Research Presentation

Fast Numerical Methods
for High Frequency Wave Scattering

Khoa Tran

Institute for Computational Engineering and Sciences
The University of Texas at Austin

khoa@ices.utexas.edu

July 8, 2010

Introduction
Recent Research Results
Hybrid Method
Future Work

Outline

1 Introduction
Motivation & Challenges
Background

2 Recent Research Results
Fast Multipole Method
GO-based Boundary Integral Equation Method

3 Hybrid Method

4 Future Work

Khoa Tran Fast Numerical Methods for High Frequency Wave Scattering

Introduction
Recent Research Results Motivation & Challenges
Hybrid Method Background
Future Work

Outline

1 Introduction
Background


3 Hybrid Method

4 Future Work


Introduction
Future Work

Motivation

High frequency wave everywhere
Wireless Communication


Introduction
Future Work

Motivation

Acoustics


Introduction
Future Work

Motivation

Acoustics
Seismology


Introduction
Future Work

Motivation

Acoustics
Seismology
Radar


Introduction
Future Work

Motivation

Acoustics
Seismology
Radar
...


Introduction
Future Work

Computational Challenge

Multiscale: Wavelength << Computational Domain

# unknowns = O (kNλ )d , k = 2π is the wave frequency,
λ
Nλ = # points per wavelength, d = # dimensions


Introduction
Future Work

Example

k = 103 −→ O(109 ) unknowns


Introduction
Future Work

Background - Wave Models

Scalar ( ) ptt = c 2 ∆p

− ∂B = × E,
Maxwell ∂t
µ0 0 ∂E = × B − µ0 J
∂t

Elastic ρutt − · σ(x , u) = 0

The equations don’t deﬁne the scales. They originate from the
geometry, initial and boundary conditions.


Introduction
Future Work

Existing Numerical Methods

Direct methods
FDTD, FEM, and DG with absorbing boundary conditions
Dimension reduction methods (frequency domain)
Asymptotic methods
Physical Optics
Geometrical Optics
Gausian Beam


Introduction
Future Work

Existing Numerical Methods

Direct methods
FDTD, FEM, and DG with absorbing boundary conditions
Dimension reduction methods (frequency domain)
Asymptotic methods
Physical Optics
Geometrical Optics
Gausian Beam

→ Background for GO-based BIE and Hybrid Method


Introduction
Future Work

Asymptotic Methods - Geometrical Optics

Eﬀective for very high frequency wave scattering
Approximate the phase φ and the amplitude A → Simple
formulation for φ and A


Introduction
Future Work

Asymptotic Methods - Bibliography

1962 Keller (Geometrical theory of diﬀraction)
1965 Maslov (Theory of perturbations and asymptotic methods)
1977 Cerveny, Julian, ... (various papers on seismic ray tracing)
1985 Melrose, Taylor (Scattering approximation for convex objects)
1995 Fatemi et al. (High frequency asymptotic expansion methods)
1996– Benamou et al. (various papers on geometrical optics)
2002 Castella et a. (High frequency limit of the Helmholtz Eq.)
2002 Engquist et al. (Wave front tracking using segment projection)
2003 Osher et al. (Wave front tracking using level set methods)
2005 Bruno et al. (GO-based BIE method)
2006 Ying, Cand´s (Phase ﬂow method)
e
2007 OR (survey on high frequency waves)


Introduction
Future Work

Asymptotic Methods - Summary

Highly eﬃcient
Sacriﬁce of prescribed accuracy


Introduction
Future Work

Dimension Reduction Methods

1 Frequency Domain −→ Time dimension elimination
2 Boundary Integral Eq. −→ Space dimension reduction

We focus on dimension reduction methods!


Introduction
Future Work

Helmholtz equation

Fourier decomposition
Wave equation − − − − − − − Helmholtz equations
− − − − − −→

& far ﬁeld condition


Introduction
Future Work

Boundary Integral Equation
Green’s function
Helmholtz equations − − − − − Boundary integral equation
−−−−→

→
1 ∂G(z, y )
µ(z) + − iγG(z, y ) µ(y )ds(y ) = −u I (z)
2 Γ ∂n(y )
on Γ = ∂Ω
G(x , y ) is the Green’s function
µ is an arﬁcial unknown function on the boundary
u is recovered using
∂G(x , y )
u(x ) = − iγG(x , y ) µ(y )ds(y ).
Γ ∂n(y )


Introduction
Future Work

Boundary Integral Equation (aka. Method of Moments)

Advantages
Dimension reduction
Finite domain – No far ﬁeld condition

1984 Moore, Pizer (Moment methods in electromagnetics)
1993 Harrington (Field computation by moment methods)
1998 Colton, Kress (Invese acoustic and electromagnetic scattering)
1999 Kress (Linear integral equations)
2009 Manolis (Recent advances in boundary element methods)
...


