Direct Methods for Inverse Scattering with Time

DIRECT METHODS FOR INVERSE SCATTERING WITH TIME

DEPENDENT AND REDUCED DATA

Jacob D. Rezac

A dissertation submitted to the Faculty of the University of Delaware in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics

Summer 2017

DEPENDENT AND REDUCED DATA

Jacob D. Rezac

Approved: Louis Rossi, Ph.D. Chair of the Department of Mathematical Sciences

Approved: George H. Watson, Ph.D. Dean of the College of Arts and Sciences

Approved: Ann L. Ardis, Ph.D. Senior Vice Provost for Graduate and Professional Education I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.

Signed: Fioralba Cakoni, Ph.D. Professor in charge of dissertation

I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.

Signed: David Colton, Ph.D. Member of dissertation committee

Signed: Houssem Haddar, Ph.D. Member of dissertation committee

Signed: Peter Monk, Ph.D. Member of dissertation committee ACKNOWLEDGEMENTS

First and foremost, I would like to thank my Ph.D supervisor, Dr. Fioraba Cakoni. She has been a knowledgable and patient collaborator and mentor, and I am greatful for her help with this thesis. I have particularly appreciated her continued support, even after moving a few hours up the road. It was a true pleasure being her student. I would also like to acknowledge my thesis committee, Drs. David Colton, Houssem Haddar, and Peter Monk for sharing their extensive knowledge and experience with me, and for always providing interesting new ideas and a welcoming environment in which to study them. In particular, I would like to thank Dr. Haddar for hosting me for two productive and enjoyable extended visits to Ecole´ Polytechnique. I would also like to thank Drs. Yehuda Braiman and Neena Iman, who hosted me at Oak Ridge National Laboratory, allowing me to work on applied problems which are unrelated to this thesis. The work done in this thesis was funded by NSF Grant DMS-1602802 and INRIA DeFI team, whose support I gratefully acknowledge. The time I spent completing my thesis was significantly improved, both mathematically and personally, by many people. In particular, thanks to my math siblings, Isaac Harris, Shixu Meng, and Irene de Teresa Trueba, and my unrelated math brother Brennan Sprinkle. Thanks to the wings crew, Kevin Aiton, Zach Bailey, Thomas Brown, Matt Hassell, Allan Hungria, Lise-Marie Imbert-Gérard,Matt McGinnis, To- natiuh Sanchez-Vizuet, and Francisco Sayas; to my friends at Ecole´ Polytechnique, Si- mone Schavi, Nicolo Castro, Mohammed Lakhal, Helle Majander, Tobias Rienmuller, and Faisal Wahid; to Food Club members Amy Jannet and Madelyn Houser and to Brunch Club members Nick Kaufman and Frances Bothfeld; former office and house mates James Alexander and Michael Depersio; and to Mike Greco and Yolanda Lin.

iv Special thanks for my family, Siobhan, Rex, Ben, Aida, Kate, Mike, Maggie, Joe, Amelia, Violet, Hannah, and Charles.

v TABLE OF CONTENTS

LIST OF TABLES ...... ix LIST OF FIGURES ...... x ABSTRACT ...... xiii

Chapter

1 FORWARD AND INVERSE WAVE SCATTERING PROBLEMS1

1.1 Wave Equations ...... 4

1.1.1 Scattering Problems ...... 8 1.1.2 Volume Integral Equations and the Born Approximation ... 9

1.2 Qualitative Methods in Inverse Scattering ...... 11 1.3 Data Reduction and Primary Contributions of this Thesis ...... 17

2 QUASI-BACKSCATTERING IN THE FREQUENCY DOMAIN 19

2.1 Introduction ...... 19 2.2 Direct Scattering Problem ...... 20

2.2.1 Forward Problem for Quasi-Backscattering Data ...... 22

2.3 Quasi-Backscattering Inverse Problem ...... 23

2.3.1 Inverse Problem for Small Obstacles ...... 23 2.3.2 A Second Range Test for Three-Dimensional Reconstructions . 27 2.3.3 Inverse Problem for Coplanar Small Obstacles ...... 28 2.3.4 Inverse Problem for Extended Obstacles ...... 29

2.4 Numerical Experiments ...... 31

2.4.1 Two-Dimensional Projections of Small Obstacles ...... 32

vi 2.4.2 Reconstruction of Third Coordinate ...... 37 2.4.3 Three-Dimensional Reconstructions ...... 39

3 DIRECT IMAGING OF SMALL SCATTERERS USING REDUCED TIME DEPENDENT DATA ...... 42

3.1 Introduction ...... 42 3.2 Forward Model and the Born Approximation ...... 46 3.3 Inverse Problem for the Time Domain Born Approximation ...... 50

3.3.1 Reconstruction of Point Scatterers from Time Domain Multistatic Data ...... 51 3.3.2 Reconstruction of Point Scatterers from Patches of Time Domain Multistatic Data ...... 56 3.3.3 Reconstruction of Point Scatterers from Time Domain Quasi-Backscattering Data ...... 57 3.3.4 Linear Sampling Method for Extended Objects Under the Time Domain Born Model ...... 61

3.4 Numerical Reconstructions ...... 63

3.4.1 MUSIC and LSM Reconstruction with Multistatic Data .... 64 3.4.2 Quasi-Backscattering Reconstructions ...... 65

4 THE BORN TRANSMISSION EIGENVALUES PROBLEM ... 73

4.1 Introduction ...... 73 4.2 Spherically Stratiﬁed Media ...... 76 4.3 Transmission Eigenvalue Free Regions ...... 81 4.4 Transmission Eigenvalues for General Shapes and Contrasts ..... 83

5 FAST METHODS FOR HELMHOLTZ EQUATION SIMULATION WITH APPLICATIONS TO TIME DOMAIN SCATTERING AND BAYESIAN INVERSE PROBLEMS .... 92

5.1 Volume Integral Equations with Galerkin and FFT-Based Techniques 93

5.1.1 Galerkin Approach ...... 93 5.1.2 Piecewise Constant Finite Element Discretization ...... 97 5.1.3 Numerical Results for Galerkin Approximation ...... 99 5.1.4 The Adaptive Integral Method (AIM) ...... 100 5.1.5 Interpolation Between Finite Element and Cartesian Grid .. 103

vii 5.1.6 Numerical Results for the Time Harmonic Problem ...... 105

5.2 Convolution Quadrature for Time Domain Equations ...... 108

5.2.1 Approximation by Independent Helmholtz Equations and Error Analysis ...... 113 5.2.2 Implementation and Numerical Results ...... 116

5.3 Application I: CQ-AIM ...... 118

5.3.1 Error Analysis ...... 118 5.3.2 Numerical Results ...... 119

5.4 Application II: Bayesian Inverse Scattering ...... 121

5.4.1 Inﬁnite Dimensional Bayesian Inverse Problems ...... 123 5.4.2 Numerical Implementation ...... 125 5.4.3 Simulations ...... 127

6 OUTLOOK AND OPEN PROBLEMS ...... 133

BIBLIOGRAPHY ...... 135

Appendix

A SEPARATION OF VARIABLES SOLUTIONS ...... 143 B COPYRIGHT PERMISSIONS ...... 145

viii LIST OF TABLES

1.1 Outline of algorithm used by many qualitative techniques. The curves d−1 Γm and Γi refer to the locations in R where receivers and transmitters are placed, respectively. We require that both curves contain D in their interior...... 12

5.1 Simulation results for time-harmonic scattering from a ball with constant index of refraction...... 107

5.2 Simulation results for time-harmonic scattering from a ball with non-constant index of refraction...... 108

5.3 Simulation results for scattering from a non-convex shape with non-constant index of refraction...... 109

5.4 Time domain simulation results for scattering from a ball with constant index of refraction...... 120

5.5 Time and memory usage for calculation Tikhonov regularized solution of contrast...... 129

ix LIST OF FIGURES

2.1 A Comparison of multi-static data (left) and quasi-backscattering data (right). Red circles correspond to device locations. The thick blue line in the right ﬁgure denotes where the quasi-backscattering set-up is moved and where each transmitting device is located. ... 20

2.2 Symmetric addition of new objects, δ = π/50 (no noise)...... 33

2.3 Symmetric addition of new objects, δ = π/50 (approximately 1% noise)...... 34

2.4 Decreasing the quasi-parameter δ. Figures have δ = π (top-left), δ = π/50 (top-right), and δ = π/100 (bottom). Approximately 5% noise...... 34

2.5 Two objects moving closer to each other, δ = π/100 (no noise). Thick bar at bottom corresponds to half of wavelength...... 35

2.6 An L-Shaped geometry which requires 3 views to see all obstacles, δ = π/30 (approximately 1% noise)...... 36

2.7 Results for co-planar obstacles, δ = π/60. Figures on left are noise-free and ﬁgures on right have approximately 1% noise. .... 37

ˆ 2.8 When JΠ⊥(zj ) is computed, peaks appear for i 6= j. Red circles show exact location of L(zj)...... 38

2.9 Reconstructions lose accuracy in the presence of 0.1% noise. Red circles show exact location of L(zj)...... 39

2.10 Three-dimensional noise-free reconstructions of point obstacles based on multiple experiments. We take 95 incident directions, 95 observation points, and use δ = π/60. The top figure is noise-free while the bottom figure has approximately 1% noise. In both figures, we display isovalues of 0.6 times the maximum value of the imaging function...... 41

x 3.1 Examples of limited aperture multistatic (left) and quasi-backscattering (right) measurements. In the limited aperture multistatic ﬁgure, the blue line represents the location of transmitters and the red line the location of receivers. In the quasi-backscattering (y) set-up, Γi is the large dashed circle, the thick solid line is Γm for a ﬁxed y ∈ Γi, and the circles on Γi not located at y represent locations (y) to which Γm will be moved...... 45

−1 −1 3.2 Plots of (ILSM(z)) (top) and (Imulti(z)) (bottom) for two diﬀerent geometries...... 66

3.3 Multistatic patch reconstructions of the same geometry of small circles, indicated by black lines. (Top) Four patches are used with 5 transmitters and receivers each. From left-to-right, the aperture of each patch decreases from π/2 to π/4 to π/8. (Bottom) The same experiment as top but with 10 patches. Each set of circles indicates the location of transmitters and receivers in each patch. Transmitters and receivers in the same patch each have the same color...... 67

3.4 Multistatic patch reconstructions of the same geometry of medium-sized ellipses, indicated by black lines. (Top) Four patches are used with 5 transmitters and receivers each. From left-to-right, the aperture of each patch decreases from π/2 to π/4 to π/8. (Bottom) The same experiment as top but with 10 patches. Each set of circles indicates the location of transmitters and receivers in each patch. Transmitters and receivers in the same patch each have the same color...... 68

−1 3.5 Plots of (Iquasi(z, τ)) for four diﬀerent geometries...... 69

ˆ −1 −1 3.6 Plots of (Iquasi(z, τ)) (top) and (Iquasi(z, τ)) (bottom), with a diﬀerent number of transmitters in each row. On the left there are 5 transmitters, in the middle there are 10 transmitters, and the right there are 15 transmitters. Time harmonic data was computed with wavenumber k = 3...... 70

−1 3.7 Backscattering reconstructions using Ibackscattering(z, τ) for two diﬀerent geometries. In both ﬁgures, 30 transmitters are used and data is measured only at the location of the transmitter. Time domain data was simulated for 14 seconds with 480 time steps. ... 71

xi 3.8 Limited aperture reconstructions using multistatic data with −1 Imulti(z, τ) (top), multistatic patch data with two patches (middle), −1 and quasi-backscattering data with Iquasi(z, τ) (bottom). In both ﬁgures, 19 transmitters are used and in the case of quasi-backscattering data, 4 receivers were used...... 72

4.1 Plot of the function d0(k) associated with m = 1 − 1.95r. When d0(k) crosses the real axis, k is a transmission eigenvalue...... 81

4.2 Sample of the region in which transmission eigenvalues can appear for a real constant contrast...... 83

5.1 Convergence rate for time harmonic scattering from a ball with constant contrast simulated with P0 ﬁnite elements...... 100

5.2 The shape of the non-convex object oﬀ which we simulate scattering. The ﬁnite element triangulation with N = 192 used is shown as well. 109

5.3 Convergence rate of CQ-Galerkin scheme for multiple meshes. ... 117

5.4 Tikhonov regularized solutions to f(m) = yobs. All images have the same color scale, and white lines indicate ﬁnite element mesh. Left column: Solutions corresponding to a Galerkin scheme h = 0.0512 (top) and h = 0.261 (middle), and the exact solution projected onto a mesh of size h = 0.0131 (bottom). Right column: Solutions corresponding to an AIM scheme h = 0.0512 (top) and h = 0.261 (middle), h = 0.0131 (bottom)...... 130

5.5 Bayesian solution to inverse scattering problem. (Top right) A sample generated by the prior distribution on a mesh of size h = 0.0512. (Top left) A sample generated by the posterior distribution on a mesh of size h = 0.0512. (Bottom) The MAP solution on a mesh of size h = 0.0512. Note that the posterior sample and MAP solution use the same color scheme, which is diﬀerent from the posterior color scheme. White lines indicate the mesh...... 132

xii ABSTRACT

This thesis is focused on the motion of acoustic waves through penetrable media, and the use of such waves to reconstruct material properties of the fluid through which the waves are moving. The reconstruction methods developed in this thesis fall under the category of qualitative inversion methods and, as such, are fast and mathematically justified. Unlike in typical qualitative methods, however, these new methods require only small amounts of scattered field data to be collected. In particular, we demonstrate that with both far field time harmonic data and near field time dependent data, the location of weakly scattering point obstacles can be reconstructed with reduced data collection requirements compared to typical qualitative schemes. We give full mathematical justification for the time harmonic method and partial justification for the time dependent method. We also analyze the transmission eigenvalue problem for weakly scattering media, proving that, under this assumption, transmission eigenvalues are discrete and can sometimes have complex part which grows without bound. Finally, we introduce a fast method for simulating time harmonic and time dependent acoustic wave scattering and apply this method to optimization schemes for reconstructing penetrable media based on scattered field data.

xiii Chapter 1

FORWARD AND INVERSE WAVE SCATTERING PROBLEMS

Much of applied mathematics research over the past 150 years has been dedicated to determining the behavior of a physical system with specific parameters. At the time, many of the mathematical and scientific breakthroughs of the era began to describe the motion of acoustic and electromagnetic waves and how they interact with their surroundings. Such was the focus on these problems that by the early 1900s, Jacques Hadamard had begun to describe the physical problems which could be solved by mathematics [55]. Indeed, this led to Hadamard’s concept of a well-posed problem, Definition 1. A problem is said to be well-posed if 1. A solution exists; 2. the solution is unique; and 3. the solution depends continuously on its initial data.

A problem which is not well-posed is called ill-posed.

This focus on well-posed problems which describe a physical phenomenon given initial conditions - which we call here a forward problem or direct problem - led to many breakthroughs in theoretical and applied mathematics (as well as, for example, in the sciences, engineering, and economics). However, there is a huge group of physically important problems which are not well-posed. Often problems in which one is given data about a physical process and asked what initial conditions or model led to that data do not satisfy one (or any!) of the criteria in Deﬁnition1. These ill-posed problems are called inverse problems, as a contrast with the forward problem. Although they are not well-posed, these problems are often vital in engineering and science and have become more mathematically tractable over the last 50 years.

1 This thesis will focus on one particular type of inverse problem, the inverse scattering problem. In an inverse scattering problem, we estimate the physical properties about hidden objects based on the way acoustic or electromagnetic waves interact with the objects. This estimation process is sometimes called, appropriately, parameter estimation or reconstruction. Although we rarely explicitly mention their applications in this thesis, these problems find use in many fields of science and engineering, including geophysical exploration, non-destructive testing of structures, and in medical devices used to scan the interior of bodies. In part due to this wide range of use, we are interested in developing methods which are fast, accurate, and mathematically justified. In this chapter, we introduce the basic theory for both forward and inverse wave scattering problems in acoustics. In Section 1.1, we derive the acoustic wave and Helmholtz equations from first principles. These will be the primary equations which we will study in later chapters. When designing methods for parameter estimation, the forward model of the physical situation represents an important piece of a priori information. As such, we will spend a significant amount of time describing the forward model of our problem. Once we understand the behavior of the physical system, we introduce the class of inversion algorithms which are the primary focus of this thesis: qualitative methods. We introduce a few qualitative methods for inverse scattering in Section 1.2. Qualitative methods are an important class of reconstruction techniques which only aim to estimate a few important parameters, rather than the entire parameter state. For example, we are often interested in estimating the size, shape, and location of a hidden object, rather than the precise value of each material property of the object at all points. Qualitative methods are significantly faster than, for example, non-linear optimization schemes, and do not require as much a priori information. However, they also do not return as much information. We will briefly introduce a nonlinear optimization-style reconstruction technique at the end of this thesis to compare with these qualitative schemes.

