavity
• A trapping domain
• Observations:
? Radar Cross Section
? GMRES number of iterations
? GMRES residual
? Localization of a resonance frequency
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Case of the unit sphereCFIE+OSRC+FMM – fixed discretization density nλ = 10
0 30 60 90 120 150 180−20
−10
0
10
20
30
40
!
Norm
alize
d RC
S
RCS Unit Sphere ; k=23.7
Mie seriesCFIE+FMMCFIE+OSRC+FMM
0 30 60 90 120 150 180−20
−10
0
10
20
30
40
!No
rmal
ized
RCS
RCS Unit Sphere ; k=47.4
Mie seriesCFIE+OSRC+FMM
46080 triangles 184320 triangles
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
CFIE+OSRC+FMM – at fixed wavenumber
0 30 60 90 120 150 180−10
−5
0
5
10
15
20
25
!
Norm
alize
d RC
S
RCS Sphere ; k=10 ; CFIE+OSRC+FMM
Mie seriesn" = 7
n" = 12
n" = 15
0 30 60 90 120 150 180−15
−10
−5
0
5
10
15
20
25
30
!No
rmal
ized
RCS
RCS Sphere ; k=15 ; CFIE+OSRC+FMM
Mie seriesn" = 8
n" = 16
n" = 32
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
GMRES convergence
0 10 20 30 40 500
50
100
150
200
k
GM
RES
itera
tions
Unit sphere, n!=10
CFIECFIE+FMMCFIE+OSRC+FMM
5 10 15 20 250
20
40
60
80
100
120
n!
GM
RES
itera
tions
Unit sphere, k=10
CFIECFIE+FMMCFIE+OSRC+FMM
Vs. k (nλ = 10) Vs. nλ (k = 10)
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
CPU costs
k Total CPU time Total CPU time Total CPU time
CFIE CFIE + SLFMM CFIE + SLFMM + OSRC
4.76 7 min 42” 13 min 47” 2 min 42”
11.85 9 h 43 min 4 h 33 min 32 min 40”
23.7 > 15 days 214 h 44 min 6 h 20 min
47.4 – – 48 h 48 min
• Asymptotic behavior of the SLFMM
• Cost related to OSRC operators << cost related to integral operators
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Case of a trapping domainCube [−1, 1]3 with the rectangular cavity [0, 1]× [−π/10, π/10]× [−π/10, π/10]
with incident wave (√
3/2, 0, 1/2).
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
RCS
0 60 120 180 240 300 360−20
−10
0
10
20
30
40
!
Norm
alize
d RC
S
RCS Trapping domain ; k=10.5
CFIE+FMMCFIE+OSRC+FMM
0 60 120 180 240 300 360−20
−10
0
10
20
30
!No
rmal
ized
RCS
RCS Trapping domain ; k=8 ; CFIE+OSRC+FMM
Referencen" = 5
n" = 12.5
n" = 26
41840 triangles, k = 10.5 Various mesh densities, k = 8
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
GMRES convergence
0 5 10 15 20 25 300
50
100
150
200
250
300
350
400
n!
GM
RES
itera
tions
Trapping domain, k=8
CFIECFIE+FMMCFIE+OSRC+FMM
0 5 10 15 20 250
100
200
300
400
500
600
k
GM
RES
itera
tions
Trapping domain, n!=10
CFIECFIE+FMMCFIE+OSRC+FMM
GMRES iterations vs nλ GMRES iterations vs k
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
GMRES convergence
0 100 200 300 40010−3
10−2
10−1
100
iteration number
GM
RES
resid
ual
Residuals, k=8
CFIE+FMM, n
!=12.5
CFIE+OSRC+FMM, n!=12.5
CFIE+OSRC+FMM, n!=26
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Eigenvalues investigation (using MELINA++ and ARPACK++)
Case of the unit sphere
• For numerical validation – comparisons with:
? the analytical eigenvalues of the operators
? the analytical eigenvalues including Pade approximation
• Behavior vs. wavenumber, mesh density and Pade order
Case of a cube with cavity
• Behavior vs. wavenumber, mesh density and Pade order
• Exhibition of a resonance frequency at k = 5
• Localization of resonance frequencies
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Case of the unit sphereDistribution of the eigenvalues, k = 11.85, nλ = 10
−2 0 2 4 6 8 10−2.5
−2
−1.5
−1
−0.5
0
0.5
Real part
Imag
inar
y pa
rt
CFIE ; k=11.85, n!=10
AnalyticalNumerical
0.85 0.9 0.95 1 1.05 1.1−0.15
−0.1
−0.05
0
0.05
0.1
Real partIm
agin
ary
part
CFIE+OSRC ; k=11.85, n!=10, Np=8
AnalyticalNumerical
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Numerical eigenvalues vs. Pade order, k = 10, nλ = 11.85
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
−0.2−0.1
00.10.20.30.4
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=10, n!=11.85, Np=2
Padé−analyticalNumerical
0.8 0.9 1 1.1 1.2 1.3
−0.2
−0.1
0
0.1
Real partIm
agin
ary
part
CFIE+OSRC ; k=10, n!=11.85, Np=4
Padé−analyticalNumerical
0.8 0.9 1 1.1 1.2 1.3
−0.2
−0.1
0
0.1
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=10, n!=11.85, Np=6
Padé−analyticalNumerical
0.8 0.9 1 1.1 1.2 1.3
−0.2
−0.1
0
0.1
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=10, n!=11.85, Np=8