Introduction
Future Work

Boundary Integral Equation - Direct Solver

1 Represent the unknown function µ on the boundary
2 Discretize the BIE (Nystrom, Galerkin)
3 Solve the linear system


Introduction
Future Work

Boundary Integral Equation - Direct Solver

1 Represent the unknown function µ on the boundary
2 Discretize the BIE (Nystrom, Galerkin)
3 Solve the linear system

Dimensions reduced but still expensive to solve at high frequency

k = 103 −→ matrix size: O(106 ) × O(106 )


Introduction
Recent Research Results Fast Multipole Method
Hybrid Method GO-based Boundary Integral Equation Method
Future Work

Outline

1 Introduction
Background


3 Hybrid Method

4 Future Work


Introduction
Future Work

State of the Arts

Two active lines of research

1 Fast multipole approach
Fast matrix-vector multiplication is key
New development of FMM for Helmholtz kernel


Introduction
Future Work

State of the Arts

Two active lines of research

1 Fast multipole approach
Fast matrix-vector multiplication is key
New development of FMM for Helmholtz kernel
2 GO-based boundary integral equation approach
Utilize asymptotic properties of the solution −→ similar to GO
→ highly eﬃcient
Does NOT compromise accuracy
Limited to certain geometries


Introduction
Future Work

FMM for Poisson kernel [Rokhlin et al.]

1 1
Assume that G = 4π |x −y |
is the Poisson kernel
Iterative matrix solver −→ need to compute
N
ui = G(xi , xj )fj
j=1

for all i = 1, .., N


Introduction
Future Work

N
ui = j=1 G(xi , xj )fj

For separated set of points

[Ying]


Introduction
Future Work


What about Use octree structure
mixed distribution? (or quadtree in 2D)

[Ying]


Introduction
Future Work

FMM for Poisson kernel - Complexity

O(N) Flops

to compute the matrix-vector multiplication, for any prescribed
accuracy, where N=#unknowns


Introduction
Future Work

FMM for Helmholtz kernel

The Helmholtz kernel is oscillatory and does NOT possess the
low rank property below


Introduction
Future Work

FMM for Helmholtz kernel

Good news: it’s possible to recover the low rank property!
↓
Multidirectional FMM [Engquist, Ying]


Introduction
Future Work

Multidirectional FMM

[Engquist, Ying]

Directional low rank property of Helmholtz kernel


Introduction
Future Work

Multidirectional FMM - Complexity

O(N log N) to compute the matrix-vector multiplication


Introduction
Future Work

Multidirectional FMM - Complexity

O(N log N) to compute the matrix-vector multiplication
O(k 2 log k) to compute the scattered wave
O(k log k) to compute the scattered wave in 2D

References
1987 Greengard, Rokhlin (original FMM paper)
1990– Rokhlin et al. (various papers on FMM for wave equation)
1997 Song et al. (various papers on FMM for Maxwell)
2004 Ying et al. (Kernel-independent adaptive FMM)
2008 Engquist, Ying (Directional FMM for oscillatory kernels)
...


Introduction
Future Work


New BIE formulation
1
µ(x ) + Kµ (x , y )µ(y )ds(y ) = f (x ), x ∈Γ
2 Γ

∂G(x ,y )
where Kµ (x , y ) = ∂n(x )
− iγG(x , y )


Introduction
Future Work


New BIE formulation
1
2 Γ

∂G(x ,y )
− iγG(x , y )
µ has physical meaning


Introduction
Future Work


New BIE formulation
1
2 Γ

∂G(x ,y )
− iγG(x , y )
µ has physical meaning
For convex object, µ admits a slow representation

µ(x ) = kµs (x )e ikα·x

(similar to GO)


Introduction
Future Work

Slow Representation

Left: µ Right: µs

Multiple wave reﬂections by a
non-convex scatterer


Introduction
Future Work

BIE for µs

BIE for µ
1
2 Γ

For x away from y , Kµ = K s e ik|x −y |
µ = µs e ikα·x


Introduction
Future Work

BIE for µs

BIE for µ
1
2 Γ

For x away from y , Kµ = K s e ik|x −y |
µ = µs e ikα·x
→ BIE for µs
1 s
µ (x ) + e ikφ(x ,y ) K s µs (y )dy = g(x ), x ∈Γ
2 Γ


Introduction
Future Work

Challenge

At each GMRES iteration, need to evaluate

e ikφ(x ,y ) K s µs (y )dy
Γ


Introduction
Future Work

Challenge

At each GMRES iteration, need to evaluate

Γ

Localized Integration Method


Introduction
Future Work

Localization Lemma

Lemma
Suppose fX is a smooth function supported on [−X , X ]. Then
A
−n
e ikφ(t) fA (t)dt = e ikφ(t) f (t)dt+O k φ( ) ∀n ≥ 1.
−A −


Introduction
Future Work

Outline of Proof

A
−n
e ikφ(t) fA (t)dt = e ikφ(t) f (t)dt +O k φ( ) ∀n ≥ 1
−A −

Change of variable −→ τ = φ(t)