2 This discussion of qualitative methods is continued in Chapter2 where we introduce a technique for reconstructing the location of small objects using less scattering data than is typically required by qualitative methods. In particular, we only collect scattered field data in a small region surrounding the location from which an incident field was transmitted, using an experimental set-up called quasi-backscattering. This is in contrast to usual qualitative methods which require scattered field data to be collected on a surface completely surrounding an object in order to reconstruct the location of the object. Chapter3 continues the discussion of reconstruction algorithms using quasi-backscattering data. In contrast to Chapter2 which makes use of time harmonic incident fields and scattered field data collected in the far field, Chapter3 makes use of time dependent incident fields with scattered field data collected in the near field. Both Chapters2 and3 include reconstruction algorithms for small and weakly- scattering objects. One of the primary advantages of typical qualitative methods is that they do not require any such assumptions. However, as Chapters2 and3 demonstrate, a weak scattering assumption can lead to a less restrictive experimental set-up. As such, in Chapter4, we discuss the justification of a qualitative method called the linear sampling method (LSM) for time dependent incident fields scattering from small and weak obstacles. In particular, we discuss the properties of an auxiliary eigenvalue problem, called the Born interior transmission problem, which would lead to the mathematical justification for the weak-scattering LSM. To end the thesis, in Chapter5, we discuss the numerical simulation of forward acoustic scattering problems. In particular, we introduce an unconditionally stable method of optimal convergence order to simulate the scattering of both time harmonic and time domain acoustic waves. This volume integral equation method uses a finite element Galerkin approximation in space and convolution quadrature in time. Un- fortunately, this method is slow and requires a large amount of memory. Hence, we also discuss a fast method to approximate some of the spatial operators in this slow method. We also apply this fast method to a different type of inversion scheme than

3 is discussed elsewhere in this thesis, a Bayesian inversion algorithm. In a Bayesian inverse problem, we look for a probability distribution to describe our reconstruction of material properties of the medium we are probing. These methods, unlike qualitative methods, require a large number of numerical simulations of acoustic wave scattering through many different fluid media. However, they result in significantly more information about the reconstructed object and in simulations do not have serious restrictions on the amount of scattered field data which is collected. Some of this thesis was originally presented in the following articles:

1. H. Haddar and J.D. Rezac, A quasi-backscattering problem for inverse acoustic scattering in the Born regime. Inverse Problems 31, 075008 (2015).

2. F. Cakoni, D. Colton and J.D. Rezac, The Born transmission eigenvalue problem. Inverse Problems 32 105014 (2016).

3. F. Cakoni and J.D. Rezac, Direct imaging of small scatterers using reduced time dependent data. Journal of Computational Physics 338, 371387 (2017).

1.1 Wave Equations We begin by deriving the acoustic wave equation, which describes the propagation of acoustic waves through media with changing material properties, such as sound speed or density. We follow [36, 43] in this derivation. Throughout the thesis, we focus on scattering from penetrable media - that is, media which allows wave to pass through it. This is opposed to obstacle scattering, which is a model which includes the assumption that scattered waves are completely reﬂected by the scattering media, completely absorbed, or only absorbed into a small amount of the object. Despite this distinction, we will often refer to scattering from penetrable media as scattering from the object deﬁned by where material properties (such as the speed of sound) change from background properties. This abuse of notation will improve readability, and references to scattering from obstacles should not be interpreted as scattering from impenetrable media. Note that while this derivation is given in R3, the resulting equations hold for R2 as well.

4 Sound waves are propagating vibrations in a fluid medium. As such, we are interested in describing the way in which the properties, such as pressure and velocity, of a small volume of fluid change in response to an acoustic disturbance. Assume that, prior to its interaction with an acoustic wave, the fluid is in an equilibrium state with

3 equilibrium pressure P0 and density ρ0 = ρ0(x), x ∈ R . Also assume that the medium of interest is inviscid and that there are no external forces on the fluid which affect sound propagation. Denote by Ω ⊂ R3 the small volume of fluid whose boundary is denoted by ∂Ω with outward pointing normal ν(x). Pressure and density are related through some equation of state which we denote as P = f(ρ, S), (1.1) where S = S(x, t) is the specific entropy of the physical system. For example, the function f could describe an ideal gas law whose form is dictated by the system’s underlying physical processes. The equilibrium state of the fluid is defined by the property that ∂f ∂t = ∇f = 0. As a simplifying assumption, assume that acoustic disturbances occur quickly enough that heat diffusion does not affect sound propagation. This adiabatic approximation is valid for sound approximation in air, for example, and is expressed by ∂S + v · ∇S = 0, (1.2) ∂t where v = v(x, t) is the velocity of the fluid. When an acoustic wave interacts with the small volume of fluid, the volume will move and change shape, and hence its density will change. This change in density, in turn, affects the pressure of the volume, which then causes fluid motion due to a pressure gradient. Since pressure and density are related by f, the relationship between pressure and fluid motion can be described through a relationship between density and fluid motion. Absent external forces, the only way to change density is through the

5 change of mass into and out of Ω. In particular, ∂ Z Z ρ(x, t) dV (x) = − ρ(x, t)v(x, t) · ν(x) ds(x) ∂t Ω ∂Ω Z = − ∇ · (ρ(x, t)v(x, t)) dx. Ω This results in the conservation of motion equation ∂ρ + ∇ · (ρv) = 0. (1.3) ∂t Finally, the way in which Ω moves due to changing pressure gradients can be described by conservation of momentum. Newton’s Second Law gives that ∂ Z Z ρ(x, t)v(x, t) dV (x) = − ρ(x, t)v(x, t)(v(x, t) · ν(x)) + P (x, t) · ν(x) ds(x) ∂t Ω ∂Ω Z = − v(x, t)∇ · (ρ(x, t)v(x, t)) + ∇P (x, t) dV (x). Ω Note that the change of momentum in Ω is due both to the change of momentum across ∂Ω and due to the external pressure changes. Hence, we have the conservation of momentum equation ∂(ρv) + v∇ · (ρv) + ∇P = 0. (1.4) ∂t In typical situations1, equilibrium values of properties of the ﬂuid are much larger than the changes caused by acoustic wave propagation. As such, we perform an asymptotic expansion of pressure, velocity, density, and entropy, and use equations (1.1)-(1.4) to describe the change in ﬂuid properties based on an acoustic disturbance. Hence, for 1, let

2 P (x, t) = P0 + P1(x, t) + O( )

2 ρ(x, t) = ρ0(x) + ρ1(x, t) + O( )

2 v(x, t) = v1(x, t) + O( )

2 S(x, t) = S0(x) + S1(x, t) + O( ).

1 An atypical situation might be an explosion or an airplane breaking the sound barrier. In these cases, nonlinear aﬀects contribute signiﬁcantly to sound propagation.

6 From the assumption, the linearized governing equations are ∂S ∂f ∂f 1 + v · ∇S = 0 P = ρ (ρ ,S ) + S (ρ ,S ) ∂t 1 0 1 1 ∂ρ 0 0 1 ∂S 0 0 ∂ρ1 ∂v1 1 + ρ0∇ · v1 = 0 + ∇P1 = 0. ∂t ∂t ρ0

Using the assumption that f(ρ0,S0) is a stationary state, ∂f ∂f 0 = ∇f = ∇ρ (ρ ,S ) + ∇S (ρ ,S ). 0 ∂ρ 0 0 0 ∂S 0 0 Combining this equation with the linearized adiabatic approximation and the time derivative of the state equation yields ∂P ∂ρ 1 = c2(x) 1 + v · ∇ρ , (1.5) ∂t ∂t 1 0 2 ∂f where c (x) = ∂ρ (ρ0(x),S0(x)) is the speed of sound in the medium. Finally, taking a time derivative of (1.5) and using the two as-yet unused linearized equations yields a wave equation for the change in pressure caused by an acoustic disturbance, 2 ∂ P1 2 1 2 = c (x)ρ0(x)∇ · ∇P1 . ∂t ρ0(x)

This is the acoustic wave equation. For simplicity, we will always set ρ0 to be constant so that ∂2P 1 = c2(x)∆P . ∂t2 1 Hence, we are only considering acoustic propagation through a medium which changes only due to changes in entropy. This could correspond to a changing temperature gradient in the medium, for example. The wave equation also governs the velocity potential, say u(x, t), which is defined by 1 ∂u v1 = ∇u and P1 = − . ρ0 ∂t Indeed, defined in this way, ∂2u = c2∆u. (1.6) ∂t2 Although the physical meaning of different variables will not be important in what follows, in the rest of this thesis we will discuss scattering problems through the velocity potential.

7 1.1.1 Scattering Problems

Assume an acoustic incident wave, ui = ui(x, t; y), x 6= y ∈ Rd (d = 2 or 3), t > 0 was emitted from a point y and is traveling through a homogeneous medium which is large enough compared to the wavelength ui that we can approximate it as all of Rd. Based on the discussion above, ui satisfies ∂2ui = c2∆ ui, x 6= y ∈ d, t > 0 (1.7) ∂t2 0 x R for some constant c0 > 0. By rescaling variables, we always set c0 ≡ 1. The primary aim of scattering theory is to understand the way in which ui is affected by an inhomogeneity in the fluid medium. Hence, assume a variable speed of sound, c ∈ L∞(Rd) so that c(x) ≥ γ > 0 for some γ ∈ R. Define n(x) = c−2(x) to be the square of the index of refraction and D = supp(1 − n(x)) to be the location of the unknown scatterer. We also introduce the contrast function m(x) = n(x) − 1, which will simplify notation below. The total acoustic field traveling through such a medium behaves according to (1.6). Separating the total field into the sum of an incident field, ui, and scattered field, us yields the governing equation ∂2us ∂2 − ∆us = −m(x) ui + us , (x, t) ∈ d × +. (1.8) ∂t2 ∂t2 R R We must ensure that at time t = 0, the incident field has not yet been affected by the inhomogeneity. As such, assume ∂us ui(x, t) = 0, x ∈ D, t ≤ 0, and us(x, 0) = (x, 0) x ∈ D. (1.9) ∂t This assumption is referred to as a causal wave assumption. Combined, (1.7)-(1.9) constitute the time dependent wave scattering problem. Thus far we have not considered the form of the incident field. We will primarily consider two types of incident field: time harmonic and fully time dependent. A time harmonic incident field is of the form

i i −ikt u (x, t; y) = Re uˆ (x)e 1t≥0.

8 where 1X is an indicator function on a set X and is only required to ﬁt the causality assumption. Note that in the following, we will always use ˆ· over an independent variable to indicate it is related to the time harmonic problem. Substituting this into (1.7) and assuming us has the same form yields the time harmonic scattering problem

∆ˆus + k2uˆs = −k2m(x) uˆi +u ˆs

∆ˆui + k2uˆi = 0.

We also add the Sommerfeld radiation condition, ∂uˆs lim r−(d−1)/2 − ikuˆs = 0, r = |x| (1.10) r→∞ ∂r which ensures that waves are outgoing rather than incoming. The other type of incident field of interest is a time dependent one. Let χ ∈ C2(D) be a causal temporal pulse function (that is, χ and its derivatives vanish for t < 0). We define the incident field originating at a point y ∈ Rd as the time convolution of χ with the fundamental solution Φ to the wave equation,  H(t − |x|)   p , d = 2 2π t2 − |x|2  Φ(x, t) = (1.11)  δ(t − |x|)  , d = 3,  4π|x| where H is the Heaviside function. For example, in R3 χ(t − |x − y|) ui(x, t; y) = , (x, t) ∈ ( 3\y) × +. 4π|x − y| R R

1.1.2 Volume Integral Equations and the Born Approximation Solutions to the scattering problems, both in the time harmonic and time dependent, can be represented in many ways. In this thesis we will focus on their volume integral equation representation. For example, working formally, convolving both sides of (1.8) against Φ deﬁned in (1.11) yields for x 6= y ∈ Rd and t > 0 that Z Z s s i u (x, t; y) = − m(z)Φ(x − z, t − τ) utt(z, τ) + utt(z, τ) dV (z) dτ. R D

9 With this form in mind, deﬁne the retarded volume potential operator V acting on

∞ d f ∈ C0 (B × R), with D ⊂ B ⊂ R , by Z Z d (V f)(x, t) := Φ(x − z, t − τ)f(τ, z) dV (z) dτ, (x, t) ∈ R × R. R D It is well-known [73] that v(x, t) = (V f)(x, t) satisﬁes

d vtt − ∆v = f in R × R.

This leads to the time domain Lippmann-Schwinger equation for us,

s s i d u (x, t; y) + (V [mu ])(x, t; y) = −(V [mu ])(x, t; y) x 6= y ∈ R , t > 0. (1.12)

Although the above has been formal, we will provide justiﬁcation and mapping properties of V in Chapters3 and5. Note that in the case of time harmonic data, we arrive at a similar conclusion. In particular,u ˆs satisﬁes

s s 2 i d uˆ (x; y) + (Vˆ [muˆ ])(x; y) = −k (Vˆ [muˆ ])(x; y), x 6= y ∈ R , (1.13)

∞ where for f ∈ C0 (B), D ⊂ B, Z ˆ 2 ˆ d (V f)(x) := −k m(z)Φk(x, z)f(z) dV (z), x ∈ R D and where   i H(1)(k|x − y|) d = 2 ˆ  4 0 Φk(x, y) := (1.14)  exp (ik|x−y|)  4π|x−y| d = 3. is the time harmonic fundamental solution. Aside from providing a solution representation for the scattered field, the Lippmann- Schwinger equations provide a helpful form for computing asymptotic expansions of the scattered field. In this thesis, we will often make strong assumptions on the speed of sound in the medium which allow us to weaken assumptions on other aspects of the scattering problem. In particular, we will make a weak scattering Born approximation in which multiple scattering effects are minor and can be ignored. Assume that n(x) = 1 + mB(x) for 1, mB = O(1), and that solutions to (1.6) take the

10 s s s s ∂us form u (x, t) = u0(x, t) + uB(x, t). The function uB = ∂ =0 is the ﬁrst term in the i s well-known Born approximation. Indeed, if u (x, t) is of the same order as u0(x, t) for s s x ∈ D, then separating into powers of yields u0 ≡ 0 and that uB satisﬁes ∂2us ∂2ui B − ∆us = −m for (x, t) ∈ d × + (1.15a) ∂t2 B B ∂t2 R R ∂us us (x, 0) = B (x, 0) = 0 for x ∈ d. (1.15b) B ∂t R

Using the volume integral equation approach introduced above yields

s i d uB(x, t; y) = −(V [mButt])(x, t; y), x 6= y ∈ R , t > 0. (1.16)

Notice that the solution represented in this way does not require us to solve an integral equation to find the scattered field, but rather just to apply an integral operator. This approximation significantly simplifies calculations at the expense of applicability of results. The above is more typically done for time harmonic data, which we will discuss in depth in Chapter2.

1.2 Qualitative Methods in Inverse Scattering A primary goal of inverse scattering theory is the reconstruction of information about unknown objects based on how acoustic or electromagnetic waves scatter oﬀ of them. Qualitative methods are able to quickly and accurately determine the shape and location of hidden objects, and require little a priori information about the objects. They are non-iterative in nature and do not require large scale wave simulations. We refer the reader to [23] for a comprehensive account of these methods for inhomogeneous media. Each qualitative method takes a similar form, as indicated by the general algorithm in Table 1.1 below. In particular, we deﬁne an indicator function, depending on z ∈ Rd (and possibly on time τ ∈ [0,T ] for some T > 0), so that the function is large when z ∈ D and small otherwise. The bulk of research in qualitative methods is dedicated to deriving indicator functions and demonstrating that they are large when z ∈ D.

11 Algorithm Frequency (time) domain sampling methods for reconstruction of obstacles Step 1 Collect scattered ﬁeld data at x ∈ Γm, y ∈ Γi (and t ∈ [0,T ]). Step 2 Select a set of sampling grid points Z (and τ ∈ [0,T ]). Step 3 Plot the indicator function for each z ∈ Z (and τ ∈ [0,T ]). Step 4 Post-process or regularize the indicator function to determine the collection of z ∈ D.

Table 1.1: Outline of algorithm used by many qualitative techniques. The curves d−1 Γm and Γi refer to the locations in R where receivers and transmitters are placed, respectively. We require that both curves contain D in their interior.

