The FMM for 3D Helmholtz Equation - CSCAMM · 2004. 4. 26. · CSCAMM FAM04: 04/19/2004 ©...
Transcript of The FMM for 3D Helmholtz Equation - CSCAMM · 2004. 4. 26. · CSCAMM FAM04: 04/19/2004 ©...
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
The FMM for 3D Helmholtz Equation
Nail Gumerov & Ramani Duraiswami
UMIACS[gumerov][ramani]@umiacs.umd.edu
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
ReferenceN.A. Gumerov & R. Duraiswami
Fast Multipole Methods for Solution of the Helmholtz Equation in Three Dimensions
Academic Press, Oxford (2004)(in process).
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Content• Helmholtz Equation• Expansions in Spherical Coordinates• Matrix Translations• Complexity and Modifications of the FMM• Fast Translation Methods• Error Bounds• Multiple Scattering Problem
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Helmholtz Equation
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Helmholtz Equation
• Wave equation in frequency domainAcousticsElectromagneics (Maxwell equations)Diffusion/heat transfer/boundary layersTelegraph, and related equationsk can be complex
• Quantum mechanicsKlein-Gordan equationShroedinger equation
• Relativistic gravity (Yukawa potentials, k is purely imaginary)• Molecular dynamics (Yukawa)• Appears in many other models
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Boundary Value Problems
nS
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Green’s Function and Identity
nS Ω
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Distributions of Monopoles and Dipoles
nS Ω
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Expansions in Spherical Coordinates
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Spherical Basis Functions
x
y
z
O
x
y
z
O
x
y
z
Oθ
φ
r
rir
iθ
iφ
x
y
z
O
x
y
z
O
x
y
z
Oθ
φ
r
rir
iθ
iφ
Spherical Coordinates
Regular Basis Functions
Singular Basis Functions
Spherical Harmonics
Associated Legendre Functions
Spherical Bessel Functions
Spherical Hankel Functions of the First Kind
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Spherical Harmonics
n
m
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Spherical Bessel Functions
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Isosurfaces For Regular Basis Functions
n
m
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Isosurfaces For Singular Basis Functions
n
m
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Expansions
Wave vector
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Matrix Translations
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Reexpansions of Basis Functions
Reexpansion Matrices
rt
R S
Or
rt
R S
Or
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Recursive Computation of Reexpansion Matrices
|m|
n’
p-1
|m*|
02p-2-|m*| 2p-2p-1
n’
m m’(n’,m’,m+1)
(n’-1,m’-1,m) (n’+1,m’-1,m)
|m|
n’
p-1
|m*|
02p-2-|m*| 2p-2p-1
n’
m m’
n’
m m’(n’,m’,m+1)
(n’-1,m’-1,m) (n’+1,m’-1,m)
n n
|m’|
|m|
|m|
|m| |m’|
|s|
|m’| < |m| |m’| > |m|2p-2-|m|
p-1
p-12p-2-|m| 2p-2-|m’|
2p-2-|m’|
0 0
p-1-|m|+|m’|
p-1-|m’|+|m|
n’
n
(n’,n-1)
(n’-1,n)
(n’,n+1)
(n’+1,n)
n
(n’,n-1)
(n’-1,n)
(l,n+1)
(n’+1,n)
n’
n’ n’
p-1
p-1n n
|m’|
|m|
|m|
|m| |m’|
|s|
|m’| < |m| |m’| > |m|2p-2-|m|
p-1
p-12p-2-|m| 2p-2-|m’|
2p-2-|m’|
0 0
p-1-|m|+|m’|
p-1-|m’|+|m|
n’
n
(n’,n-1)
(n’-1,n)
(n’,n+1)
(n’+1,n)
n
(n’,n-1)
(n’-1,n)
(l,n+1)
(n’+1,n)
n’
n’ n’
p-1
p-1
p4 elements of the truncated reexpansion matrices can be computed for O(p4) operations recursively
Gumerov & Duraiswami,SIAM J. Sci. Stat. Comput.25(4), 1344-1381, 2003.
