Compressed Absorbing Boundary Conditions for the Helmholtz Equation Rosalie Bélanger-Rioux

105 Pages·2014·2.19 MB·English

Compressed Absorbing Boundary Conditions for
the Helmholtz Equation
by
Rosalie B´elanger-Rioux
B.Sc., McGill University (2009)
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
(cid:13)c Massachusetts Institute of Technology 2014. All rights reserved.
Author ..............................................................
Department of Mathematics
May 8, 2014
Certified by..........................................................
Laurent Demanet
Assistant Professor
Thesis Supervisor
Accepted by.........................................................
Michel Goemans
Chairman, Department Committee on Graduate Theses
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Compressed Absorbing Boundary Conditions for the
Helmholtz Equation
by
Rosalie B´elanger-Rioux
Submitted to the Department of Mathematics
on May 8, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Absorbing layers are sometimes required to be impractically thick in order to offer
an accurate approximation of an absorbing boundary condition for the Helmholtz
equation in a heterogeneous medium. It is always possible to reduce an absorbing
layer to an operator at the boundary by layer-stripping elimination of the exterior
unknowns, but the linear algebra involved is costly. We propose to bypass the elimi-
nation procedure, and directly fit the surface-to-surface operator in compressed form
from a few exterior Helmholtz solves with random Dirichlet data. We obtain a concise
description of the absorbing boundary condition, with a complexity that grows slowly
(often, logarithmically) in the frequency parameter. We then obtain a fast (nearly
linear in the dimension of the matrix) algorithm for the application of the absorb-
ing boundary condition using partitioned low rank matrices. The result, modulo a
precomputation, is a fast and memory-efficient compression scheme of an absorbing
boundary condition for the Helmholtz equation.
Thesis Supervisor: Laurent Demanet
Title: Assistant Professor
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Acknowledgments
I would like to thank my advisor, Laurent Demanet, for his advice and support
throughout my time as a graduate student.
Also, I would like to acknowledge professors I took classes from, including my the-
sis committee members Steven Johnson and Jacob White, and qualifying examiners
Ruben Rosales and again Steven Jonhson. I appreciate your expertise, availability
and help.
I would like to thank collaborators and researchers with whom I discussed various
ideas over the years, it was a pleasure to learn and discover with you.
Thanks also to my colleagues here at MIT in the mathematics PhD program, in-
cluding office mates. It was a pleasure to get through this together.
Thanks to my friends, here and abroad, who supported me. From my decision to
come to MIT, all the way to graduation, it was good to have you.
Finally, I thank my loved ones for being there for me always, and especially when I
needed them.
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Contents
1 Introduction 15
1.1 Applications of the Helmholtz equation in an unbounded domain . . . 16
1.2 A two-step numerical scheme for compressing ABCs . . . . . . . . . . 18
1.2.1 Background material . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.2 Matrix probing . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.3 Partitioned low-rank matrices . . . . . . . . . . . . . . . . . . 20
1.2.4 Summary of steps . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Background material 23
2.1 The Helmholtz equation: free-space problem . . . . . . . . . . . . . . 24
2.1.1 Solution in a uniform medium using the Green’s function . . . 24
2.1.2 SolutioninaheterogeneousmediumusingAbsorbingBoundary
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.3 The Dirichlet-to-Neumann map as an ABC . . . . . . . . . . . 26
2.2 Discrete absorbing boundary conditions . . . . . . . . . . . . . . . . . 27
2.2.1 Surface-to-surface ABCs . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 Absorbing layer ABCs . . . . . . . . . . . . . . . . . . . . . . 28
2.2.3 Complexity of ABCs in heterogeneous media . . . . . . . . . . 29
2.3 The Helmholtz equation: exterior and half-space problems . . . . . . 30
2.3.1 The exterior problem . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 The half-space problem . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 Solution to the half-space problem in a uniform medium using
the Green’s function . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.4 The analytical half-space DtN map . . . . . . . . . . . . . . . 34