The primary idea behind each of these indicator functions is that a specified function, which depends on z, is in the range of an operator depending on us if and only if z ∈ D. There are numerous specific examples of qualitative methods, such as the MUltipe SIgnal Classification (MUSIC) method [32, 42, 66, 69,7, 58] and the factorization method [69]. The first to be developed, however, is known as the linear sampling method [35]. It is also the most thoroughly researched and hence there are linear sampling-type methods to find obstacles using time harmonic and time dependent data in the context of acoustic, electromagnetic, and elastic media (among other physical settings). We describe the linear sampling method here as a specific introduction to qualitative methods. To explain the LSM in more detail, consider far field scattering from a time harmonic incident field transmitted towards an unknown object D ⊂ Rd from directions yˆ ∈ Sd−1, the surface of a ball in Rd. Scattered field data is then collected onx ˆ ∈ Sd−1. The notation ˆ· on a dependent variable indicates a normalized vector. This is an unfortunate conflict of notation with ˆ· for time harmonic fields, though there should be no confusion between dependent variable fields and independent variable vectors. Here, far field scattering refers to the fact that transmitters and receivers are placed far (in terms of number of wavelengths) from D, which is a common assumption in the time-harmonic case. Under this assumption, it is neccessary to analyze the

12 behavior of the ﬁeld far from the obstacle; in particular, it can be shown that

eik|x| uˆs(x) = uˆ (ˆx) + O(|x|−(d+1)/2), as |x| → ∞ |x|(d−1)/2 ∞

d−1 where the so-called far ﬁeld pattern u∞ is a function of observation directionsx ˆ ∈ S . ˆ An important example of a far ﬁeld pattern is that of the fundamental solution Φk:

ˆ −ikx·yˆ Φ∞(x;y ˆ) = γde √ iπ/4 −1 where γd = e / 8πk for d = 2 and γd = (4π) for d = 3. We will use the far field formulation and the far field pattern of the fundamental solution in particular in Chapter2. Also important when discussing far field problems is the Herglotz wave function, defined for functions g ∈ L2(Sd−1) by Z ikx·dˆ ˆ ˆ d vg(x) := e g(d) ds(d) x ∈ R . Sd−1 This is a linear combination of incident fields emitted from directions on Sd−1 and will prove helpful in the analysis of some inverse scattering problems below. In the case of far field scattering, we takeu î(z;y ˆ) = eikz·yˆ, z ∈ Rd. The linear sampling method proceeds by exploiting properties of solutions to

2 d−1 the far ﬁeld equation, gz ∈ L (S ),

ˆ ˆ (F gz)(ˆx) = Φ∞(ˆx, z), (1.17) where the far field operator Fˆ is defined by Z ˆ (F gz)(ˆx) := uˆ∞(ˆx;y ˆ)gz(ˆy) ds(ˆy) Sd−1 and where we take z ∈ Z for a set of points Z ⊂ Rd which contains D (or, in practice, which we think contains D). A key step in the justification of the linear sampling algorithm is to factor the far field operator so that Fˆ = GˆHˆ where the operators Gˆ and Hˆ have function-analytic properties which allow us to relate regularized solutions to (1.17) to D. More precisely, ˆ 2 2 2 ˆ we define H : L (S ) → L (D) by Hg := vg D, where vg is the Herglotz wave function

13 ˆ 1 2 introduced above. To deﬁne G, introduce the functionw ˆ ∈ Hloc(R ) to be the unique solution to the scattering problem

∆w ˆ + k2nwˆ = −k2mϕˆ ∂wˆ lim r−(d−1)/2 − ikwˆ = 0, r = |x| r→∞ ∂r forϕ ˆ ∈ L2(D). Then the operator Gˆ : {ϕˆ ∈ L2(D) : ∆ϕ ˆ + k2ϕˆ = 0} → L2(Sd−1) is ˆ i s ˆ deﬁned by Gv :=w ˆ∞. Note that ifϕ ˆ =u ˆ , thenw ˆ = u and so G maps tou ˆ∞. Indeed, it is this fact, along with the form of Fˆ that leads to the factorization Fˆ = GˆHˆ . A key point in the justiﬁcation of the LSM is that

ˆ ˆ Φ∞(·, z) ∈ R(G) if and only if z ∈ D.

Indeed, from this statement we see the relationship between D and Fˆ which is vital to the LSM. While such a range test does not hold for Fˆ, we can construct approximate solutions to the far ﬁeld equation which serve as useful indicators to the location of D. Indeed, by construction, Fˆ g is the far ﬁeld corresponding to a linear combination

d−1 of incident fields emitted from S weighted by g. As such, for z ∈ D, gz satisfies the homogeneous far field equation if and only if

∆ˆv + k2nvˆ = 0 in D

2 ∆vg + k vg = 0 in D ˆ vˆ − vg = Φk(·, z) on ∂D ∂vˆ ∂v ∂Φˆ − g = k (·, z) on ∂D, ∂ν ∂ν ∂ν wherev ˆ =w ˆ +ϕ ˆ using the notation from above. This is only true, in general, for very specific choices of D, n, and k. However, theoretical conditions on D, n, and k exist under which we can find approximate solutions to the far field equation which allow us

14 to image D. Loosely speaking, these conditions follow from understanding the interior transmission problem, which consists of ﬁndingv, ˆ wˆ, and nonzero k ∈ C which satisfy

∆w ˆ + k2nwˆ = 0 in D

∆ˆv + k2vˆ = 0 in D (1.18)

∂wˆ ∂wˆ wˆ =v, ˆ ∂ν = ∂ν on ∂D.

We call k a transmission eigenvalue if there exists a nontrivial solution to (1.18). If n and D are such that k is not a transmission eigenvalue, then one of the two following items holds [23]: α 2 d−1 • If z ∈ D then there exists a sequence gz ∈ L (S ) such that ˆ α ˆ 2 ˆ α 2 lim kF gz − Φ∞(·, z)k 2 d−1 = 0 and lim kHgz kL2(D) < ∞, α→0 L (S ) α→0 or

α 2 d−1 ˆ α ˆ 2 • if z∈ / D then for all g ∈ L ( ) such that limα→0 kF g −Φ∞(·, z)k 2 d−1 < , z S z L (S ) ˆ α 2 limα→0 kHgz kL2(D) = ∞.

This suggests a method for finding the shape of D by using the process described in Table 1.1: after collecting the scattered far field data, proceed by finding a regularized α 2 d−1 ˆ α ˆ d−1 solution gz ∈ L (S ) to (F gz )(ˆx) = Φ∞(ˆx, z) forx ˆ ∈ S . By the result above, ˆ ˆ −1 the indicator function IH,D,ˆ LSM(z) = kHgzkL2(D) will be large when z ∈ D and small otherwise. ˆ Note immediately that this method is extremely problematic in that IH,D,ˆ LSM(z) is a function of D which is unknown (and indeed, an operator H which cannot be com- ˆ α −1 puted from the data we have). In practice, the indicator function ILSM = kg k 2 d−1 z L (S provides satisfactory results. Another serious drawback of this method is that the

α above result does not indicate how to construct gz . Typically, Tikhonov regularization is used and the far ﬁeld equation is replaced by the regularized equation

ˆ∗ ˆ α ˆ∗ ˆ (αI + F F )gz = F Φ∞(·, z) for some sufficiently-small α. While this method seems to work in practice, it is not clear that the regularized solution satisfies the same blow-up properties as the theoretical solution to the far field equation. A more recently-developed technique, the

15 generalized linear sampling method (GLSM) combines aspects of both the linear sampling method and the factorization method and provides a theoretical and numerical technique for overcoming many of these downsides of the linear sampling method. Moreover, while the theoretical assumptions required for the application of the GLSM have partially limited its use, it allows for complete theoretical justification of a wide variety of scattering problems. The GLSM uses a second factorization of the far field operator as well as a more carefully constructed regularized solution, an indicator func- ˆ tion similar to ILSM is fully justified method for locating D in the GLSM framework. More details are available in [10, 11, 23]. Note that all of this can also be done for near field problems where the far field equation (1.17) is replaced by the near field equation

ˆ ˆ (Ngz)(x) = Φk(x, z), z ∈ Z, (1.19) and where the near ﬁeld operator Nˆ is deﬁned by Z ˆ s (Ngz)(x) := uˆ (x; y)gz(y) ds(y). Γm

i ˆ In near field scattering, we takeu ˆ (x, y) = Φk(x, y) to be a time harmonic pulse and let x ∈ Γi and Γm simply be curves in the exterior of D. See, for example, [54], for details in the acoustic scattering case. As discussed above, indeed, in order for the linear sampling method to be theoretically justified, n and k must be so that (1.18) is uniquely solvable. We will discuss this problem in depth for the weakly-scattering case in Chapter4. Note now that, until this year, a lack of understanding of the behavior of complex transmission eigenvalues has prevented full justification of the linear sampling method for penetrable media using time dependent data. However, [94] gives an optimal description of the growth transmission eigenvalues in the complex plane, leading to full justification of the technique under appropriate assumptions on n and D. Although this work is recent, we will discuss it more in the context of our own results in Chapter4.

16 1.3 Data Reduction and Primary Contributions of this Thesis The above remarks have hardly considered the real-world usage of qualitative methods for inverse scattering problems. Technological advances have increased the reliability and affordability of sensors for detecting scattered field data. Nonetheless, qualitative methods have lagged behind in their requirement of large amounts of scattered field data, often requiring transmitters and receivers to completely surround an object of interest. On the other hand, reconstruction methods based on nonlinear- optimization schemes are often successful with less scattered field data, but require significant amounts of computing power, time, and a priori data in order to return satisfactory results. The main contributions of this thesis are related to addressing these problems. The linear sampling method discussed above was developed under the assumption of multistatic scattered field data - that is, every location from which an incident field is transmitted is also a location at which scattered field data is collected. Most examples in the literature have made a further restriction on the available data: multistatic data is available on a full aperture curve, completely surrounding the object of interest. This assumption requires access to an area surrounding the entire object of interest, which is infeasible for large objects or ones in difficult-to-reach places. It also increases the cost of imaging, as large devices consisting of transmitters and receivers must be constructed. The quasi-backscattering approach introduced in Chapters2 and3 of this thesis have been developed by the author and his colleagues in order to combat these problems. As will be described in detail in the relevant chapters, the quasi-backscattering approach makes use of a small device composed of transmitters and receivers which can be moved around the object. In this way, only a small region surrounding the object needs to be accessed at once. Note that, this improvement in experimental geometry requires the theoretical assumption of weakly-scattering objects. While this is a strong assumption, experimental evidence provided in Chapters2 and3 suggest the weak-scattering assumption is not completely in affect.

17 Another method for reducing spatial data collection requirements has been the use of time domain or multi-frequency data. For example, the time domain linear sampling method developed in [31, 52, 53] experimentally requires fewer spatial data collection points than the single-frequency linear sampling method in order to success- fully reconstruct the location and shape of an object. The multi-frequency methods developed in [54] is strongly related to the time domain linear sampling method (without the requirement of causal data) and can be expected to have similar reductions in spatial data collection requirements. Inspired by these examples, we considered the time domain quasi-backscattering problem in Chapter3. Indeed, as will be discussed in detail there, acceptable reconstructions are achievable with orders-of-magnitude fewer spatial data points than single frequency methods. We also contributed results on the theoretical justification of the time domain linear sampling method for penetrable media under the Born approximation; as shown in Chapter3, the time domain linear sampling method for scattering in the Born regime produces accurate reconstructions with very few transmitters and receivers. However, its justification requires results for the Born transmission eigenvalue problem. As discussed above, in the case of non-weakly scattering data, these theoretical questions were not satisfactorily an- swered until very recently [94] where only non-absorbing media considered. To this end, Chapter4 is only a first step in justifying the Born time domain linear sampling problem for penetrable, and possibly absorbing, media. Finally, in Chapter5, we address the problem of time-consuming numerical simulations leading to slow object reconstructions from non-linear optimization schemes. In particular, the numerical scheme described in that chapter simulates acoustic wave scattering from penetrable media in an unbounded domain with computational complexity O(MN log N), where M is the number of desired time steps and N the number of points in a triangulation of the scattering object. Although this method is not highly accurate, it is one of the only fast methods for simulating time domain scattering through penetrable media on an unstructured mesh which exactly models the unbounded spatial domain inherent in scattering problems.

18 Chapter 2

QUASI-BACKSCATTERING IN THE FREQUENCY DOMAIN

2.1 Introduction In this chapter we propose a data collection geometry in which to frame the inverse scattering problem of locating unknown obstacles from far field measurements of time harmonic scattering data. The measurement geometry, which we call a quasi- backscattering set-up, requires less data than traditional multistatic configurations. We demonstrate that the data collected can be used to locate inhomogeneities in problems in which the Born approximation applies. In particular, we are able to image a two-dimensional projection of the location of a small obstacle by checking if a test function, corresponding to a point in R2, belongs to the range of a measurable operator. Combining several projections then allows us to identify the location of the small inclusions in R3. We also show how this algorithm can be extended to the case of extended spherical inclusions. The quasi-backscattering inversion scheme we describe in this chapter makes use of a particular experimental set-up; one device acts as a transmitter and a line of receivers extends in one-dimension a small distance from the transmitter. Fig- ure 2.1 demonstrates the difference between a usual multistatic set-up and the quasi- backscattering geometry. As the figure demonstrates, the quasi-backscattering geometry requires significantly less data than the multi-static geometry, which can be beneficial in practical applications. In Section 2.2, the direct scattering problem is formulated and the quasi-backscattering data setting is explicated. In Section 2.3, we introduce and analyze the inversion proce- dure capable of identifying two-dimensional projections of small objects’ locations. We

19 Figure 2.1: A Comparison of multi-static data (left) and quasi-backscattering data (right). Red circles correspond to device locations. The thick blue line in the right ﬁgure denotes where the quasi-backscattering set-up is moved and where each transmitting device is located. then extend the algorithm to the case of extended spherical inclusions. In Section 2.4, extensive numerical experimentations are presented in order to show the performance of this new algorithm. We end by explaining how one can obtain three-dimensional locations from two-dimensional projections.

2.2 Direct Scattering Problem We begin by discussing the mathematical formulation for the problem of acoustic incident plane waves scattering against inhomogeneous media in three-dimensions which was discussed in Chapter1. This problem has been studied extensively and more information about the related direct and inverse problems can be found in, e.g., [21, 36, 69]. Assume a plane wave incident field with a fixed wave number k is generated far from the area of an inhomogeneity. Such an incident field is described byu î(x, dˆ) =

20 ˆ eikd·x for x ∈ R3 and dˆ∈ S2. Recall from Chapter1 that the total field û(x) satisfies

2 3 ∆û + k n(x)û = 0 in R , (2.1a) uˆ(x) =u î(x, dˆ) +u ˆs(x), (2.1b) ∂uˆs lim r − ikuˆs = 0, (2.1c) r→∞ ∂r whereu ˆs(x) is the scattered field, r = |x| is the Euclidean magnitude of x, n(x) is the bounded refractive index of the inhomogeneous medium, and (2.1c) is the Sommerfeld radiation condition which holds uniformly with respect tox ˆ = x/|x|. For wave numbers such that Im k ≥ 0 and compactly supported refractive indices in L∞, it is known that

1 3 (2.1a)–(2.1c) has a unique solution in Hloc(R ). Recall that the contrast function as m(x) = 1 − n(x) is such that m(x) is non- zero only on a compact set D ⊂ R3 which contains the inhomogeneity. As discussed in Chapter1, the Lippmann-Schwinger equation for time-harmonic scattering from plane waves is, Z ikx·dˆ 2 ˆ uˆ(x) = e − k m(y)Φk(x, y)û(y) dy. (2.2) D This gives an exact expression for the unique solution to (2.1a)-(2.1c) where, as in Chapter1, 1 eik|x−y| Φˆ (x, y) = , x 6= y k 4π |x − y| is the fundamental solution to the Helmholtz equation in R3. Assuming [69] Z 2 ˆ k max |m(y)Φk(x, y)| 1, y∈D D which ensures a Neumann series solution to (2.2) converges, the first term of this series gives the Born approximation Z B ikx·dˆ 2 ˆ ikdˆ·y uˆ (x) = e − k m(y)Φk(x, y)e dy. (2.3) D Formally, this is Fourier transform of the time domain Born approximation introduced in Chapter1,(1.16). The inverse problem in which we are interested is to find information about D

s given data about the asymptotic behavior ofu ˆB(x), the Born approximation to the

21 scattered field. As in the full-scattering problem, we are able to explicitly characterize the asymptotic behavior of the scattered field because of the Sommerfeld radiation condition, (2.1c). Specifically,

eik|x| 1 uˆs (x) = uˆB (ˆx, dˆ) + O , |x| → ∞ B |x| ∞ |x|2

∞ ˆ whereu ˆB (ˆx, d) is the Born approximation to what is known as the far ﬁeld pattern. Using (2.3), Z s 2 ˆ ikdˆ·y uˆB(x) = −k m(y)Φk(x, y)e dy D and we conclude that

2 Z B ˆ k ik(dˆ−xˆ)·y 2 uˆ∞(ˆx, d) = − e m(y) dy, xˆ ∈ S . (2.4) 4π D

2.2.1 Forward Problem for Quasi-Backscattering Data The above derivations have not ﬁxed the measurement geometry. We now re- strictx ˆ and dˆ to the quasi-backscattering experimental set-up. In what follows, let xˆ = −dˆ+ ηeˆ where η ∈ [−δ, δ] for a small constant δ ande ˆ ∈ S2 is a ﬁxed unit vector which is orthogonal to dˆ. The traditional backscattering set-up corresponds with δ = 0. Using the orthogonality of dˆ withe ˆ and the fact that both are unit vectors, a Taylor expansion about η = 0 yields ˆ ˆ −d + ηeˆ −d + ηeˆ ˆ 2 xˆ = = p = −d + ηeˆ + O(η ). ˆ 1 + η2 −d + ηeˆ

ˆ As such, we choosex ˆ in this way as an approximation tox ˆ = −d+ηeˆ up to O(η2). k−dˆ+ηeˆk For this reason we continue to use the notationx ˆ, although it is no longer normalized. Substituting this choice ofx ˆ into (2.4) gives

2 Z B ˆ ˆ k 2ikdˆ·y −ikηeˆ·y ˆ 1 uˆ∞(−d + ηe,ˆ d) = − e e m(y) dy, d ∈ S (ê) (2.5) 4π D where S1(ê) := {dˆ ∈ S2; dˆ · eˆ = 0}. Following the typical approach of sampling methods in inverse scattering problems, we introduce the quasi-backscattering far field

22 operator, Fˆ : L2([−δ, δ]) → L2(S1(ˆe)) which we will use extensively in solving the inverse problem. In particular, Fˆ is deﬁned as

Z δ ˆ ˆ B ˆ ˆ ˆ 1 (F g)(d) = uˆ∞(−d + ηe,ˆ d)g(η) dη, d ∈ S (ˆe). (2.6) −δ

2.3 Quasi-Backscattering Inverse Problem We now turn our attention to the inverse problem of reconstructing the location of inhomogeneities from the quasi-backscattering far field data. We first consider the case of obstacles which are small compared to the wavelength of the incident wave and which are sufficiently far from one another. In Section 2.3.4 , we use the analysis for this case as the basis for finding the centers of extended spherical obstacles. The key result of this section is Theorem1 which will allow us to locate obstacles by testing if a specific function is in the range of the quasi-backscattering far field operator.