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Translations
Ω1
Ω2
x*1
x*2R (R|R)
x R
Ωr(x*)
x*
(R|R)
y x*+tt
r
Ωr1(x*+t)
r1
xiΩ1
Ω2x*1
x*2
S
(S|S)
S
xi
x*x*+t
(S|S)
y
t
Ωr1(x*+t) Ωr(x*)
rr1
xiΩ1
x*1x*2
S
(S|S)
S
xi
x*
x*+t
(S|R)
y
t
Ωr1(x*+t)
r
r1
R|R S|S S|R
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Problem:
• For the Helmholtz equation absolute and uniform convergence can be achieved only for
p > ka. For large ka the FMM with constant p isvery expensive (comparable with straightforward methods);inaccurate (since keeps much larger number of terms than required, which causes numerical instabilities).
a
ExpansionDomain
D
2a=31/2D
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Model of Truncation Number Behavior for Fixed Error
p
ka0 ka*
p*
In the multilevel FMM we associate its own plwith each level l:
“Breakdown level”
Box size at level l
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Complexity of Single Translation
Translation exponent
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Spatially Uniform Data Distributions
Constant!
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Complexity of the Optimized FMM for Fixed kD0 and Variable N
100000
1000000
1E+07
1E+08
1E+09
1E+10
1E+11
1E+12
1000 10000 100000 1000000Number of Sources
Num
ber o
f Mul
tiplic
atio
ns
nu=1, lb=2nu=1.5, lb=2nu=2, lb=2nu=1, lb=5nu=1.5, lb=5nu=2, lb=5
Straightforward, y=x2
3D Spatially Uniform Random Distribution
N=M
y=ax
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Optimum Level for Low Frequencies
0.00001
0.0001
0.001
0.01
0.1
1
10
100
2 3 4 5 6 7
Max Level of Space Subdivision
Num
ber o
f Mul
tiplic
atio
ns, x
10e1
1N=M=10000003D Spatially Uniform Random Distribution
Direct SummationTranslation
Total
nu=1
1.5
2
2
1.5
1
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Volume Element Methods
Ns
wavelength
D0 = D0 k/(2π) wavelengths = N1/3 sourcesCritical Translation Exponent!
computational domain
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What Happens if Truncation Number is Constant for All Levels?
“Catastrophic Disaster of the FMM”
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Surface Data Distributions
Critical Translation Exponent!
Boundary Element Methods:
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Optimum Level of Space Subdivision
0
1
2
3
4
5
6
7
8
1000 10000 100000 1000000Number of Sources
Opt
imum
Max
Lev
el
nu=1nu=1.5nu=2
lb=20
1
2
3
4
5
6
7
8
1000 10000 100000 1000000Number of Sources
Opt
imum
Max
Lev
el
nu=1nu=1.5nu=2
lb=5
lb=2 lb=5
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Performance of the MLFMM for Surface Data Distributions
100000
1000000
1E+07
1E+08
1E+09
1E+10
1E+11
1E+12
1000 10000 100000 1000000Number of Sources
Num
ber o
f Mul
tiplic
atio
ns
nu=1, lb=2nu=1.5, lb=2nu=2, lb=2nu=1, lb=5nu=1.5, lb=5nu=2, lb=5
Straightforward, y=x2
Uniform Random Distributionover a Sphere Surface
N=M
y=ax
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Effective Dimensionality of the Problem
Computational domain
1 cube
8 cubes
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Fast Translation Methods
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Translation Methods• O(p5): Matrix Translation with Computation of Matrix Elements Based on Clebsch-
Gordan Coefficients;• O(p4) (Low Asymptotic Constant): Matrix Translation with Recursive Computation of
Matrix Elements• O(p3) (Low Asymptotic Constants):
Rotation-Coaxial Translation Decomposition with Recursive Computation of Matrix Elements;Sparse Matrix Decomposition;
• O(p2logβp)Rotation-Coaxial Translation Decomposition with Structured Matrices for Rotation and Fast Legendre Transform for Coaxial Translation;Translation Matrix Diagonalization with Fast Spherical Transform;Asymptotic Methods;Diagonal Forms of Translation Operators with Spherical Filtering.
