The FMM for 3D Helmholtz Equation - CSCAMM · 2004. 4. 26. · CSCAMM FAM04: 04/19/2004 ©...

© Duraiswami & Gumerov, 2003-2004CSCAMM FAM04: 04/19/2004

The FMM for 3D Helmholtz Equation

Nail Gumerov & Ramani Duraiswami

UMIACS[gumerov][ramani]@umiacs.umd.edu


ReferenceN.A. Gumerov & R. Duraiswami

Fast Multipole Methods for Solution of the Helmholtz Equation in Three Dimensions

Academic Press, Oxford (2004)(in process).


Content• Helmholtz Equation• Expansions in Spherical Coordinates• Matrix Translations• Complexity and Modifications of the FMM• Fast Translation Methods• Error Bounds• Multiple Scattering Problem


Helmholtz Equation


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


Boundary Value Problems

nS


Green’s Function and Identity

nS Ω


Distributions of Monopoles and Dipoles

nS Ω


Expansions in Spherical Coordinates


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


Spherical Harmonics

n

m


Spherical Bessel Functions


Isosurfaces For Regular Basis Functions

n

m


Isosurfaces For Singular Basis Functions

n

m


Expansions

Wave vector


Matrix Translations


Reexpansions of Basis Functions

Reexpansion Matrices

rt

R S

Or

rt

R S

Or


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.


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


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


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


Complexity of Single Translation

Translation exponent


Spatially Uniform Data Distributions

Constant!


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


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


Volume Element Methods

Ns

wavelength

D0 = D0 k/(2π) wavelengths = N1/3 sourcesCritical Translation Exponent!

computational domain


What Happens if Truncation Number is Constant for All Levels?

“Catastrophic Disaster of the FMM”


Surface Data Distributions

Critical Translation Exponent!

Boundary Element Methods:


Optimum Level of Space Subdivision

0

1

2

3

4

5

6

7

8


Opt

imum

Max

Lev

el

nu=1nu=1.5nu=2

lb=20

1

2

3

4

5

6

7

8


Opt

imum

Max

Lev

el

nu=1nu=1.5nu=2

lb=5

lb=2 lb=5


Performance of the MLFMM for Surface Data Distributions

100000

1000000

1E+07

1E+08

1E+09

1E+10

1E+11

1E+12


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


Effective Dimensionality of the Problem

Computational domain

1 cube

8 cubes


Fast Translation Methods


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.


O(p3) Methods


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


Sparse Matrix Decomposition

Matrix-vector products with these matrices computed recursively


O(p2logβp) Methods


Fast Rotation Transform

Euler angles

Diagonal matrices

Diagonal matrices

Block-Toeplitz matrices

Complexity: O(p2logp)


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.


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.


Method of Signature Function(Diagonal Forms of the Translation Operator)

Regular Solution

Singular Solution


Final Summation and Initial Expansion


The FMM with Band-Unlimited Signature Functions (O(p2) method)

0.1

1

10

100

1000

10000


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


Deficiencies• Low Frequencies;• High Frequencies;• Constant p;• Instabilities after two or three levels of translations.


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.


Error Bounds


Source Expansion Errors

rs

r

ab

rs

r

ab0 0

rs

r

ab

rs

r

ab0 0


Approximation of the Error


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


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


High Frequency FMM Error


Multiple Scattering Problem


Problem

Boundary Conditions:

100 random spheres 1000 random spheres100 random spheres 1000 random spheres


Scattered Field DecompositionT-Matrix Method

Expansion Coefficients

Singular Basis Functions Hankel Functions

Spherical Harmonics

Vector Form:

dot product


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


Reflection Method & Krylov Subspace Method (GMRES)

Reflection (Simple Iteration) Method:

General Formulation (used in GMRES)

Iterative Methods


100 random spheres (MLFMM)ka=1.6 ka=4.8ka=2.8

R G B


Surface Potential Imaging


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


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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The FMM for 3D Helmholtz Equation - CSCAMM · 2004. 4. 26. · CSCAMM FAM04: 04/19/2004 ©...

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