2.3.1 Inverse Problem for Small Obstacles

3 Assume there are M obstacles with supports described by Dj ⊂ R , j = 1,...,M, embedded in a homogeneous background. Let the contrast be deﬁned by PM the weighted sum of characteristic functions m(x) = j=1 mj1Dj where mj are constants.

3 If Dj = zj + RjΩj are small obstacles centered at a point zj ∈ R with size and shape described by Rj and Ωj respectively, then using (2.5) we obtain that up to 4 O(max(Rj) ) error terms,

M ˆ B ˆ ˆ X 2ikd·zj −ikηe·zj ˆ 1 uˆ∞(−d + ηe,ˆ d) ' τje e , d ∈ S (ˆe), η ∈ [−δ, δ]. (2.7) j=1

k2 Here, τj = − 4π mj|Ωj|, where |Ωj| indicates the volume of Ωj, are constants related to the strength of each scatterer. Combining (2.6) and (2.7) reduces the quasi- backscattering operator to

δ M δ Z ˆ Z ˆ ˆ B ˆ ˆ X 2ikd·zj −ikηe·zj (F g)(d) = uˆ∞(−d + ηe,ˆ d)g(η) dη = τje e g(η) dη. −δ j=1 −δ

23 To further simplify the far ﬁeld operator, we write each obstacle’s location in terms of its components parallel toe ˆ and perpendicular toe ˆ. For a ﬁxede ˆ, we write zj =

Π⊥(zj) + L(zj)ê, j = 1,...,M where Π⊥ maps onto the plane orthogonal toe ˆ and where L isolates the component of a vector which is parallel toe ˆ. For example, if eˆ = (0, 0, 1) and z1 = (1, 2, 3), we would have Π⊥(z1) = (1, 2, 0) and L(z1) = 3. Note that for the sake of notational conciseness we will sometimes treat Π⊥(z) as a vector in R2. Decomposing the locations of obstacles in this way, we can write the far field operator as M δ ˆ Z ˆ ˆ X 2ikd·Π⊥(zj ) −ikL(zj)η (F g)(d) = τje e g(η) dη. (2.8) j=1 −δ Since Fˆ can be computed from the measurable far field pattern data, we use it to solve the inverse scattering problem. Indeed, Theorem1 gives conditions under which we can relate the range of Fˆ to the location of a small obstacle. Such a characterization is typical for sampling-type methods such as the linear sampling or factorization schemes, as well as the MUSIC algorithm. Before stating this theorem, we prove two short lemmas which are required for its proof.

2 3 Lemma 1. Assume eˆ ∈ S is ﬁxed and let zj ∈ R , j = 1,...,M be distinct points whose components in the direction of eˆ diﬀer (i.e., L(zi) 6= L(zj), i 6= j). Then A =

{η 7→ e−ikL(zj )η, j = 1,...,M} is a linearly independent sequences of functions for η ∈ [−δ, δ].

Proof. We would like to show that the Wronskian matrix of A, denoted by W , is non-singular. A short calculation shows that det(W ) = c(η)det(V ) where c(η) = PM j−1 exp −ikη j=1 L(zj) is a function which never vanishes and V(i,j) = ωi for ω =

−ikL(zi). Since V is a Vandermonde matrix, it has a non-zero determinant so long as

ωi 6= ωj for each i 6= j, which is true by the assumption on L(zi), i = 1,...,M.

2 3 Lemma 2. Assume eˆ ∈ S is ﬁxed, let zj ∈ R , j = 1,...,M be distinct points, and 3 let z∗ ∈ R be any point perpendicular to eˆ and distinct from each Π⊥(zj). Then

24 ˆ 2ikdˆ·z B = {d 7→ e , z = z∗, Π⊥(z1),..., Π⊥(zM )} is a linearly independent sequences of functions for dˆ∈ S1(ˆe).

The proof of this lemma follows the idea of Theorem 4.1 in [69]. This theorem of Kirsch and Grinberg implies that the above can also be proven for a ﬁnite number ˆ 1 of dj, ηj ∈ S with a similar but more technical argument.

Proof. To show that B is linearly independent, assume

M ˆ ˆ 2ikd·z∗ X 2ikd·Π⊥(zj ) ˆ 1 β0e + βje = 0 for d ∈ S (ˆe). j=1 The left-hand-side of the above equation is, up to a constant multiple, the far ﬁeld pattern of the function

M ˆ X ˆ x 7→ β0Φk(x, z∗) + βjΦk(x, Π⊥(zj)) j=1 where i Φˆ (x, z) = H(1)(2k|x − z|), x 6= z (2.9) k 4 0 is the (radiating) fundamental solution of the Helmholtz equation in R2 with wave (1) number 2k and H0 is a Hankel function. As such, since the far ﬁeld pattern vanishes, Rellich’s lemma and unique continuation show that

M ˆ X ˆ β0Φk(x, z∗) + βjΦk(x, Π⊥(zj)) = 0 for x∈ / {z∗, Π⊥(z1),..., Π⊥(zM )}. j=1

Taking the limit as x approaches each of z∗ and Π⊥(zj), j = 1,...,M shows immediately that B is a linearly independent sequence of functions for each dˆ∈ S1(ˆe).

With these lemmas in hand, we are ready to prove the key theorem for small obstacles.

2 ˆ 1 3 Theorem 1. Assume eˆ ∈ S is ﬁxed and d ∈ S (ˆe). Let zj ∈ R for j = 1,...,M 3 and let z ∈ R be orthogonal to eˆ. If the components of each zj parallel to eˆ are ˆ ˆ 2ikdˆ·z not equal (i.e., L(zi) 6= L(zj), i 6= j), then φz(d) = e ∈ R(F ) if and only if z ∈ {Π⊥(zj), j = 1,...,M}.

25 Proof. Let z∈ / {Π⊥(zj), j = 1,...,M} be orthogonal toe ˆ. Assume by contradiction that there exists some g(η) ∈ L2([−δ, δ]) such that (Fˆ g)(dˆ) = e2ikdˆ·z. From the deﬁnition of Fˆ, this would imply

M ˆ ˆ 2ikd·z X 2ikd·Π⊥(zj ) e = cje , j=1

R δ −ikL(zj )η where cj = τj −δ e g(η) dη are constants. However, this is a contradiction with ˆ ˆ 2ikd·Π⊥(ζ) the linear independence of {d 7→ e , ζ = z, Π⊥(z1),..., Π⊥(zM )}, which shows ˆ ˆ that if φz ∈ R(F ) then z ∈ {Π⊥(zj), j = 1,...,M}.

To prove the second half of the theorem, assume L(zi) 6= L(zj), i 6= j and ˆ ˆ∗ ⊥ ˆ z ∈ {Π⊥(zj), j = 1,...,M}. We will show that φz ∈ N (F ) = R(F ) which gives the result since Fˆ is a ﬁnite rank operator with closed range. A short calculation gives that M Z ˆ ˆ∗ X ikL(zj )η ˆ −2ikd·Π⊥(zj ) ˆ (F h)(η) = τje h(d)e ds(d). 1 j=1 S (ˆe) If h ∈ N (Fˆ∗), then

M Z ˆ X ikL(zj )η ˆ −2ikd·Π⊥(zj ) ˆ τje h(d)e ds(d) = 0. 1 j=1 S (ˆe)

The linear independence of {eikL(zj )η, j = 1,...,M} proven in Lemma1 gives that for each j = 1,...,M,

Z ˆ ˆ −2ikd·Π⊥(zj ) ˆ ˆ ˆ ˆ 0 = h(d)e ds(d) = h(d), φΠ⊥(zj )(d) 1 L2( 1(ê)) S (ê) S where (·, ·) indicates the inner-product on L2( 1(ê)). As such, φˆ ∈ N (Fˆ∗)⊥ for L2(S1(ê)) S z each z ∈ {Π⊥(zj), j = 1,...,M}, which gives the result.

The proof of Theorem1 in fact implies a slightly stronger result.

Corollary 1. With no restrictions on L(zj) and the same hypotheses on eˆ as in The- ˆ ˆ orem1, if φz ∈ R(F ) then Π⊥(z) ∈ {Π⊥(zj), j = 1,...,M}.

26 Another corollary to Theorem1 is that, for the appropriate restrictions on ˆ 2 1 Π⊥(zj), P φz = 0 if and only if z ∈ {Π⊥(zj), j = 1,...,M} where P : L (S (ˆe)) → R(Fˆ)⊥ is the orthogonal projection onto the orthogonal complement of the range of Fˆ. −1 ˆ ˆ This suggests that the function I(z) = P φz for each z perpendicular toe ˆ within a region of interest will be large when z is near Π⊥(zj), j = 1,...,M and small otherwise. This is exactly the MUSIC-type algorithm which we will use to locate the centers of small objects. ˆ To construct the imaging function I(z), let (uk, σk, vk), k = 1, 2,... be the ˆ 2 1 singular system for F where the left singular functions are uk ∈ L (S (ˆe)) and the 2 ˆ right singular functions are vk ∈ L ([−δ, δ]). Since R(F ) is spanned by the left singular functions uk which correspond to singular values σk = 0, we can write

∞ 2!−1 ˆ X ˆ I(z) = φz, uk , (2.10) L2(S1(ˆe)) k=r+1 where r is the number of non-zero singular values. Numerical results showing that Iˆ(z) is large near obstacles are given in Section 2.4.

2.3.2 A Second Range Test for Three-Dimensional Reconstructions

Assume that the range test described above has been performed so that {Π⊥(zj), j = ˆ ˆ ˆ 1,...,M} are known. From Theorem1, φΠ⊥(zk)(d) ∈ R(F ) for a given k = 1,...,M. 2 As such, there is a gΠ⊥(zk) ∈ L ([−δ, δ]) such that

M ˆ ˆ ˆ ˆ X 2ikd·Π⊥(zj ) 2ikd·Π⊥(zk) (F gΠ⊥(zk))(d) = cje = e j=1

δ R −ikL(zk)η where, as before, cj = τj −δ gΠ⊥(zk)(η)e dη. By linear independence, cj = τjδjk, where δjk is the Kronecker delta function. This suggests a second indicator function which can be used to ﬁnd L(zj) when Π⊥(zj) are already known. Formally,

Z δ −1 ˆ −ikL(z)η JΠ⊥(zk)(z) = gΠ⊥(zk)(η)e dη (2.11) −δ

27 is arbitrarily large when z = L(zj), j 6= k. This argument is formal and we have no ˆ guarantee that JΠ⊥ (zk)(z) is small away from z = zj, j 6= k. Nevertheless, in the ˆ numerical examples in Section 2.4.2 below, JΠ⊥(zk) indicates the location of L(zj), as expected when Π⊥(zj) are known accurately and gz(η) is calculated using Tikhonov regularization and the Morozov discrepancy principle. As the numerical simulations will demonstrate, however, calculating L(zj) in this manner is not robust to noise.

2.3.3 Inverse Problem for Coplanar Small Obstacles Due to the hypotheses on Theorem1, the algorithm outlined above does not necessarily locate an object in the case that L(zi) = L(zj) for some i 6= j ∈ 1,...,M. This problem can be easily alleviated: in all proofs we have assumed a ﬁxede ˆ ∈ S2.

Since L(zj) is a function ofe ˆ, we can perform multiple quasi-backscattering experiments with differente ˆ directions to solve the problem. Indeed, we recommend this for purely geometric reasons as well. Since the quasi-backscattering technique gives only two- dimensional projections of the locations of scatterers, two obstacles which lie on top of each other with respect toe ˆ (i.e., Π⊥(zi) = Π⊥(zj) but L(zi) 6= L(zj)) will appear as the same obstacle in the reconstruction. Multiple experiments corresponding to differente ˆ directions helps to fix this problem as well. In Section 2.4.3, we outline a technique for using data from multiple experiments with differente ˆ directions to reconstruct obstacles in three-dimensions. Before continuing, we note that the algorithm outlined above does not necessarily identify obstacles if L(zi) = L(zj) for all i 6= j. In particular, under these conditions, we show below that there is no obvious reason which suggests that Iˆ(z) will be arbitrarily large at z = Π⊥(zj), j = 1,...,M. Indeed, the numerical simulations in Section 2.4 indicate that the reconstruction of co-planar obstacles is sensitive to noise.

Assume that L(zi) = L(zj) for each i, j = 1,...,M. Since we can shift the origin with a change of variables, we set each L(zi) = 0 without loss of generality. In

28 this case, the far ﬁeld operator becomes

M δ ˆ Z ˆ ˆ X 2ikd·Π⊥(zj ) ˆ 1 (F g)(d) = τje g(η) dη, d ∈ S (ˆe), j=1 −δ and Fubini’s Theorem gives that for η ∈ [−δ, δ],

M M ! Z ˆ ˆ∗ X −2ikd·Π⊥(zj ) ˆ ˆ ˆ X ˆ (F h)(η) = τj e h(d) ds(d) = h(d), τjφΠ⊥(zj ) . 1 j=1 S (ê) j=1 L2(S1(ê)) ˆ ˆ ˆ ˆ Let h(d) = uk(d) for a fixed k where uk(d) is a left singular function of F corresponding ˆ∗ to a singular value σk = 0. Since uk ∈ N (F ),

M ! ˆ X ˆ ˆ∗ uk(d), τjφΠ⊥(zj ) = (F uk)(η) = 0. j=1 L2(S1(ˆe)) However, we cannot conclude from the above equation that

ˆ ˆ uk(d), φΠ⊥(zj ) = 0, j = 1,...,M. L2(S1(ˆe))

2.3.4 Inverse Problem for Extended Obstacles We now adapt the arguments given in the previous section to the problem of ﬁnding extended obstacles. We show that, for small δ, the arguments given in Theorem1 apply directly to locating the center of extended spherical obstacles. While the spherical nature of the extended obstacles does not seem to be required, it is not clear that we can uncover more information than the location of the center of these obstacles.

In this section, assume there are M obstacles Dj again of the form Dj = zj +

RjΩj, where zj are the obstacles’ center, Ωj their shape, and Rj their size. Now, however, assume each Ωj = B(0; 1) is a ball centered at zero of radius one and that Rj is of similar size as the wavelength or larger. Assume, as before, that the contrast is

29 PM deﬁned by m = j=1 mj1Dj where mj are constants. We will begin our discussion by B ˆ calculatingu ˆ∞(ˆx, d) under these assumptions. From (2.4) we have 2 M Z k X ˆ uˆB (ˆx, dˆ) = − m eik(d−xˆ)·y dy ∞ 4π j j=1 Dj 2 M Z k X ˆ ˆ = − m eik(d−xˆ)·zj eik(d−xˆ)·y dy. 4π j j=1 B(0;Rj ) To simplify this expression into a more useful one, we state the following lemma.