Introduction
Future Work

Outline of Proof

A
−n
−A −

Integration by parts n times
Note that fA − f is compactly supported


Introduction
Future Work

Outline of Proof

A
−n
−A −

Integration by parts n times
Note that fA − f is compactly supported
Error estimates using Taylor expansions


Introduction
Future Work

Localization Lemma - Application
Localized Integration

Γ

Only need to compute the integral over the small neighborhoods of
critical points of φ( , y )


Introduction
Future Work

Localized Integration - Other Considerations

Singular Integrator
Special weighted Trapezoidal rule


Introduction
Future Work

Localized Integration - Other Considerations

Singular Integrator
Special weighted Trapezoidal rule
Shadow Boundary
Change of variable
0.1

0.05

0

−0.05

−0.1

−0.15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Introduction
Future Work

Numerical Results

L2 errors of µs
Frequency
30 secs 150 secs
1000 6.8e-4 7.1e-6
2000 4.4e-4 5.6e-6
4000 4.8e-4 4.5e-6
8000 6.4e-4 6.7e-6
16000 5.9e-4 5.7e-6
32000 3.2e-4 8.5e-6


Introduction
Future Work

Remarks

Highly eﬃcient


Introduction
Future Work

Remarks

Highly eﬃcient
Drawback: Slow decomposition
inc (x )
µ(x ) = kµs (x )e ikφ

only valid for convex objects (extension is possible but
diﬃcult)

[Bruno et al.]


Introduction
Hybrid Method
Future Work

Motivation

Many objects of interest contain a large convex part and a
complicated small structure


Introduction
Hybrid Method
Future Work

Analysis

Multidirectional FMM GO-based BIE Method

Eﬃciency: O(k 2 log k) Eﬃciency: O(1)
Geometry: arbitrary Geometry: convex


Introduction
Hybrid Method
Future Work

Analysis

Multidirectional FMM GO-based BIE Method

Eﬃciency: O(k 2 log k) Eﬃciency: O(1)
Geometry: arbitrary Geometry: convex

→ Hybrid Method


Introduction
Hybrid Method
Future Work

Solution

1

0.5

0

−0.5
µs

−1

−1.5

−2

−2.5

−3
0 1 2 3 4 5 6
x


Introduction
Hybrid Method
Future Work

Analysis (cont.)

Use partition of unity to decompose the boundary


Introduction
Hybrid Method
Future Work

Analysis (cont.)

Denser point sampling on Γ1


Introduction
Hybrid Method
Future Work

Analysis (cont.)

Denser point sampling on Γ1

Apply FMM on Γ1 & Localized Integration on Γ2


Introduction
Hybrid Method
Future Work

Algorithm

1 Point Sampling using transformation
map


Introduction
Hybrid Method
Future Work

Algorithm

map
2 Precomputations


Introduction
Hybrid Method
Future Work

Algorithm

map
2 Precomputations
3 GMRES Solver: at each step
Solution Interpolation


Introduction
Hybrid Method
Future Work

Algorithm

map
2 Precomputations
Run FMM once on Γ1


Introduction
Hybrid Method
Future Work

Algorithm

map
2 Precomputations
Run FMM once on Γ1
Localized integration on Γ2 for each xi


Introduction
Hybrid Method
Future Work

Complexity

O(kL log(kL))

L = |Γ1 |


Introduction
Hybrid Method
Future Work

Numerical Results

CPU time CPU time of Error of
k of DFMM hybrid method hybrid method
50 57s 13s (20 iterations) 2.3 e-4
100 2m 12s 14s (21 iterations) 3.5 e-4
200 5m 23s 21s (22 iterations) 4.1 e-4
400 13m 40s 33s (25 iterations) 4.2 e-4
800 35m 39s 52s (27 iterations) 5.4 e-4
1600 1h 32m 82s (29 iterations) 4.7 e-4
3200 3h 57m 128s (32 iterations) 4.9 e-4
6400 10h 40m 203s (35 iterations) 5.7 e-4

1
L= √
k


Introduction
Hybrid Method
Future Work

Remarks

Two very recent and highly eﬃcient methods are involved


Introduction
Hybrid Method
Future Work

Remarks

Two very recent and highly eﬃcient methods are involved
Smooth transition between the 2 regions using paritition
of unity
→ Allows the full 2-way coupling in the overlapping region


Introduction
Hybrid Method
Future Work

Future Work

More realistic hybrid applications
Parallel DFMM with applications to wave scattering in
cluttered environments


Introduction
Hybrid Method
Future Work

Wave Scattering in Cluttered Environment

Too large for BIE and DFMM
on a single computer
Need parallel algorithm

[Li]


Introduction
Hybrid Method
Future Work

Parallel Algorithm using DFMM - Challenges

MPI implementation
New parallel data structure
Data distribution
Data communication


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