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
O(p3) Methods
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Rotation - Coaxial Translation Decomposition (Complexity O(p3))
z
y
y
xx
yx
z
yx
z
zp4
p3p3
p3
z
y
y
xx
yx
zyx
z
yx
z
zp4
p3p3
p3
From the group theory follows that general translation can be reduced to
0.01
0.1
1
10
10 100
Truncation Number, p
CP
U T
ime
(s)
Full Matrix Translation
Rotational-Coaxial Translation Decomposition
y=ax4
y=bx3
kt=86
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Sparse Matrix Decomposition
Matrix-vector products with these matrices computed recursively
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
O(p2logβp) Methods
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Fast Rotation Transform
Euler angles
Diagonal matrices
Diagonal matrices
Block-Toeplitz matrices
Complexity: O(p2logp)
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Fast Coaxial Translation
Diagonal matricesLegendre andtransposed Legendre matrices
Fast multiplication of the Legendre and transposed Legendre matrices can be performed via the forward and inverse FAST LEGENDRE TRANSFORM (FLT) with complexity O(p2log2p)
Healy et al Advances in Computational Mathematics 21: 59-105, 2004.
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Diagonalization of General Translation Operator
Diagonal matricesMatrices for the forward and inverse and Spherical Transform
FAST SPHERICAL TRANSFORM (FST) can be performed with complexity O(p2log2p)
Healy et al Advances in Computational Mathematics 21: 59-105, 2004.
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Method of Signature Function(Diagonal Forms of the Translation Operator)
Regular Solution
Singular Solution
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Final Summation and Initial Expansion
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
The FMM with Band-Unlimited Signature Functions (O(p2) method)
0.1
1
10
100
1000
10000
1000 10000 100000 1000000Number of Sources
CP
U T
ime
(s)
y = ax 2
y = bx
Straightforward
FMM
O(p )3
O(p )2
O(p )+err bound2
kD =10, M=N,Spatially Uniform Random Distributions
0
Pre-Set StepO(p )2
O(p )3
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Deficiencies• Low Frequencies;• High Frequencies;• Constant p;• Instabilities after two or three levels of translations.
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Methods to Fix:• Use of Band-limited functions;• Error control via band-limits;• Requires filtering procedures (complexity O(p2log2p) or O(p2logp))
with large asymptotic constants;• The length of the representation is changed via
interpolation/anterpolation procedures.
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Error Bounds
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Source Expansion Errors
rs
r
ab
rs
r
ab0 0
rs
r
ab
rs
r
ab0 0
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Approximation of the Error
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
We proved that for source summation problems the truncation numbers can be selected based on the above
chart when using translations with rectangularlytruncated matrices
r
ab 0
rsa/2t
b/2
r
ab 0
rsa/2t
b/2
ra
b
0rs
ta
ra
b
0rs
ta
r
ab
rs
a/2t0
r
ab
rs
a/2t0
S|S S|R R|R
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Low Frequency FMM Error
a
b
O
A
B
ab
C
B
a
O
A
Q
R|R, S|S S|R
a
b
O
A
B
ab
C
B
a
O
A
Qa
b
O
A
B
ab
C
B
a
O
A
Q
R|R, S|S S|R
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
High Frequency FMM Error
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Multiple Scattering Problem
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Problem
Boundary Conditions:
100 random spheres 1000 random spheres100 random spheres 1000 random spheres
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Scattered Field DecompositionT-Matrix Method
Expansion Coefficients
Singular Basis Functions Hankel Functions
Spherical Harmonics
Vector Form:
dot product
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Solution of Multiple Scattering Problem
T-Matrix Method
4
2
1
6
53
Incident Wave
Scattered Wave
Coupled System of Equations:
(S|R)-TranslationMatrix
“Effective” Incident Field
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Reflection Method & Krylov Subspace Method (GMRES)
Reflection (Simple Iteration) Method:
General Formulation (used in GMRES)
Iterative Methods
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
100 random spheres (MLFMM)ka=1.6 ka=4.8ka=2.8
R G B
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Surface Potential Imaging
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Performance Test
1
10
100
1000
10000
100000
10 100 1000 10000 100000Number of Scatterers
CP
U T
ime
(s)
Volume Fraction = 0.2, ka=0.5
Periodically-Random Spatial Distributionof Spheres of Equal Size
Reflection +FMMStraightforward
y=cx3
y=ax
Reflection+Direct
y=bx2
© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
Computable Problems on Desktop PC
MLFMM
Met
hod
Number of Scatterers101 102 103100 104 105
BEM
Multipole Straightforward
Multipole Iterative
MLFMM
Also strongly depends on ka !
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© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004
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