Lemma 3. For a constant R > 0 and any two vectors x, y ∈ R3, Z ix·y 4π e dy = 3 (sin(R|x|) − R|x| cos(R|x|)) . B(0;R) |x| Proof. Under the change of coordinates y 7→ ryˆ where r = |y|, Z Z R Z eix·y dy = r2 eirx·yˆ ds(ˆy) dr. (2.12) B(0;R) 0 S2 R irx·yˆ It is known that 2 e ds(ˆy) = 4πj0(rx) where j0 is the spherical Bessel function of S sin(x) order zero [77]. Since j0(x) = x , an integration-by-parts gives the result. ˆ We are interested in the above result for x = k(d − xˆ) and R = Rj. With this choice of parameters, M ˆ ˆ ˆ ! ˆ sin(kRj|d − xˆ|) − kRj|d − xˆ| cos(kRj|d − xˆ|) B ˆ 2 X ik(d−xˆ)·zj uˆ (ˆx, d) = −k mje . ∞ ˆ 3 j=1 (k|d − xˆ|) Returning to the quasi-backscattering approach and lettingx ˆ = −dˆ +eη ˆ , a Taylor expansion about η = 0 gives that for j = 1,...,M, sin(kR |dˆ− xˆ|) − kR |dˆ− xˆ| cos(kR |dˆ− xˆ|) sin(2kR ) − 2kR cos(2kR ) j j j = j j j + O(η2). (k|dˆ− xˆ|)3 (2k)3

ˆ ˆ R δ B ˆ If we again define the quasi-backscattering far field operator as (F g)(d) = −δ u∞(d, η)g(η) dη, we find M δ ˆ Z ˆ ˆ X L 2ikd·zj −ikηeˆ·zj 2 (F g)(d) = τj e e g(η) dη + o(δ ) j=1 −δ M δ ˆ Z X L 2ikd·Π⊥(zj ) −ikL(zj)η 2 = τj e e g(η) dη + o(δ ) j=1 −δ

30 L where τj = −mj(sin(2kRj) − 2kRj cos(2kRj))/8k. Here, the asymptotic analysis follows from an application of Cauchy-Schwarz. Up to constants and o(δ2), the quasi- backscattering operator for extended spheres is identical to (2.8), the quasi-backscattering operator for small obstacles. As such, using the same technique described in Section 2.3.1, we can ﬁnd two-dimensional projections of the centers of extended spherical

L 1 2 3 5 obstacles. Note, incidentally, that τj = − 3 k mjRj + O(Rj ), which matches the expression for τj used in the case of small spheres.

2.4 Numerical Experiments In this section, we give numerical results demonstrating the effectiveness of the above technique. In all experiments, we approximate obstacles by spheres with a small radius. Specifically, the radius for each obstacle is 1/500 units. We will use simulated forward data which is corrupted by random noise. Using the formula given in (2.4) ˆ we simulateu ˆ∞(ˆx, d) using numerical integration. Numerically integrating (2.6) gives ˆ ˆ a discrete representation of the far field matrix, Fij, which is corrupted by Fij(1 + γξ) where ξ is a uniform random variable in [−1, 1] and γ is a constant related to the level of noise. To calculate the indicator function, we compute the singular value decomposition of Fˆ = USV∗ and use U to calculate a discrete version of (2.10). The approximate imaging function is regularized by computing with all but the first ten singular vectors (i.e., r = 9 in (2.10)). In all examples, we take k = 15 to be the wave number. Other parameters are given for each experiment. The experimental parameters discussed above merit a few comments. The first is related to our use of ten singular vectors in reconstructions. Typically when using a MUSIC-type algorithm, the number of singular vectors is related to the number of unknown obstacles, which is estimated by the numerical rank of the far field operator. However, in our numerical experiments we have found such a technique to be sensitive to added noise. As such, we took the number of singular vectors as an upper bound of the number of obstacles. The results do not change noticeably when using the same number of singular vectors as there are obstacles. The second comment is related to

31 the relatively-high wave number used in these experiments. In the case of extended obstacles, low wave numbers are used to ensure that the Born approximation of the far field is valid. However, because we assume our objects are very small (a radius of 1/500 units), we are justified in using a higher wave number. We present three types of numerical inversions. In Section 2.4.1, we show two- dimensional projections of small obstacles. In Section 2.4.2, we generate the third unknown coordinate assuming the first two are known. Finally, we use multiplee ˆ- directions to generate full three-dimensional reconstructions for small obstacles in Sec- tion 2.4.3.

2.4.1 Two-Dimensional Projections of Small Obstacles We give several numerical examples in this section which help demonstrate both the strengths and weaknesses of the quasi-backscattering technique. In all reconstructions, darker colors correspond to higher values of the imaging function which correspond with the predicted locations of the obstacles. A small red circle in each picture corresponds to the true location of each obstacle. Note that the size of the dark areas near obstacle locations do not correspond to an estimate of obstacle size, but are merely an artifact of the way in which reconstructions are displayed. In all reconstructions, we use 802 sampling points uniformly chosen in the unit-square. In all experiments we use 95 incident directions and for each incident direction we use 95 locations for xˆ between −δ and δ. We call these points between −δ and δ the observation points. This is a large number of both incident directions and observation points and, indeed, acceptable results are achievable with far fewer. However, we prefer to focus these experiments on the affect geometric and physical parameters have on reconstructions. The first example, given in Figures 2.2 and 2.3, shows the algorithm differenti- ating between multiple small obstacles, added one at a time. The obstacles are located at z1 = (−0.25, −0.25, −0.5), z2 = (0.25, 0.25, −0.25), z3 = (0.25, −0.25, 0.25), and z4 = (−0.25, 0.25, 0.5).

32 1 12 1 14 10 12 0.5 0.5 10 8 8 y 0 6 y 0 6 −0.5 4 −0.5 4 2 2 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x

1 14 1 15 12 0.5 0.5 10 10 8

y 0 y 0 6 −0.5 4 −0.5 5 2 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x

Figure 2.2: Symmetric addition of new objects, δ = π/50 (no noise).

For the next example, we show the aﬀect of δ on reconstructions. While the motivation for the quasi-backscattering set-up comes from a Taylor expansion about η = 0 (and hence small δ), the experiments in Figure 2.4 show that in the presence of noise, the reconstruction technique is not stable for too small of δ, in particular when many obstacles are present. All three ﬁgures have z1 = (0, −0.5, 0.25), z2 =

(0, 0.5, −0.75), z3 = (0.5, 0, −0.25), and z4 = (−0.5, 0, 0.75).

33 1 14 1 14 12 12 0.5 0.5 10 10 8 8

y 0 y 0 6 6 −0.5 4 −0.5 4 2 2 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x

1 12 1 6 10 0.5 0.5 5 8 4 y 0 6 y 0 3 −0.5 4 −0.5 2 2 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x

Figure 2.3: Symmetric addition of new objects, δ = π/50 (approximately 1% noise).

1 12 1 3.5 10 0.5 0.5 3 8 2.5

y 0 6 y 0 2 −0.5 4 −0.5 1.5 2 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x

1 3 0.5 2.5

y 0 2

−0.5 1.5

−1 −1 −0.5 0 0.5 1 x

Figure 2.4: Decreasing the quasi-parameter δ. Figures have δ = π (top-left), δ = π/50 (top-right), and δ = π/100 (bottom). Approximately 5% noise.

In the experiment in Figure 2.5 we show the resolution achievable by the quasi- backscattering technique. Often, inversion schemes based on the Born approximation or a Fourier transform are limited to a half-wavelength resolution. Though we do not

34 show this rigorously, the numerical example in Figure 2.5 suggests such a limitation for the quasi-backscattering technique. Indeed, we see that the method is unable to diﬀerentiate between obstacles once they are within half a wavelength of each other. In this case, the technique gives a large range of possible locations, containing the true centers of the obstacles. In this experiment, there is a constant 0.2 unit distance between the z-coordinate of the obstacles.

1 10 1 8 0.5 8 0.5 6 6

y 0 y 0 4 4 −0.5 −0.5 2 2 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x

1 1 14 8 12 0.5 0.5 10 6 8

y 0 y 0 4 6 −0.5 −0.5 4 2 2 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x

Figure 2.5: Two objects moving closer to each other, δ = π/100 (no noise). Thick bar at bottom corresponds to half of wavelength.

The final two experiments of this type show the need to take multiple experiments with differente ˆ directions when the underlying geometry of the obstacles is complicated. Figure 2.6 shows reconstructions from three differente ˆ directions of three small obstacles which would form an approximate “L”-shape if they were connected with straight lines. In particular, z1 = (−0.25, −0.25, −0.25), z2 =

(0.25, −0.24, −0.25), and z3 = (−0.25, −0.26, 0.25). Due to the geometry of the obstacles, takinge ˆ = (0, 0, 1) ore ˆ = (1, 0, 0) only gives reconstructions of two of the three obstacles. By takinge ˆ = (0, 1, 0), however, we are able to ﬁnd all three obstacles.

35 Finally, we apply the quasi-backscattering algorithm to the reconstruction of co-planar obstacles – that is, obstacles which violate the assumptions in Theorem1. As Figure 2.7 shows, in the absence of noise, reconstructions are acceptable. However, under the addition of noise, the reconstructions become less clean. Changinge ˆ so that

L(zi) 6= L(zj) results in more acceptable reconstructions. The ﬁgures are located at z1 = (0.75, 0.75, 0.25) and z2 = (−0.25, −0.25, 0.25).

1 20 1 30 25 0.5 15 0.5 20 z y 0 10 0 15 10 −0.5 5 −0.5 5

−1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x y

(a) XY -plane (b) YZ-plane

1 4 3.5 0.5 3

z 0 2.5 2 −0.5 1.5

−1 −1 −0.5 0 0.5 1 x

Figure 2.6: An L-Shaped geometry which requires 3 views to see all obstacles, δ = π/30 (approximately 1% noise).

36 1 4 1 1.8 3.5 0.5 0.5 1.6 3

y 0 2.5 y 0 1.4 2 −0.5 −0.5 1.2 1.5

−1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x

(a) XY plane (b) XY plane

1 30 1 25 25 0.5 0.5 20 20 15

z 0 15 z 0 10 10 −0.5 −0.5 5 5 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x

Figure 2.7: Results for co-planar obstacles, δ = π/60. Figures on left are noise-free and ﬁgures on right have approximately 1% noise.

2.4.2 Reconstruction of Third Coordinate We now show two reconstructions of the third coordinate of a small obstacle, assuming the other two coordinates are known. We use the indicator function given by (2.11) where gz(η) is calculated using Tikhonov regularization plus the Morozov discrepancy principle. In both reconstructions, we take δ = π/50 and 377 observation points and incident directions. Though this is an unrealistically-large number of observation points and incident directions, we will show that the indicator function is still sensitive to noise. In both reconstructions,e ˆ = (0, 0, 1) so that we are generating the z-coordinate in a typical Cartesian plane. For this reason, we explore another technique for three-dimensional reconstructions in Section 2.4.3 below.

In Figure 2.8, let z1 = (−0.24, −0.24, −0.75), z2 = (0.26, −0.24, 0), and z3 = (0.26, 0.26, 0.75). Adding no noise and assuming the two-dimensional projections of

37 zj, j = 1, 2, 3 are known exactly, the ﬁgure demonstrates we are able to construct L(zj) under ideal circumstances.

0.1 0.06

0.04 z z 0.05 J J 0.02

0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 z z

(a) Jˆζ (z), ζ = Π⊥(z1) (b) Jˆζ (z), ζ = Π⊥(z2) 0.03

0.02 z J 0.01

0 −1 −0.5 0 0.5 1 z

ˆ Figure 2.8: When JΠ⊥(zj ) is computed, peaks appear for i 6= j. Red circles show exact location of L(zj).

We consider a more realistic scenario in Figure 2.9. Let z1 = (−0.25, −0.25, −0.75), z2 = (0.25, −0.25, 0), and z3 = (0.25, 0.25, 0.75). However, we have added 0.1% noise and assume we guess Π⊥(z1) = (−0.24, −0.24), Π⊥(z2) = (0.26, −0.24), and

Π⊥(z3) = (0.26, 0.26). We see that even under small perturbations, the accuracy of the reconstructions is dramatically decreased.

38 0.1 0.2

z 0.05 z 0.1 J J

0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 z z

(a) Jˆζ (z), ζ = Π⊥(z1) (b) Jˆζ (z), ζ = Π⊥(z2) 0.01

z 0.005 J

0 −1 −0.5 0 0.5 1 z

Figure 2.9: Reconstructions lose accuracy in the presence of 0.1% noise. Red circles show exact location of L(zj).

2.4.3 Three-Dimensional Reconstructions The inversion schemes described above do a good job locating obstacles within the two-dimensional plane perpendicular to the selectede ˆ direction. Given data from multiple experiments with multiplee ˆ directions, we are better able to ﬁnd the full three- dimensional coordinates of an obstacle or set of obstacles. As discussed above, there are many scenarios in which reconstructing obstacles with multiplee ˆ is encouraged. In this section, we show that multiplee ˆ directions can be used to calculate three-dimensional reconstructions of obstacle locations. The creation of three-dimensional images from a selection of two-dimensional projections has been thoroughly studied in the image processing literature and we do not attempt to use state-of-the-art techniques here. Instead, we perform multiple quasi- backscattering experiments on the same obstacle set-up, interpolate the results from each experiment onto a ﬁxed sampling grid, and average the results. We regularize

39 our results for eache ˆ before computing the averaged result. In particular, we apply a total variation minimization algorithm (see [28, 29, 86]) which emphasizes changes in gradient and hence sharpens edges. We next locally normalize each two-dimensional projection over a 5 × 5 grid of sampling points to further sharpen edges. After these regularization steps are performed, we average on a sampling grid as described. We compute forward and inverse data for this section as we did in Section 2.3.1. Here, however, we varye ˆ. Speciﬁcally, we take 30 values ofe ˆ from a circle in the XY -plane. The results are given as three-dimensional contour plots of the imaging function. The contour which is plotted is α max Iˆ(z) where α is a value between 0 and 1. In Figure 2.10, we demonstrate the techniques described above to compute three-dimensional object reconstructions. In particular, we consider three small objects located at the points z1 = (−0.5, −0.5, −0, 5), z2 = (0.5, −0.5, −0.5), and z3 = (−0.5, −0.5, 0.5). Notice that these are in a geometry which forms an “L”-shape, as in Figure 2.7. As demonstrated above, when we use two-dimensional projection techniques, we require at least threee ˆ directions to locate all objects for such a geometry. By taking moree ˆ directions, however, we are able to give a full three-dimensional image of the geometry.

40 Figure 2.10: Three-dimensional noise-free reconstructions of point obstacles based on multiple experiments. We take 95 incident directions, 95 observation points, and use δ = π/60. The top figure is noise-free while the bottom figure has approximately 1% noise. In both figures, we display isovalues of 0.6 times the maximum value of the imaging function.

41 Chapter 3

DIRECT IMAGING OF SMALL SCATTERERS USING REDUCED TIME DEPENDENT DATA

In this chapter, we introduce qualitative methods for locating small objects using time dependent acoustic near field waves. These methods have reduced data collection requirements compared to typical qualitative imaging techniques. As in Chapter2, we only collect scattered field data in a small region surrounding the location from which an incident field was transmitted. The new methods are partially theoretically justified and numerical simulations demonstrate their efficacy. We show that these reduced data techniques give comparable results to methods which require full multistatic data and that these time dependent methods require less scattered field data than their time harmonic analogs.

3.1 Introduction We propose two schemes in this chapter which signiﬁcantly reduce the amount of data required for accurate reconstructions. In both schemes, we use a small array of transmitters and receivers constructed so that data is collected only in a small region. Incident waves are emitted from the transmitters, collected by the nearby receivers, and the entire device is moved to a new location where the experiment is repeated. In one scheme, we allow the device to contain many transmitters and receivers, collecting multistatic data only in patches with a small aperture. In the other scheme, a quasi-backscattering set-up, the array contains one transmitter and a small number of receivers in a small neighborhood of the transmitter. We must increase the amount of a priori information we assume about the object in order to justify these

42 methods theoretically. As in Chapter2, we will assume objects are small and weakly scattering. The quasi-backscattering data collection scheme proposed here is somewhat similar to the time harmonic study initiated in Chapter2 and studied further in [56], though the reconstruction method and applicability of the method here differs. Of particular importance here is that the algorithms described below directly use causal time dependent near field data and require no Fourier or Laplace transformation into frequency domain data. In many applications, ranging from medical imaging to non- destructive testing, time dependent data is readily obtained. Moreover, as our numerical examples will demonstrate, using time dependent data allows us to use significantly fewer transmitters and receivers than time harmonic data. Most previous studies of similar problems make use of time harmonic far field data with one or multiple frequencies. In some applications, far field data is required due to physical constraints on how near to an object sensors can be placed. Nonetheless, near field data is sometimes easier to obtain in practice, and typically results in higher resolution reconstructions. Fur- thermore, the type of data collection scheme suggested here is readily implementable in practice. For example, a device with transmitters and receivers concentrated in a small region was built in [45] to collect scattered field data for potential industrial applications. To make the above comments precise, assume scattering is caused by time dependent acoustic waves propagating through a medium with a variable speed of sound, c ∈ L∞(Rd)(d = 2 or 3) so that c(x) ≥ γ > 0 for some γ ∈ R. We assume a constant i background speed of sound, c0 = 1. Let u (x, t; y) indicate the incident field emitted from a point y ∈ Rd evaluated at a point x ∈ Rd\{y} and time t ∈ R+. Recall from Chapter1 that such an incident field satisfies the free space acoustic wave equation,

i i d + utt − ∆xu = 0 for x ∈ R \{y}, t ∈ R .