2.4 Eliminating the unknowns: from any ABC to the DtN map . . . . . . 36
2.4.1 Eliminating the unknowns in the half-space problem . . . . . . 36
2.4.2 Eliminating the unknowns in the exterior problem . . . . . . . 38
3 Matrix probing for expanding the Dirichlet-to-Neumann map 41
3.1 Introduction to matrix probing . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Setup for the exterior problem . . . . . . . . . . . . . . . . . . 42
3.1.2 Matrix probing . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.3 Solving the Helmholtz equation with a compressed ABC . . . 45
3.2 Choice of basis matrices for matrix probing . . . . . . . . . . . . . . . 45
3.2.1 Oscillations and traveltimes for the DtN map . . . . . . . . . 46
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3.2.2 More on traveltimes . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Probing tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.2 Using the probed DtN map in a Helmholtz solver . . . . . . . 56
3.3.3 Grazing waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.4 Variations of p with N . . . . . . . . . . . . . . . . . . . . . . 59
3.4 Convergence of probing for the half-space DtN map: theorem . . . . . 61
3.4.3 Chebyshev expansion . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.4 Bound on the derivatives of the DtN map kernel with oscilla-
tions removed . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.5 Bound on the error of approximation . . . . . . . . . . . . . . 64
3.5 Convergence of probing for the half-space DtN map: numerical confir-
mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5.1 Uniform medium . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Partitionedlow-rankmatricesforcompressingtheDirichlet-to-Neumann
map 67
4.1 Partitioned low-rank matrices . . . . . . . . . . . . . . . . . . . . . . 68
4.1.1 Construction of a PLR matrix . . . . . . . . . . . . . . . . . . 68
4.1.2 Multiplication of a PLR matrix with a vector . . . . . . . . . 70
4.1.3 Structure and complexity . . . . . . . . . . . . . . . . . . . . 72
4.2 Using PLR matrices for the DtN map’s submatrices . . . . . . . . . . 77
4.2.1 Choosing the tolerance . . . . . . . . . . . . . . . . . . . . . . 77
4.2.2 Minimizing the matrix-vector application time . . . . . . . . . 78
4.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 The half-space DtN map is separable and low rank: theorem . . . . . 81
4.3.1 Treating the Hankel function . . . . . . . . . . . . . . . . . . . 82
4.3.2 Treating the 1/kr factor . . . . . . . . . . . . . . . . . . . . . 83
4.3.4 Dyadic interval for the Gaussian quadrature . . . . . . . . . . 85
4.3.5 Finalizing the proof . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.9 The numerical low-rank property of the DtN map kernel for
heterogeneous media . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Thehalf-spaceDtNmapisseparableandlowrank: numericalverification 89
4.4.1 Slow disk and vertical fault . . . . . . . . . . . . . . . . . . . 90
4.4.2 Constant medium, waveguide, diagonal fault . . . . . . . . . . 91
4.4.3 Focusing and defocusing media . . . . . . . . . . . . . . . . . 92
5 Conclusion 95
A Summary of steps 97
(cid:93)
A.1 Matrix probing: D → D . . . . . . . . . . . . . . . . . . . . . . . . . 98
(cid:93)
A.2 PLR compression: D → D . . . . . . . . . . . . . . . . . . . . . . . . 99
A.3 Using D in a Helmholtz solver . . . . . . . . . . . . . . . . . . . . . . 99
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List of Figures
2-1 The exterior problem: Ω is in grey, there is a thin white strip around
Ω, then an absorbing layer in blue. . . . . . . . . . . . . . . . . . . . 32
2-2 The half-space problem: there is thin white strip below the bottom
side of Ω, and an absorbing layer (in blue) surrounds that strip on
three sides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3-1 Color plot of c(x) for the uniform medium. . . . . . . . . . . . . . . . 49
3-2 Color plot of c(x) for the Gaussian waveguide. . . . . . . . . . . . . . 49
3-3 Color plot of c(x) for the Gaussian slow disk. . . . . . . . . . . . . . . 49
3-4 Color plot of c(x) for the vertical fault. . . . . . . . . . . . . . . . . . 49
3-5 Color plot of c(x) for the diagonal fault. . . . . . . . . . . . . . . . . 49
3-6 Color plot of c(x) for the periodic medium. . . . . . . . . . . . . . . . 49
3-7 Condition numbers for the (1,1) block, c ≡ 1. . . . . . . . . . . . . . 53
3-8 Condition numbers for the (2,1) block, c ≡ 1. . . . . . . . . . . . . . 53
3-9 Approximation error (line) and probing error (with markers) for the
blocks of D, c ≡ 1. Circles are for q = 3, squares for q = 5, stars for
q = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3-10 Approximation error (line) and probing error (with markers) for the
blocks of D, c is the waveguide. Circles are for q = 3, squares for q = 5,
stars for q = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3-11 Approximation error for the (1,1) blocks of D, c is the slowness disk,
comparing the use of the normal traveltime (circles) to the fast march-
ing traveltime (squares). . . . . . . . . . . . . . . . . . . . . . . . . . 55
3-12 Approximation error for the (2,1) blocks of D, c is the slowness disk,
comparing the use of the normal traveltime (circles) to the fast march-
ing traveltime (squares). . . . . . . . . . . . . . . . . . . . . . . . . . 55
3-13 Approximation error (line) and probing error (with markers) for the
blocks of D, c is the fault. Circles are for q = 3, squares for q = 5,
stars for q = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3-14 Approximation error (line) and probing error (with markers) for the
sub-blocks of the (1,1) block of D, c is the fault. Circles are for q = 3,
squares for q = 5, stars for q = 10. . . . . . . . . . . . . . . . . . . . . 56
3-15 Real part of the solution, c is the slow disk. . . . . . . . . . . . . . . 57
3-16 Real part of the solution, c is the waveguide. . . . . . . . . . . . . . . 57
3-17 Real part of the solution, c is the vertical fault with source on the left. 57
3-18 Real part of the solution, c is the vertical fault with source on the right. 57
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3-19 Real part of the solution, c is the diagonal fault. . . . . . . . . . . . . 59
3-20 Imaginary part of the solution, c is the periodic medium. . . . . . . . 59
3-21 Error in solution u, c ≡ 1, moving point source. Each line in the plot
(cid:93)
corresponds to using D from a different row of Table 3.2. . . . . . . . 60
3-22 Probing error of the (1,1) block of the DtN map for the slow disk, fixed
FD error level of 10−1, increasing N. This is the worst case, where p
follows a weak power law with N. . . . . . . . . . . . . . . . . . . . . 60
3-23 Probing error of the (1,1) block of the DtN map for the waveguide,
fixed FD error level of 10−1, increasing N. Here p follows a logarithmic
law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3-24 Probingerrorofthehalf-spaceDtNmap(q = 1, 10trials, circlemarkers
and error bars) compared to the approximation error (line), c ≡ 1,
L = 1/4, α = 2, n = 1024, ω = 51.2. . . . . . . . . . . . . . . . . . . . 65
3-25 Condition numbers for probing the half-space DtN map, c ≡ 1, L =
1/4, α = 2, n = 1024, ω = 51.2, q = 1, 10 trials. . . . . . . . . . . . . 65
4-1 Weak hierarchical structure, N = 8. . . . . . . . . . . . . . . . . . 74
Rmax
4-2 Strong hierarchical structure, N = 16. . . . . . . . . . . . . . . . . 74
Rmax
4-3 Corner PLR structure, N = 8. . . . . . . . . . . . . . . . . . . . . 74
Rmax
4-4 Matrix-vector complexity for submatrix (1,1) for c ≡ 1, various R .
max
Probing errors of 10−5, 10−6. . . . . . . . . . . . . . . . . . . . . . . . 78
4-5 Matrix-vector complexity for submatrix (2,1) for c ≡ 1, various R .
max
Probing errors of 10−5, 10−6. . . . . . . . . . . . . . . . . . . . . . . . 78
4-6 PLR structure of the probed submatrix (1,1) for c ≡ 1, R = 2.
max
Each block is colored by its numerical rank. . . . . . . . . . . . . . . 79
4-7 PLR structure of the probed submatrix (1,1) for c ≡ 1, R = 8.
max
Each block is colored by its numerical rank. . . . . . . . . . . . . . . 79
4-8 PLR structure of the probed submatrix (1,1) for c ≡ 1, R = 32.
max
Each block is colored by its numerical rank. . . . . . . . . . . . . . . 79
4-9 PLR structure of the probed submatrix (2,1) for c ≡ 1, R = 4.
max
Each block is colored by its numerical rank. . . . . . . . . . . . . . . 79
4-10 Maximum off-diagonal ranks with N for the slow disk, various (cid:15). FD
error of 10−3, r = h. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
0
4-11 Maximum off-diagonal ranks with N for the slow disk, various (cid:15). FD
error of 10−3, r = 4h. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
0
4-12 Maximum off-diagonal ranks with ε for the slow disk, various N. FD
error of 10−3, r = h. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
0
4-13 Maximum off-diagonal ranks with ε for the slow disk, various N. FD
error of 10−3, r = 4h. . . . . . . . . . . . . . . . . . . . . . . . . . . 91
0
4-14 Maximum off-diagonal ranks with N for the diagonal fault, various (cid:15).
FD error of 10−2, r = h. . . . . . . . . . . . . . . . . . . . . . . . . 91
0
4-15 Maximum off-diagonal ranks with ε for the diagonal fault, various N.
FD error of 10−2, r = h. . . . . . . . . . . . . . . . . . . . . . . . . . 91
0
10