Efficient computation of acoustical scattering from N spheres via the

fast multipole method accelerated flexible generalized minimal residual method

Nail A. Gumerov Ramani Duraiswami

Institute for Advanced Computer StudiesUniversity of Maryland at College Park

www.umiacs.umd.edu/~gumerovwww.umiacs.umd.edu/~ramani

This study has been supported by NSF

OutlineIntroductionProblem FormulationMethod of Solution

Multipole Reexpansion (T-matrix) MethodIterative MethodsFast Multipole Method

Results of ComputationsConclusion

Introduction

Multiple Scattering ProblemsSound propagation in disperse media (particles, bubbles, etc.)Modeling of scattering from environment (humans, animals, fish, etc.)Electromagnetic scattering problems (microwaves, optics, etc.)Efficient parametrization in inverse problems (tomography, etc.)

Introduction

Why Multipole Methods?

Introduction

BEM Mesh5402 nodes10800 elementsDiscretization # Dn=30

Can be usedto compute the fieldonly for ka < 25 (for human head < 16.5 kHz)

Run Time for one frequencyon Dual Processor1 GHz Pentium III ~ 1 day.

Required Maximum Frequency to Compare with ExperimentalHRIR (22 or 44 kHz), 200 frequencies

Formal Requirement:ka << Dn

Scattering computation with BEM for a single object

Why Multipole Methods?

Meshless for spherical scatterersFast

Needs MeshRelatively Slow

Introduction

Multipole Methods BEM

Problem Formulation

Equations and Boundary Conditions

Helmholtz Equation

Impedance Boundary Conditions

Field Decomposition

Sommerfield Radiation Condition

4

2

1

6

5

3

Incident Wave

Formulation

Wave EquationFourierTransform

Multipole Reexpansion(T-Matrix) Method

Scattered Field Decomposition

T-Matrix Method

Expansion Coefficients

Singular Basis Functions Hankel Functions

Spherical Harmonics

Vector Form:

dot product

Incident Field Decompositionand T-matrix for a Single Sphere

T-Matrix Method

Regular Basis Functions Bessel Functions

Analytical Solution of the Problem:

T-matrix

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

Reexpansions/Translations

T-Matrix Method

q

p

M

O

rp

rq

r’pr’q

r’pq

r q

p

M

O

rp

rq

r’pr’q

r’pq

r

Two Spheres: Convergence with Respect to Truncation Number

T-matrix Method

-30

-20

-10

0

10

0 10 20 30 40 50Truncation Number

HR

TF (d

B)

ka1=30

5110

20

Two spheres,θ1 = 60o, φ1 = 0o,rmin/a1 = 2.3253.

Three Spheres Comparisons ofBEM & MultisphereHelmholtz

T-matrix Method

BEM: 5184 triangular elementsMH: Ntrunc = 9 (100 coefficients for each sphere)

-12

-9

-6

-3

0

3

6

9

12

-180 -90 0 90 180

Angle φ1 (deg)

HR

TF (d

B)

BEMMultisphereHelmholtz

θ1 = 0o

30o

60o

90o

60o

30o

90o

120o

120o

150o

150o

180o

Three Spheres, ka1 =3.0255.

Conclusions on T-matrix MethodWe used recursive computation of translation matrices (Chew, 1992; Gumerov & Duraiswami, 2001). In some cases speed up of computations 103-104

times compared to BEM.But… Computational Complexity is O(N3P3)= O(N3

p6), where P= p2 is the total length of the vector of expansion coefficients. Method is not suitable for large N and ka.Details can be found in our paper JASA 112(6), 2002, 2688-2701.

T-matrix Method

Iterative Methods

Reflection MethodReflection (Simple Iteration) Method:

Iterative Methods

Convergence of Reflection Iteration Method

Iterative Methods

Exponential Convergence

Problems with the Reflection Technique

Large number of particlesLarge volume fractionsLarger ka

Krylov Subspace Method (GMRES)

Iterative Methods

Diagonal Matrix

The product of this matrix byAn arbitrary input vector can bedone fast with the FMM

FGMRES

Iterative Methods

LA = E LM-1(MA)=E

Unpreconditioned Right Preconditioner

1). Internal Loop:Solve M-1F=GRequires N(1)

iter multiplications MG2). External Loop:Requires N(2)

iter multiplications LG

Cost: C(1)·N(1)iter+ C·N(2)

iter

To converge requiresNiter multiplications LG,where G is an input vector

Cost: C·Niter

Substantial speed up if M≈L and C(1) « C

Multilevel Fast Multipole Method

Some Facts on the Fast Multipole Methods (FMM)

Introduced by Rokhlin & Greengard (1987,1988) for computation of 2D and 3D fields for Laplace Equation;Reduces complexity of matrix-vector product from O(N 2) to O(N) or O(NlogN) (depends on data structure);Hundreds of publications for various 1D, 2D, and 3D problems (Laplace, Helmholtz, Maxwell, Yukawa Potentials, etc.);Application to acoustical scattering problems (Koc & Chew, 1998; JASA);We developed modification of the FMM for solution of the Helmholtz equation: Level dependent truncation number + Fast translation operators based on rotation-coaxial translation decomposition.