43 The resulting scattered ﬁeld, us(x, t; y), satisﬁes

−2 s s −2 i d + c (x)utt − ∆u = −(c (x) − 1)utt (x, t) ∈ R × R (3.1a)

s s d u (x, 0) = ut (x, 0) = 0 x ∈ R . (3.1b)

Define n(x) = c−2(x) to be the index of refraction and D = supp(1 − n(x)) to be the location of the unknown scatterer. We will be more precise about n and D below. Let χ ∈ C2(D) be a causal temporal pulse function (that is, χ and its derivatives vanish for t < 0). We define the incident field originating at a point y ∈ Rd as the time convolution of χ with the fundamental solution Φ to the wave equation. Recall from Chapter1 that  H(t − |x|)   p , d = 2, 2π t2 − |x|2  Φ(x, t) = (3.2)  δ(t − |x|)  , d = 3,  4π|x| where H is the Heaviside function. For example, in R3 χ(t − |x − y|) ui(x, t; y) = , (x, t) ∈ ( 3\y) × +. 4π|x − y| R R s + The inverse problem is to find D from u (x, t; y) for x ∈ Γm, y ∈ Γi, t ∈ R where the d−1 measurement and incident locations, Γm and Γi respectively, are sets in R which do not intersect with D. For example, in a full aperture multistatic set-up, Γm = Γi =

∂BR(0), where ∂BR(0) is the boundary of a ball of radius R > 0 centered at the origin where R is large enough that D ⊂ BR(0). In the limited aperture case Γm, Γi ⊂ ∂BR(0)

(possibly Γm = Γi). See Figure 3.1 (left) for a sample of a limited aperture multistatic geometry. In this chapter, we will primarily use reduced data. First, we use a series of limited aperture multistatic arrays which are moved around the obstacles. For example,

3 let Γi = Γm be patches with a small area in R . We collect multistatic data with this patch and then move the entire array to a new location and collect data again. The second type of reduced data is a quasi-backscattering set-up. To describe this data set-

d−1 up, let Γi ⊂ R be the curve on which we will place transmitting devices. We again

44 assume we can collect data only with a small device which moves around Γi. Denote (y) by δ > 0 a small constant. For each ﬁxed y ∈ Γi, data is collected on Γm := Γi ∩Bδ(y), d where BR(x) is the ball of radius R > 0 centered at x ∈ R . See Figure 3.1 (right) for a sample set-up geometry in R2. Note that this set-up requires more data than the related backscattering data, in which each transmitter has just one associated receiver, and both are located at the same point.

s us u D D Bδ(y) (y) Γm ui Γm ui y y Γi D D Γi us us

Figure 3.1: Examples of limited aperture multistatic (left) and quasi-backscattering (right) measurements. In the limited aperture multistatic ﬁgure, the blue line represents the location of transmitters and the red line the location of receivers. In the (y) quasi-backscattering set-up, Γi is the large dashed circle, the thick solid line is Γm for (y) a ﬁxed y ∈ Γi, and the circles on Γi not located at y represent locations to which Γm will be moved.

There exist many qualitative methods for solving inverse scattering problems with multistatic time domain or multifrequency data. In [52, 53], a qualitative method known as the linear sampling method is used to approximate the shape of D using causal multistatic time-domain scattering data. In these papers, the theoretical jus- tiﬁcation of the method remains incomplete due to technical problems involving an associated problem called the interior transmission problem described in Chapter1. However, a new result on transmission eigenvalues [94] alleviates this diﬃculty. This will be discussed in more detail in Section 3.3.4. This is in contrast to the time domain linear sampling method for scattering from bounded objects with Dirichlet, Neumann,

45 or Robin boundary conditions whose theory is fully described in [31, 57]. The multifrequency linear sampling method, which can be seen as time dependent technique with non-causal waves, is studied in [54]. In [84, 91], it was shown that, under certain conditions, a potential function related to speed of sound can be calculated based on backscattered time domain data collected in the far field. While these require less data than we do, they solve a slightly different problem than we do here and do not provide a method for constructing the potential. Time reversal methods, described for example in the review article [44], are also popular for solving inverse scattering problems with time dependent data. The time harmonic backscattering problem for small and weak scatterers was studied in [50] using multiple frequencies. A number of recent reconstruction algorithms have been proposed, e.g. [1, 51, 80], which reduce data requirements by using only one incident source and scattered field data with receivers surrounding the objects. Such approaches result in fast data collection, since there is only one experiment required, but require that the objects can be simultaneously surrounded by receivers. We take a different approach to data reduction here, assuming there is enough time to perform many experiments, but that the objects cannot be completely surrounded by receivers at the same time. This approach is useful in the case of imaging large regions or in cases where it is costly to place many receivers at once.

3.2 Forward Model and the Born Approximation We begin by discussing the well-posedness of (3.1). This is well known, and to discuss it precisely we follow [13, 52, 73], introducing some space-time Sobolev spaces described through the Fourier-Laplace transform. This will allow us to introduce and state the well-posedness of a time domain weak scattering approximation and its frequency domain counterpart. This approximation is the Born approximation described in Chapter1. These will be used in Section 3.3.1 to validate a multistatic MUSIC-type algorithm in the time domain.

46 As discussed in the introduction to this chapter, we are able to reduce the amount of data required for reconstructions by making a priori assumptions on the contrast n and the scatterer D. In particular, we will make a weak scattering Born approximation in which multiple scattering eﬀects are minor and can be ignored. As- sume that n(x) = 1 + mB(x) for 1, mB = O(1), and that solutions to (3.1) s s s take the form u (x, t) = u0(x, t) + uB(x, t). Recall from Chapter1 that the function s ∂us uB = ∂ =0 is the well-known Born approximation which satisﬁes ∂2us ∂2ui B − ∆us = −m for (x, t) ∈ d × + (3.3a) ∂t2 B B ∂t2 R R ∂us us (x, 0) = B (x, 0) = 0 for x ∈ d. (3.3b) B ∂t R

This time domain Born approximation should be considered as a linearization of the scattered field with respect to the strength of scatterers, rather than as the first term of a series solution to (3.1) in the way that the time harmonic Born approximation sometimes is; as is discussed in Remark 4.5 of [70], terms associated with higher order terms in are not necessarily well-defined in any reasonable spaces. We follow the same process for solving (3.3) as in Chapter1 and take a space-

i time convolution of mB(x)utt with Φ(x, t). This results in a time domain Lippmann- Schwinger equation,

s s i d uB(x, t; y) + (V [mBuB])(x, t; y) = −(V [mBu ])(x, t; y) x 6= y ∈ R , t > 0.

Recall that the retarded volume potential operator V is deﬁned by Z Z d (V f)(x, t) := Φ(x − z, t − τ)f(τ, z) dV (z) dτ, (x, t) ∈ R × R R D where Φ(x, t) is given by (3.2). Later we will also use the related single layer potential,

SΓ, deﬁned by Z Z d (SΓf)(x, t) := Φ(x − y, t − τ)f(τ, y) ds(y) dτ, (x, t) ∈ (R \Γ) × R R Γ where Γ is some closed surface.

47 In order to make these equations precise, we recall the appropriate space-time Sobolev spaces, following [52, 73]. To this end, we ﬁrst introduce the Fourier-Laplace transform. Let ω = η + iσ for η, σ ∈ R with σ > σ0 > 0 for some σ0 ∈ R. We use the notation Cσ0 = {ω ∈ C : Im(ω) ≥ σ0 > 0} to deﬁne this half-plane. Let X be a Hilbert space. The set of temporal, smooth, and compactly supported in [0, ∞)

+ ∞ X-valued functions is denoted by D(R ; X) = C0 (R; X). The associated X-valued distributions on the real line which vanish for time t < 0 are denoted by D0(R+; X) and the corresponding tempered distributions by S0(R+; X). Deﬁne

0 + 0 + −σt 0 + Lσ(R ,X) := {f ∈ D (R ,X): e f ∈ S (R ,X), for some σ(f) < ∞} to be the space of functions with well-deﬁned Fourier-Laplace transforms. Indeed, the 0 + ˆ Fourier-Laplace transform of f = f(x, t) ∈ Lσ(R ,X), denoted by f(x, ω) is given by Z ∞ ˆ f(x, ω) = f(x, t) exp (iωt) dt,ω ∈ Cσ0 0

d for σ0 = σ0(f) and x ∈ R , t ∈ R. Note that fˆ(x, ω) = F(e−σtf)(η) where F represents the typical Fourier transform on causal functions, so many properties of the Fourier transform will transfer to the Fourier-Laplace transform with little change. We can now deﬁne the Hilbert space for p ∈ N0, σ ∈ R,

Z 2 p + 0 + 2p ˆ Hσ(R ,X) := f ∈ Lσ(R ; X): |ω| f(·, ω) ds < ∞ . X R+iσ By Parseval’s theorem, the norm of this space is equivalent to

Z ∞ p 2 2 −2σt ∂ f(·, t) kfk p + = e dt Hσ(R ;X) p 0 ∂t X where we have used the fact that f and its derivatives vanishes for t < 0. For more details see e.g. [60]. With this notation in hand, we have the following result about the solvability of (3.3), where σ > σ0 > 0 for a σ0 depending on the speciﬁcs of the problem.

48 p + 2 Theorem 2 ([73], Theorem 3.2). For r = 0, 1, 2 and p ∈ R, V : Hσ(R ,L (D)) → p+1−r + r d Hσ (R ,H (R )) is a bounded linear operator. Moreover, if v = V (f) for some p + 2 p + 1 d f ∈ Hσ(R ,L (D)) then v(t) = 0 for t < 0 and v ∈ Hσ(R ,H (R )) satisﬁes

p−1 + 2 d vtt − ∆v = f in Hσ (R ,L (R )).

Theorem2 allows us to write

s i uB(x, t; y) = −(V mutt)(x, t) (3.4) Z Z i d + = − m(z)Φ(x − z, t − τ)utt(z, τ; y) dV (z) dτ, (x, t) ∈ R × R . R D For later, we introduce the bounded linear solution operator for (3.3),

p + 2 p+1−r + r d G : Hσ(R ,L (D)) → Hσ (R ,H (R )). (3.5)

i s which takes u to uB with (3.4). Here σ, p, and r are as in Theorem2. From the above, the solution of the Born wave equation satisﬁes

s i p + 1 d p + 2 kuBkHσ(R ,H (R )) ≤ Cku kHσ(R ,L (D)). (3.6)

See also [87] and Chapter5 for a discussion of these properties in both R2 and R3. Taking the Fourier-Laplace transform of V gives an equivalent formulation in ∞ ˆ the frequency-domain. In particular, for f ∈ C0 (D) deﬁne the operator V by Z ˆ ˆ d (V f)(x; ω) = Φω(x, z)f(z) dV (z), x ∈ R D ˆ where Φω(·, ·) is the fundamental solution of the Helmholtz equation with wavenumber

ω ∈ Cσ0 . for some σ0 > 0. It can be shown [73] that Vˆ : L2(D) → H2(Rd) and that ifv ˆ = Vˆ f thenv ˆ satisﬁes

2 d ∆ˆv + ω vˆ = f in R .

Hence,

s 2 s 2 i ∆ˆuB + ω uˆB = −ω muˆ (3.7)

49 has the solution Z s 2 ˆ i uˆB(x, ω; y) = −ω m(z)Φω(x, z)û (z; y) dV (z). (3.8) D This is identical to the plane wave scattering case considered in Chapter2, with a different incident wave. If Im(ω) = 0 then the Fourier-Laplace transform becomes the standard Fourier transform, (3.7) becomes the usual equation for a Born approximation to the time harmonic scattered field with wavenumber Re(ω), and (3.8) is the first term of the Born series.

3.3 Inverse Problem for the Time Domain Born Approximation We now discuss time dependent imaging algorithms, two for multistatic data and one for quasi-backscattering data. We first introduce a MUSIC-type method for imaging weak and small scatterers which is fully justified theoretically. As a specific case of this method, we describe a reconstruction algorithm using multistatic patch data. Next, we introduce a MUSIC-type method for quasi-backscattering data. As the numerical results in Section 3.4 suggest, both techniques can be used to find obstacles from time domain data. Finally, we discuss the linear sampling method for extended weak scatterers with multistatic data and why it lacks full justification. Each algorithm we develop below takes a similar form. In particular, we define an indicator function, depending on z ∈ Rd (and possibly on time τ ∈ [0,T ] for some T > 0), so that the function is large when z ∈ D and small otherwise. As such, the bulk of this section is dedicated to deriving indicator functions and demonstrating that they are large when z ∈ D. The primary idea behind each of these indicator functions is that a specified function, which depends on z, is in the range of an operator depending on us if and only if z ∈ D. For multistatic data, we are interested in the near field equation

ξ + (Nmultigz,τ )(y, t) = `z,τ (y, t), (y, t) ∈ Γi × R (3.9)

50 2 + 2 2 + 2 for each z ∈ Z, where the near ﬁeld operator Nmulti : Lσ(R ,L (Γm)) → Lσ(R ,L (Γi)), σ > 0 is deﬁned by Z Z s + (Nmultig)(y, t) = uB(x, t − τ; y)g(x, τ) ds(x) dτ, (y, t) ∈ Γi × R . (3.10) R Γm

In Section 3.3.4 below we will give more details about the mapping properties of Nmulti. Furthermore, Z ξ `z,τ (y, t) := Φ(y − z, t − τ − t0)ξ(t0) dt0, (3.11) R

∞ is the convolution of a smooth compactly supported ξ ∈ Cc (R) with the fundamen- 3 ξ tal solution of the wave equation given by (3.2). For example, in R , `z,τ (y, t) = ξ(t − τ − |y − z|) . The idea for quasi-backscattering data is similar. 4π|y − z|

3.3.1 Reconstruction of Point Scatterers from Time Domain Multistatic Data Assume now that D is composed of M weak point scatterers located at the points d PM zj ∈ R , j = 1,...,M. Let the contrast mB be of the form mB(x) = j=1 mj1Dj (x) where mj are constant. In this section we collect multistatic data and introduce a MUSIC-type algorithm for locating small objects based on near ﬁeld time domain

d−1 data. Hence, let Γi ⊂ R be the curve from which incident fields are transmitted d−1 and let Γm ⊂ R be the curve on which the resulting scattered field is measured. We assume the curves do not intersect D and that they are either closed curves or open subsets of analytic curves. Below we will take Γi = Γm. In the above configuration, the near field operator (3.10) takes the form

M X (N g)(y, t) = − m Φ(z − y, ·) ∗ Sχ¨ g) (z , ·) (t), (y, t) ∈ Γ × +, multi j j Γm j i R j=1 (3.12) where (Sχ¨ g)(x, t) = (¨χ(·) ∗ (S g)(x, ·)) (t) (3.13) Γm Γm

51 and ∗ indicates a time convolution. Hence, the point scattering near ﬁeld equation (3.9) becomes

M X − m Φ(z − y, ·) ∗ Sχ¨ g) (z , ·) (t) = `ξ (y, t), (y, t) ∈ Γ × +. j j Γm j y,τ i R j=1

The Fourier-Laplace transform Nmulti in this point scattering context yields the fre- 2 2 quency domain weakly scattering near ﬁeld operator, Nbmulti : L (Γm) → L (Γi) deﬁned as M Z X ˆ ˆ (Nbmultig)(y, ω) = − mjχb¨(ω)Φω(zj, y) Φω(x, zj)ˆg(x, ω) ds(x), ω ∈ Cσ0 j=1 Γm (3.14) for some σ0 > 0. ξ Similarly, the Fourier-Laplace transform of `z,τ is

ξ\ ˆ iωτ ˆ `z,τ (y, t)(y, ω) = ξ(ω)e Φω(y, z),

ˆ where Φω is the fundamental solution for the Helmholtz equation. Thus the transformed point scattering near ﬁeld equation reads

M X ˆ ˆ d αjΦω(y, zj) = βΦω(y, z), y ∈ Γi, z ∈ R (3.15) j=1 where α = −m χ¨(ω) R Φˆ (x, z )ˆg(x, ω) ds(x), β = ξˆ(ω)eiωτ are constants depending j j b Γm ω j ong ˆ, zj, τ, mj,Γm, and ω. Point scattering approximations of the type derived here can also be derived through high frequency truncation of asymptotic expansions, as explained in [4]. For ﬁxed ω, τ = 0, and ξˆ(ω) ≡ 1, (3.15) leads the MUSIC algorithm in the ˆ frequency domain; the following lemma allows us to characterize the range of Nmulti.