Padé−analyticalNumerical
0.8 0.9 1 1.1 1.2 1.3
−0.2
−0.1
0
0.1
Real part
Imag
inar
y pa
rtCFIE+OSRC ; k=10, n
!=11.85, Np=10
Padé−analyticalNumerical
0.8 0.9 1 1.1 1.2 1.3
−0.2
−0.1
0
0.1
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=10, n!=11.85, Np=12
Padé−analyticalNumerical
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Analytical eigenvalues vs. Pade order, k = 10, nλ = 11.85
0.85 0.9 0.95 1 1.05 1.1−0.1
0
0.1
0.2
0.3
0.4
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=10, n!=11.85, Np=2
AnalyticalPadé−analytical
0.85 0.9 0.95 1 1.05 1.1−0.1
−0.05
0
0.05
0.1
0.15
Real partIm
agin
ary
part
CFIE+OSRC ; k=10, n!=11.85, Np=4
AnalyticalPadé−analytical
0.85 0.9 0.95 1 1.05 1.1−0.1
−0.05
0
0.05
0.1
0.15
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=10, n!=11.85, Np=6
AnalyticalPadé−analytical
0.85 0.9 0.95 1 1.05 1.1−0.1
−0.05
0
0.05
0.1
0.15
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=10, n!=11.85, Np=8
AnalyticalPadé−analytical
0.85 0.9 0.95 1 1.05 1.1−0.1
−0.05
0
0.05
0.1
0.15
Real part
Imag
inar
y pa
rtCFIE+OSRC ; k=10, n
!=11.85, Np=10
AnalyticalPadé−analytical
0.85 0.9 0.95 1 1.05 1.1−0.1
−0.05
0
0.05
0.1
0.15
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=10, n!=11.85, Np=12
AnalyticalPadé−analytical
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Performance of Pade approximation
O
−2θε
θp
2θε
1+
θε = arg(kε)
set of eigenvalues of(
[I] +[∆Γ]
k2ε
)on red line
θp = rotation angle of the usual branch-cut.
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Condition number
2 4 6 8 10 12 141.1
1.2
1.3
1.4
1.5
1.6
1.7
Padé order Np
Cond
ition
num
ber
CFIE+OSRC ; k=10, n!=11.85
AnalyticalPadé−analytical
5 10 15 200
100
200
300
400
k
Cond
ition
num
ber
n!=10
CFIECFIE+OSRC
5 10 15 20 250
50
100
150
200
250
n!
Cond
ition
num
ber
k=10
CFIECFIE+OSRC
a) Vs. Pade order b) Vs. k, nλ = 10 c) Vs. nλ, k = 10
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Case of the trapping domainDistribution of the eigenvalues, k = 8, nλ = 9.6, Np = 8
−2 0 2 4 6 8 10 12−2
−1.5
−1
−0.5
0
0.5
Real part
Imag
inar
y pa
rt
CFIE ; k=8, n!=9.6
0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
Real partIm
agin
ary
part
CFIE+OSRC ; k=8, n!=9.6, Np=8
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Condition number
5 10 15 20 250
50
100
150
n!
Cond
ition
num
ber
k=4.6
CFIECFIE+OSRC
5 10 15 20 250.1
0.15
0.2
0.25
n!
Smal
lest−m
agni
tude
eig
enva
lue
k=4.6
CFIECFIE+OSRC
5 10 15 20 250
5
10
15
n!
Larg
est−
mag
nitu
de e
igen
valu
e
k=4.6
CFIECFIE+OSRC
2 4 6 8 10 120
50
100
150
200
k
Cond
ition
num
ber
n!=10
CFIECFIE+OSRC
2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
k
Smal
lest−m
agni
tude
eig
enva
lue
n!=10
CFIECFIE+OSRC
2 4 6 8 10 120
5
10
15
k
Larg
est−
mag
nitu
de e
igen
valu
e
n!=10
CFIECFIE+OSRC
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Condition number vs. Pade order
2 4 6 8 10 12 140
1
2
3
4
5
6
7
8
9
Np
CFIE+OSRC ; k=4.2, n!=10
Condition numberLargest mag. eig.Smallest mag. eig.
2 3 4 5 6 7 80
0.5
1
1.5
2
2.5
3
3.5
4
Np
CFIE+OSRC ; k=6.4, n!=10
Condition numberLargest mag. eig.Smallest mag. eig.
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Resonance frequency, k = 5
0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=5, Np=8
0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=5.2, Np=8
0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
Real part
Imag
inar
y pa
rt
CFIE+OSRC ; k=5.4, Np=8
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Localization of resonances
3 4 5 6 7 8 9 10 110
500
1000
1500
2000
2500
3000
3500
k
Cond
ition
num
ber
Resonance localization
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012
OSRC-Preconditioning and FMM
Conclusion and perspectives
• Efficiency of the preconditioning:
? low effect on the cost per iteration of the resolution
? Strong increase of the convergence speed
• Instructive eigenvalues investigation:
? very low condition number for the OSRC-CFIE
? analysis of the effect of the Pade approximation
• Improvements on the strategy:
? Multilevel FMM with FastMMLib
? An alternative to Pade approximation
• Maxwell’s equations
? a short-term task to be done
? join us for the talk given by Marion this afternoon !!!
Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012