MLFMM

Far and Near Fields

MLFMM

Neighborhood(Near Field)

Far Field

Computation of the Far Field (1)

MLFMM

xixc

(n,L)

y

1). Set Data Structure (hierarchically subdivide space with an oct-tree)

2). (S|S)-translate S-expansions for all scatterers in a box at the finest level to the center of the box and sum up (determine contribution to Far Field for each box at the finest level).

3). Recursively (S|S)-translate S-expansions to the center of the parent box and sum up (determine contribution to Far Field for each box at all courser levels).

UpwardPass

(From the finestto the coarsest

level)

Computation of the Far Field (2)

MLFMM

4). (S|R)-translate S-expansions for boxes which are inside the parent neighborhood but outside the box neighborhood to the center of the box (convert S-expansion to R-expansion).

5). (R|R)-translate R-expansions from the center of the box to the center of its child boxes (determine Far Field for each box at all levels).

DownwardPass

(From the coarsest

to the finest level)

Steps 4 and 5 performed one after the other recursively

6). (R|R)-translate R-expansions from the center of the boxes at the finest level to the centers of the spheres.

Complexity of MLFMM

MLFMM

(For translation cost p3= P3/2 ):

(For each level of 3D space subdivision the computational work is approximately the same)

Computable Problems on Desktop PC

MLFMM

Met

hod

Number of Scatterers101 102 103100 104 105

BEM

Multipole Straightforward

Multipole Iterative

FMM

Also strongly depends on ka !

Preconditioned FGMRES

FGMRES

Results of Computations

Range of Parameters

Number of Spheres: 1-105;ka: 0.1-50;Random and regularly spaced grids of spheres;Polydispersity: 0.5-1.5 (ratio to the mean radius);Volume fractions: 0.01-0.25

Results

4 spheres (T-matrix straightforward)

Results

Vector of the incidentplane wave

Imaging plane

Scattererska=15.2

Incident Field Total Field Scattered Field

100 random spheres (MLFMM)

Results

ka = 1.6

Plane Wave Plane Wave

ka = 2.8ka = 1.6

Plane Wave Plane Wave

ka = 2.8

100 random spheres (MLFMM)

Results

ka=1.6 ka=4.8ka=2.8

R G B

1000 random spheres

ka=1

Results

10000 random spheres

ka = 0.75kD0 = 90

Results

A posteriori Error Check

Results

A posteriori Error Check

Results

d/a - 1

ka = 1

Error

Truncation number

Interparticle distance

In contact

GMRES vs FGMRES

Results

Volume Fraction = 0.2ka = 0.5, D0/a = 60, p2=225

3.2GHz, Xeon,3.5 MB RAM

Performance Test

Results

0.1

1

10

100

1000

10000

10 100 1000 10000

Number of Scatterers

CP

U T

ime

(s)

Total

Matrix-Vector Multiplication

External Loop

Internal Loop

y = ax

y = bx

y = cx 1.25

FMM+FGMRESy=cx1.25

Volume fraction = 0.2,ka = 0.5, p2=225

3.2GHz, Xeon,3.5 MB RAM

Conclusions

We developed, implemented, and tested a T-matrix method for solution of multiple scattering problem accelerated by the Flexible GMRES and the FMM for the main system matrix and preconditioner multiplication. A modification of the FMM that uses level dependent truncation number and fast translation operators was developed. O(NlogN) complecxity of the method was proven theoretically and in the numerical tests.The convergence speed of the method depends on the volume fraction of scatterers, frequencies and the scatterer sizes. For dense systems the best performance was achieved using the right dense preconditioners which includes far field interactions.We performed study on the dependence of the error on the truncation numbers, which depend on the frequency, size and interparticle distances, and can be used for efficient error control.

Future work

Comparisons with continuum (averaging) theories and theories of wave propagation in random media;Computations of acoustic fields in disperse systems (bubbly liquids, particulate systems);Comparisons with experimental data.

THANK YOU !