Lemma 4. Let Γi be a closed curve or an open subset of an analytic curve and let d zj ∈ R , j = 1,...,M be distinct points which do not lie on Γi. Then the sets of ˆ functions {y 7→ Φω(y, zj): j = 1, . . . , M, y ∈ Γi} are linearly independent for any

ω ∈ Cσ0 with σ0 = σ0(Φ) > 0.

52 Proof. Let aj ∈ C be constants so that

M X ˆ ajΦω(y, zj) = 0 y ∈ Γi. (3.16) j=1

ˆ Since Φω(y, z) are solutions to the Helmholtz equation, they are real analytic on y away from y = z. Without loss of generality, assume Γi is a closed curve. Otherwise, we can analytically continue (3.16) to the analytic curve of which Γi is a subset. Note that the left hand side of (3.16) is a radiating solution to the Helmholtz equation outside of Γi. Hence, by unique continuation and uniqueness of the exterior d Dirichlet problem with boundary Γi,(3.16) is in fact true for all y ∈ R \{z1, z2,...,z M }. ˆ Due to the singularity of Φω(y, z) at y = z, letting y → zj, we see that each aj must vanish identically.

The above lemma enables us to characterize the range of the ﬁnite rank operator ˆ Nmulti which in turn leads to a test to locate the point scatterers. Note that the proof of this theorem is very similar to the proof of the equivalent theorem for far ﬁeld data given in [69]. We include this proof for completeness.

Theorem 3. Assume ω ∈ Cσ0 for some σ0 > 0, D = {z1, z2, ··· zM } and that Γi and Γm (not necessarily the same) are closed curves or open subsets of analytic curves ˆ ˆ which do not intersect D. Then Φω(·, z) ∈ Range(Nmulti) if and only if z = zj some j = 1,...,M.

Proof. Assume by contradiction that z = z0 ∈/ {z1, . . . , zM }, and that there is some 2 ˆ ˆ ˆ g ∈ L (Γm) so that (Nmultig)(y) = Φω(y, z0) for each y ∈ Γi. By deﬁnition of Nmulti,

M ˆ X ˆ Φω(y, z0) = αjΦω(y, zj), y ∈ Γi. j=1

This is a contradiction with the linear independence shown in Lemma4. Hence, if ˆ ˆ Φω(·, z) ∈ Range(Nmulti) then z ∈ {z1, z2 . . . , zM }.

53 ˆ ˆ ∗ ⊥ Now assume z ∈ {z1, z2, . . . , zM }. We will show that Φω(·, z) ∈ Kern(Nmulti) = ˆ ˆ ˆ ˆ Range(Nmulti) (notice that Nmulti is ﬁnite rank so Range(Nmulti) = Range(Nmulti)). We ˆ ∗ 2 2 explicitly calculate the adjoint of Nmulti : L (Γi) → L (Γm) as M Z ˆ ∗ X ˆ ˆ ˆ (Nmultih)(x) = − mjχ¨(ω)Φω(x, zj) Φω(y, zj)h(y) ds(y). j=1 Γi ˆ ∗ As such, if h ∈ Kern(Nmulti) then M Z X ˆ ˆ mjχ¨ˆ(ω)Φω(x, zj) Φω(y, zj)h(y) ds(y) = 0, for x ∈ Γm j=1 Γi and by the assumption on Γm, the linear independence shown in Lemma4 gives Z ˆ 0 = Φω(y, zj)h(y) ds(y). Γi ˆ ˆ ∗ ⊥ ˆ Hence each Φω(·, zj) ∈ Kern(Nmulti) = Range(Nmulti).

While we have primarily proven Theorem3 in order to prove a similar result for the time dependent case, it also gives a MUSIC-type inversion scheme for multistatic weakly scattering near ﬁeld time harmonic data. In particular, both Lemma4 and

Theorem3 follow in an identical way for real positive values of ω ∈ R, ω > 0. Then us is time harmonic acoustic scattering data from a point source incident ﬁeld. Let P ˆ ⊥ : Nmulti 2 ˆ ⊥ L (Γi) → R(Nmulti) be the orthogonal projection onto the orthogonal complement of ˆ ˆ the range of Nmuti. Theorem3 gives that P ˆ ⊥ Φω(·, z) = 0 if and only if z = zj. Nmulti ˆ In a typical MUSIC application, the function I(z) = P ˆ ⊥ Φs(y, z) Nmulti 2 L (Γi) serves as an indicator function to locate D: if D ⊂ Z for some set of sampling points −1 Z ⊂ Rd, then (I(z)) is large for each z ∈ D and small otherwise. We change this ˆ ˆ slightly here and test the angle between Range(Nmulti) and Φω(y, z) for each z ∈ Z. ˆ ˆ When the angle between these is very small, we assert that Φω(y, z) ∈ Range(Nmulti) and hence that z = zj for j = 1,...,M. We ﬁnd numerically that this results in a more stable reconstruction algorithm than the typical approach. To calculate this angle, introduce ˆ ˆ Φω(y, z),P ˆ Φω(y, z) Nmulti 2 ˆ L (Γi) Jmulti(z) := , kΦˆ (y, z)k 2 kP ˆ Φˆ (y, z)k 2 ω L (Γi) Nmulti ω L (Γi)

54 ˆ ˆ where P ˆ is the projection operator onto the range of Nmulti. Note that Jmulti ≤ 1 Nmulti ˆ ˆ ˆ with equality if and only if Φω(z) ∈ Range(Nmulti). Then the angle between the Φω(z) ˆ ˆ and the range of the near field operator is Imulti(z) = arccos Re Jmulti(z) . As −1 ˆ the numerical results demonstrate in Section 3.4, Imulti(z) is large if and only if ˆ z = zj. Note that Jmulti is similar in form to the indicator function introduced for time harmonic scattering in [64]. However, the two functions are derived in a very different fashion - the indicator function in [64] is not related to the range of the near field operator - and as far as the authors can tell, their similarity is only coincidental. The range test in the frequency domain formulated in Theorem3 can now be used to obtain a range test for time domain scattering.

Theorem 4. Assume D = {z1, z2, ··· zM } and that Γi and Γm (not necessarily the same) are closed curves or open subsets of analytic curves which do not intersect D.

ξ d ∞ + ξ Deﬁne `z,τ (x, t) as in (3.11) with z ∈ R and τ > 0 and ξ ∈ C0 (R ). Then `z,τ ∈

Range(Nmulti) if and only if z ∈ {z1, . . . , zM }, where Nmulti is given by (3.12).

ξ Proof. Assume `z,τ ∈ Range(Nmulti). This is true if and only if there exists some gz,τ ξ so that (Nmultigz,τ )(y, t) = `z,τ (y, t), which by Parseval’s equality is true if and only if

∞ Z 2 0 = e−2σt (N g )(y, t) − `ξ (y, t) dt, σ > 0 multi z,τ z,τ L2(Γ ) −∞ i Z ∞+iσ 2 1 ˆ ξ = Nmultigˆz,τ (y, ω) − `b (y, ω) dω. (3.17) z,τ 2 2π −∞+iσ L (Γi) This holds true if and only if

2 ˆ ˆξ Nmultigˆz,τ (y, ω) − ` (y, ω) = 0,ω ∈ σ. z,τ 2 C L (Γi) ˆ ˆ Note that by analyticity χ¨(ω) = 0 and ξ(ω) = 0 only for a discrete set of ω ∈ Cσ0 with ξ σ > σ0 > 0. Hence, recalling (3.11) and (3.14) we now have that `z,τ ∈ Range(Nmulti) if and only if

M X ˆ ˆ d αjΦω(y, zj) = βΦω(y, z), y ∈ Γi, z ∈ R ,ω ∈ Cσ0 . j=1

55 where α = −m χ¨(ω) R Φˆ (x, z )ˆg (x, ω) ds(x) and β = ξˆ(ω)eiωτ But this is exactly j j b Γm ω j z,τ the range test from Theorem3, and so is true if and only if z ∈ {z1, . . . , zM }.

As in the frequency domain case, this leads to an inversion scheme for time dependent multistatic data. Indeed, to calculate the angle between Φ(y − z, t − τ) and

Range(Nmulti), introduce

(Φ(y − z, t − τ),PN Φ(y − z, t − τ)) 2 multi L (Γi×R) Jmulti(z, τ) := , 2 2 kΦ(y − z, t − τ)kL (Γi×R)kPNmulti Φ(y − z, t − τ)kL (Γi×R)

where PNmulti is the projection operator onto the range of Nmulti. Then the angle between the Φ(y − z, t − τ) and the range of the near ﬁeld operator is Imulti(z, τ) = arccos (Jmulti(z, τ)). Note that, unlike in the frequency domain case, we do not need to take a the real part of Jmulti since the time domain values are inherently real-valued.

As seen in Section 3.4, the indicator function Imulti(z, τ) := arccos(Jmulti(z, τ)) ≈ 0 if and only if z = zj. It is not completely clear how the sampling time τ aﬀects reconstructions. How- ever, numerical examples in Section 3.4 suggest that its choice is not very important for scattering from small and weak scatterers. On the other hand, numerical experiments with large obstacles, for which the above theory is not justiﬁed, show that a good choice of τ results in reconstruction of both the shape and location of an object. Poor choice of τ for large objects only allows the reconstruction of the location of the objects.

3.3.2 Reconstruction of Point Scatterers from Patches of Time Domain Multistatic Data A key point in the reduction of data collection requirements is that the above theorems make very weak assumptions about the geometry of Γi and Γm. As such, both Γi and Γm can be chosen as, e.g., sectors of a circle with a very small aperture. However, because of errors in data collection and limitations in measurement accuracy, this is not feasible in practice. Nonetheless, numerical simulations suggest that a patch

56 of multistatic data, in which Γi = Γm are, e.g., sectors of a circle with a very small aperture gives some indication of the hidden objects. These observations lead to a simple technique for limiting data collection requirements in obstacle reconstruction: collect multistatic data on a small patch array of transmitters and receivers, then repeatedly move the array around the obstacles and repeat the experiment. Once this data is collected, reconstruct the obstacles from each experiment and post-process these reconstructions to give one reconstruction incorpo- rating each experiment. The simplest post-processing is to compute a weighted average of each reconstruction, though more complex processes may be applied. As is shown in Section 3.4, this patch data and post-processing step results in acceptable reconstructions. Indeed, in an error-free case it is theoretically justiﬁed. However, in practice it requires a possibly time consuming reconstruction process for each set of patch data. Each of these reconstructions may further require regularization and choice of regularization parameters. Furthermore, a simple average of each reconstruction does not take into account that reconstructions should be similar, as they come from the same objects. More sophisticated post-processing algorithms could cer- tainly alleviate this problem, but we do not explore them here. In the next section, we propose a method which requires only one reconstruction using even less data. Unlike the multistatic patch method, however, we are unable to fully justify its theory.

3.3.3 Reconstruction of Point Scatterers from Time Domain Quasi-Backscattering Data For simplicity we assume that the transmitters are distributed on the boundary

Γi := SR of a large ball BR centered at the origin containing the scatterer D ⊂ BR and (y) for each transmitting point y ∈ Γi the scattered field is measured at Γm := SR ∩Bδ(y), where Bδ(y) is a small ball centered at y of radius δ. We will first consider briefly the full backscattering case when δ → 0. As numerical results demonstrate below, the quasi-backscattering setting for δ > 0 produces better reconstructions than the full backscattering case.

57 Consider the weak scattering near ﬁeld backscattering operator for (y, t) ∈ Γi × R+, Z s (Nbackscatteringg)(y, t) = u (y, t − τ; y)g(τ) dτ R M Z X mj = − χ¨(t − τ − 2|y − z |)g(τ) dτ. (4π|y − z |)2 j j=1 j R Taking the Fourier-Laplace transform yields

M X mj (Nˆ g)(y, ω) = −ω2 χˆ(ω)ˆg(ω) exp (2iω|y − z |) backscattering (4π|y − z |)2 j j=1 j M 2 X ˆ 2 = −ω mjχˆ(ω)ˆg(ω)Φω(y, zj) j=1 M X ˆ 2 = αj(ω)Φω(y, zj), j=1

2 where αj(ω) = −ω mjχˆ(ω)ˆg(ω) and ω ∈ Cσ0 for σ0 > 0. Notice the similarities between Nbbackscattering and Nbmulti and deﬁne the sampling function ξ −1 ˆ 2 ψz,τ (y, t) = F [Φω(y, z)] ∗ ξ(· − τ) (3.18)

∞ −1 where ξ ∈ C0 and F here denotes the inverse of the Fourier-Laplace transform deﬁned in Section 3.2. Using the same arguments as in the proof of Theorem4 for the multistatic case, the form of Nbbackscattering suggests that

ξ ξ ! ψz,τ (y, t),PNbackscattering ψz,τ (y, t) 2 L (Γi×R) Ibackscattering(z, τ) := arccos ξ ξ 2 2 kψz,τ (y, t)kL (Γi×R)kPNbackscattering ψz,τ (y, t)kL (Γi×R) acts as an indicator for the location of zj, where PNbackscattering is the projection operator onto the range of Nbackscattering.

As shown in Figure 3.7, reconstructions using Ibackscattering are quite recogniz- able, but less clear than their quasi-backscattering equivalents. Hence, we suggest collecting quasi-backscattering data with δ > 0. This diﬀers from the results originally introduced for the quasi-backscattering context in Chapter2, and in [56] in which a (y) speciﬁc relationship between Γi and Γm is required in order to reconstruct zj under

58 the assumption that |x − zj| 1. Here we do not require any such relationship or that data is collected suﬃciently-far from the point scatterers. However, unlike in [56, 59], we are unable to prove an exact characterization of which components of {zj} can be reconstructed. Indeed, other than an intuitive argument that more data leads to better reconstructions, we are unsure why the quasi-backscattering reconstructions are superior to the backscattering reconstructions. The quasi-backscattering time domain weakly scattering near ﬁeld operator

p + 2 p + 2 Nquasi : Hσ(R ,L (SR)) → Hσ(R ,L (SR)), σ > 0, p ∈ R is deﬁned by Z Z s + (Nquasig)(y, t) = uB(x, t − τ; y)g(x, τ) ds(x) dτ, (y, t) ∈ Γi × . (y) R R Γm

For a collection of point sources with strength mj centered at points zj, j = 1,...,M,

Nquasi takes the form

M X χ¨ + (Nquasig)(y, t) = − mj Φ(zj − y, ·) ∗ S (y) g) (zj, ·) (t), (y, t) ∈ Γi × R Γm j=1 (3.19) and the Fourier-Laplace transform of the quasi-backscattering near ﬁeld equation reads

M Z X ˆ ˆ ˆ iωτ ˆ 2 mjχb¨(ω)Φω(y, zj) Φω(x, zj)g(x) ds(x) = −ξ(ω)e Φω(y, z) y ∈ Γi, z ∈ Z, (y) j=1 Γm (3.20) for ω ∈ Cσ0 , σ > σ0 > 0. Unfortunately, as opposed to the backscattering or multistatic cases, in (3.20) the term Z ˆ J(y) := Φω(x, zj)g(x) ds(x) (3.21) (y) Γm ˆ is not proportional to Φω(y, zj). As such, the arguments used to justify Theorem3 ˆ no longer apply. We would like to show that J(y) = cΦω(y, zj) + O(δ) where c is (y) independent of y. To this end, for a small patch Γm on the sphere of radius R, say with diameter δ > 0, where δ is small compared to |y − zj| for j = 1,...,M, up to 2 (y) 3 2 order δ we can replace Γm by the tangent plane in R or line in R at y; let us denote

59 δ δ ⊥ ⊥ it by Ty . Hence x ∈ Ty and we have that x = y + ηy where 0 < η < δ and y are the components of y parallel to the tangent plane. Simple asymptotic calculations, which for sake of the argument, we present here only in the R3 case, give

⊥ (zj · y ) 2 |x − zj| = |y − zj| + η + O δ and |y − zj| ⊥ −1 −1 (zj · y ) 2 |x − zj| = |y − zj| 1 − η 2 + O δ |y − zj| and hence, up to order O(δ2)

⊥ Z iωη(zj ·y ) ⊥ (zj · y ) ˆ |y−zj | ⊥ ⊥ J(y) ≈ Φω(y, zj) e 1 − η 2 g(y + ηy )d(y + ηy ). |ηy⊥|<δ |y − zj|

⊥ Hence we obtain the desired result if (zj · y ) = 0 for all y ∈ Γi and j = 1,...,M. There is a special measurement conﬁguration which works only in R3 and is detailed in [56]. Here, the measurement geometry is speciﬁed so that the integral in the above equation is independent of y. Indeed, in this set-up, Γi is set to be a line parallel to some directionv ˆ and for each y ∈ Γi, the corresponding measurements are taken on a line parallel tov ˆ⊥ passing through y. In this set-up,

M X ˆ 2 ˆ iωτ ˆ 2 mjχb¨(ω)Φω(y, zj) = −ξ(ω)e Φω(y, z) y ∈ Γi, z ∈ Z j=1

⊥ provides an exact range test for zj whose projections tov ˆ diﬀer. See Theorem 1 in [56] for more details. In the far ﬁeld this setup becomes simpler and is studied in [59].

2 3 Nevertheless, for our data configuration in both R and R ,(3.20) shows that if z = zj 2 δ for some j = 1,...,M, then we can find a g := gzj ,y ∈ L (Ty ) that solves exactly (3.20). Arguing heuristically, taking g as an approximating sequence of the Dirac delta function at y suggest that a range test as in the framework built up in the previous sections allowing us to introduce an indicator function for finding zj. As the numerical examples below demonstrate, these assumptions on the geometry of Γm and Γi do do not seem to be active. Indeed, based on the discussion above,

60 after taking the inverse Fourier-Laplace transform of (3.20), we introduce an indicator function to measure the angle between ψz,τ and Range(Nquasi). Let

ξ ξ ψz,τ (y, t),PNquasi ψz,τ (y, t) 2 L (Γi×R) Jquasi(z, τ) := ξ ξ , 2 2 kψz,τ (y, t)kL (Γi×R)kPNquasi ψz,τ (y, t)kL (Γi×R) where PNquasi is the projection operator onto the range of Nquasi. We demonstrate in

Section 3.4 that the indicator function Iquasi(z, τ) := arccos(Jquasi(z, τ)) ≈ 0 if and only if z = zj. In exactly the same way, we can derive a time harmonic indicator function for quasi-backscattering data. In particular, time harmonic reconstructions are computed with   2 2  Φˆ (y, z),P ˆ Φˆ (y, z) ω Nquasi ω 2 ˆ L (Γi) Iquasi(z) = arccos Re .   2 2  kΦˆ (y, z)k 2 kP Φˆ (y, z)k 2 ω L (Γi) Nˆquasi ω L (Γi)

We will show below that time dependent reconstructions out perform time harmonic reconstructions when an incident ﬁeld is transmitted from only few points.

3.3.4 Linear Sampling Method for Extended Objects Under the Time Do- main Born Model Before showing numerical reconstructions of point objects, we discuss one technique, the linear sampling method, for extending the above multistatic results to extended obstacles which are weakly scattering. Full justification the linear sampling for the weak scattering case - as well as full justification under a strongly scattering model - requires new results about an associated interior transmission problem which we are unable to prove. Note that in the case of strongly scattering media, [94] provides the necessary results about the interior transmission problem for full justification of the linear sampling problem. We are unable to apply these results to the case of the Born regime. To introduce the linear sampling method for small objects, we follow [52], in which the linear sampling method for strongly scattering time domain data for inhomogeneous media is first investigated. As detailed in Chapter1, the primary idea

61 in linear sampling is to ﬁnd a regularized solution to the near ﬁeld equation (3.9). Then I (z, τ) = kg k−1 is used as an indicator function. The values I are LSM z,τ L2(D,R+) LSM d then used to suggest where D is located; the z ∈ R so that ILSM(z) is large are our reconstruction of D.

s By making use of the volume integral equation representation of uB, we can factor Nmulti into the product of two well-studied operators. In particular, Z Z s (Nmultig)(y, t) = uB(x, t − τ; y)g(x, τ) ds(x) dτ R Γm Z Z Z Z = − m(z)Φ(x − z, t − τ − s)× R Γm R D i utt(z, s, y) dV (z) ds g(x, τ) ds(x) dτ Z Z Z Z i = − m(z)utt(z, s; y) Φ(x − z, t − τ − s)× R D R Γm g(x, τ) ds(x) dτ dV (z) ds Z

= − m(z) (Φ(z − y, ·) ∗ χ¨(·) ∗ (SΓm g)(z, ·)) (t) dV (z) D Z Z = − m(z)Φ(z − y, t − τ) Sχ¨ g (z, τ) dV (z) dτ. Γm R D Sχ¨ is deﬁned by (3.13). Γm This yields the following factorization

N g = γ GSχ¨ g, multi Γi Γm

where G is the wave equation solution operator deﬁned by (3.5), and γΓi is the trace operator restricting the solution to Γi. Using (3.6), and combining results for the multiple scattering linear sampling theory [52] with the Born transmission eigenvalue results in Chapter4, it can be shown that the weakly scattering near ﬁeld operator

p + 2 p + 1/2 Nmulti : Hσ(R ,L (Γm)) → Hσ(R ,H (Γi)), p ∈ R and σ > 0 is bounded, injective, and has dense range. Note that when p = 0 we have the mapping properties used above. By a contradiction argument it is also possible to show as in

p + 2 [52] that for z∈ / D any approximate solution of (3.9) is such that kgz,τ kHσ(R ,L (Γm))

62 blows up as the regularization parameter in the equation goes to zero. However a complete justification of the linear sampling method, namely to describe the behavior of the approximate solution of (3.9) for z ∈ D, one need to find causal solution to the so-called interior transmission problem for the Born problem. This problem is to find w(x, t) and v(x, t) satisfying

+ wtt(x, t) − ∆w(x, t) = −m(x)v (x, t) ∈ D × R

+ vtt(x, t) − ∆v(x, t) = 0 (x, t) ∈ D × R

ξ + w(x, t) = `z,τ (x, t)(x, t) ∈ ∂D × R ξ ∂w(x,t) ∂`z,τ (x,t) + ∂ν = ∂ν (x, t) ∈ ∂D × R

w(x, 0) = wt(x, 0) = v(x, 0) = vt(x, 0) = 0 x ∈ D.

The solution of this problem remains open both for the Born approximation model and the full multiple scattering model. Under the Born approximation, Fourier or Fourier- Laplace analysis fail to work for this problem since the transformed homogeneous problem (known as the transmission eigenvalue problem) is non-selfadjoint and its complex eigenvalues may have unbounded imaginary part, though in the full scattering case it is known from [94] the conditions under which transmission eigenvalues have bounded complex part. Moreover, there are either infinitely many real eigenvalues or a sequence of complex eigenvalues may approach the real axis (see Chapter4 or [25]). Nonetheless, numerical implementation of the linear sampling method, i.e. finding a −1 p + 2 regularized solution of (3.9) shows that kgz,τ kHσ(R ,L (Γm)) remains finite inside D and is very small outside D. As such, the linear sampling method can be applied to image weak scatterers, as demonstrated in Figure 3.2.

3.4 Numerical Reconstructions To simulate forward scattering, we use the convolution quadrature-volume integral equation approach introduced in [73]. We use a Galerkin semi-discretization in

s space, as discussed in Chapter5. With this, we simulate values of u (xj, tk; yi) at some discrete values xj ∈ Γm, yi ∈ Γi, and at tk ∈ [0,T ], where T > 0 is some ﬁnal time. We

63 choose T so that at least 99% of the energy of us has left the computational domain. For simplicity, we always take n = 1 outside of D and n = 1.1 inside of D.

Using these simulations we can calculate discrete approximations to Nmulti and

Nquasi, which we denote as Nmulti and Nquasi, respectively. As we explain below, we are more interested in a partial singular value decomposition (SVD) of these near ﬁeld operators than in the matrices themselves. Since the near ﬁeld operator is a convolution, this decomposition can be calculated quickly and without explicitly forming Nmulti or

Nquasi by using a fast Fourier transform, as described in [57]. In order to avoid inverse crimes, we calculate the full multiple scattering data, not the Born approximation to this data. We further avoid inverse crimes by adding noise to reconstructions, replacing us with (1 + ρ)us where is what we refer to below as the noise level and ρ is a uniform random variable in [−1, 1]. Our reconstructions use 10-25 transmitters. This less than one-half of the amount of transmitter locations as are typically used in sampling-type schemes, though our reconstructions are of similar accuracy. This is likely both due to the increased amount of a priori data we assume about our obstacles and the fact that time domain data contains more information than the single frequency time harmonic data which is usually used. Unless otherwise noted in ﬁgure captions, obstacles are indicated by black lines. Red dots indicate location of transmitters and black dots the location of the receivers for each transmitter. Time domain data was simulated for 18 seconds with 600 time steps and with the impulse function χ(t) = sin(4t) exp(−1.6(t − 3)2). All ﬁgures have 5% added noise. All reconstructions are rescaled to [0, 1] for comparison purposes.

3.4.1 MUSIC and LSM Reconstruction with Multistatic Data

We ﬁrst use Imulti and ILSM to ﬁnd reconstructions using multistatic data. To ∗ this end, we compute Nmulti = USV , the SVD of Nmulti. Then, it is well-known ∗ −1 ∗ that the projection operator can be written as PNmulti = Nmulti(NmultiNmulti) Nmulti = −1 US (S∗S) S∗U, so long as each of the inverse matrices exists. By using the singular value decomposition, we avoid the need to construct the possibly-large matrix Nmulti

64 and can easily regularize using a spectral cut-off method, by looking for a gap in the singular values on the diagonal of S and using only the large singular values in reconstruction. We calculate Imulti(z, τ) in this way in order to reconstruct D. We calculate ILSM by solving (3.9) with a truncated SVD as in [52]. We calculate these multistatic indicator functions for two geometries in R2. This is shown in Figure 3.2, where forward data is calculated for 10 incident points and 10 measurement points. For the point obstacles on the right, we use a sampling time of τ = 1, though the reconstructions seem acceptable for any 0 < τ < T . For the larger obstacle on the left, we set τ = 5.4. By choosing τ in this way, we reconstruct both the location and shape of D, while some values of τ only reconstruct the location. It is also an open problem to automatically chose τ for linear sampling methods under the full multiple scattering model. In these figures, large values indicate the reconstructed location of objects. We now restrict our figures to the averages of small multistatic patches. We demonstrate in Figure 3.3 that small objects and be reconstructed well if either the aperture of each patch is sufficiently large or if there are a large number of patches used. In Figure 3.4, we show the same experimental set-up used for larger objects, showing that reconstructions are worse, but still acceptable.

3.4.2 Quasi-Backscattering Reconstructions In Figures 3.5-3.8 we use quasi-backscattering data to reconstruct the location of a number of objects. To construct Iquasi, we again use the SVD of the discrete near ﬁeld operator to calculate the angle between the test functions and their projection onto the range of Nquasi. We again regularize with a spectral cut-oﬀ. In Figure 3.5 we demonstrate the feasibility of the proposed technique with

−1 different geometries. In this figure, we plot only the values of (Iquasi(z)) larger than a cut-off threshold chosen visually. In some ways, this figure then represents an idealized set of reconstructions. However, there are a number of algorithms which make this choice automatically, described for example in [9, 59]. For each figure, we use 20

65 −1 −1 Figure 3.2: Plots of (ILSM(z)) (top) and (Imulti(z)) (bottom) for two different geometries. transmitters, each with 4 receivers, and set δ = π/100. For each image, we use the temporal sampling point τ = 1, though as before this choice does not seem to seriously affect reconstructions. In Figure 3.6 we demonstrate the dependence of reconstructions on the number of transmitter locations. We show both time harmonic and time dependent reconstructions. As expected, reconstructions become more accurate when more transmitter locations are used. Furthermore, time dependent reconstructions are more accurate than time harmonic reconstructions, until a sufficient number of transmitters are used. In the limiting case as δ → 0, the quasi-backscattering set-up becomes a pure

66 Figure 3.3: Multistatic patch reconstructions of the same geometry of small circles, indicated by black lines. (Top) Four patches are used with 5 transmitters and receivers each. From left-to-right, the aperture of each patch decreases from π/2 to π/4 to π/8. (Bottom) The same experiment as top but with 10 patches. Each set of circles indicates the location of transmitters and receivers in each patch. Transmitters and receivers in the same patch each have the same color. backscattering set-up. In Figure 3.7 we show numerical examples of pure backscattering. There are a total of 30 transmitter locations, each with 1 receiver located at the same point. While reconstructions are not as sharp as, e.g., the full aperture multistatic reconstructions, there is a clear indication of object location. Finally, we compare limited aperture reconstructions from multistatic data to limited aperture reconstructions from patch and quasi-backscattering data. As expected, Figure 3.8 demonstrates that the multistatic reconstructions are superior to the patch and quasi-backscattering reconstructions, which are somewhat similar. In- deed, the quasi-backscattering algorithm results in reconstructions which are noisier and, the case of three point obstacles, only clearly reconstruct the two obstacles nearest the transmitter and receiver arrays, incorrectly indicating an extra obstacle in the

67 Figure 3.4: Multistatic patch reconstructions of the same geometry of medium-sized ellipses, indicated by black lines. (Top) Four patches are used with 5 transmitters and receivers each. From left-to-right, the aperture of each patch decreases from π/2 to π/4 to π/8. (Bottom) The same experiment as top but with 10 patches. Each set of circles indicates the location of transmitters and receivers in each patch. Transmitters and receivers in the same patch each have the same color. bottom right. Patch data reconstructions are also noisier than multistatic reconstructions, but do not have the same issues as quasi-backscattering reconstructions do with three point obstacles. The patch reconstructions do not separate the larger ellipses as eﬀectively as the quasi-backscattering reconstructions. Note that both the multistatic and patch reconstructions required a careful selection of τ = 5 in order to produce optimal results, while the quasi-backscattering data did not need any such choice.

68 −1 Figure 3.5: Plots of (Iquasi(z, τ)) for four diﬀerent geometries.

69 ˆ −1 −1 Figure 3.6: Plots of (Iquasi(z, τ)) (top) and (Iquasi(z, τ)) (bottom), with a diﬀerent number of transmitters in each row. On the left there are 5 transmitters, in the middle there are 10 transmitters, and the right there are 15 transmitters. Time harmonic data was computed with wavenumber k = 3.

70 −1 Figure 3.7: Backscattering reconstructions using Ibackscattering(z, τ) for two diﬀerent geometries. In both ﬁgures, 30 transmitters are used and data is measured only at the location of the transmitter. Time domain data was simulated for 14 seconds with 480 time steps.

71 −1 Figure 3.8: Limited aperture reconstructions using multistatic data with Imulti(z, τ) (top), multistatic patch data with two patches (middle), and quasi-backscattering data −1 with Iquasi(z, τ) (bottom). In both ﬁgures, 19 transmitters are used and in the case of quasi-backscattering data, 4 receivers were used.

72 Chapter 4

THE BORN TRANSMISSION EIGENVALUES PROBLEM

As discussed previously, central to the justiﬁcation of the linear sampling method is the interior transmission problem which consists of ﬁnding v, w ∈ L2(D), and nonzero k ∈ C which satisfy

∆w + k2nw = 0 in D

∆v + k2v = 0 in D (4.1)

∂w ∂w w = v, ∂ν = ∂ν on ∂D.

We call k a transmission eigenvalue if there exists a nontrivial solution to (4.1). As before, the squared refractive index of a penetrable obstacle is represented by the

(possibly complex-valued) function n ∈ L∞(D), Re(n) > 0, Im(n) ≥ 0, and D ⊂ R3 is a bounded domain with connected complement deﬁned by D = supp(n−1). Denote the boundary of D by ∂D which we assume to be C1 and let ν be the outward unit normal vector to ∂D. In this chapter, we analyze problem (4.1) under the Born approximation, deriving properties which are important when studying the time domain linear sampling algorithm in the Born regime.

4.1 Introduction The justification of the linear sampling method requires that the set of associated transmission eigenvalues be discrete. The location of real transmission eigenvalues can be determined from scattered far field data [22] and this knowledge can be used to obtain information about the properties of n [24], [47]. Though typically discussed for time-harmonic problems, the linear sampling method can be justified for transient

73 problems in the time domain if the transmission eigenvalues in the frequency domain form a discrete set and lie in a strip parallel to the real axis [52]. Conditions under which the latter is true were derived for the eigenvalues with spherically symmetric eigenfunctions for spherically stratified media [37]. The general case was proven recently in [94]. The question of the discreteness and location in the complex plane of transmission eigenvalues for the Born approximation model, which is central to the justification of the Born approximation linear sampling method in the time domain, motivated the current investigation. For the justification of the Born approximation in the time domain we refer the reader to Chapter3,[70] and the references therein, whereas for the mathematical understanding of inverse scattering theory in the Born regime see [65], [68], [78] and the references therein. In this chapter we study an approximation to (4.1) in the case that n − 1 is small in magnitude. In particular, assume n(x) = 1 + m(x) for 1 and kmk∞ = 1 ∂w and that solutions take the form w(x) = w0(x) + u(x). The function u = ∂ =0 is the

ﬁrst term in the well-known Born approximation. When v is of the same order as w0, (4.1) becomes

∆u + k2u = −k2mv in D (4.2)

∆v + k2v = 0 in D (4.3)

∂u u = ∂ν = 0 on ∂D. (4.4)

i In what follows we always take m(x) = m1(x)+ k m2(x) for real-valued m1, m2 ∈ ∞ L (D) with m2 ≥ 0 in D. When m2 is not identically zero, this corresponds to scattering from an absorbing medium embedded in free space with speed of sound normalized to one. Note that this form of contrast appears after taking the Fourier- Laplace transform of the wave equation with damping term. We will always assume that there is some α > 0 so that k ∈ C|z|>α := {z ∈ C : |z| > α > 0}.

2 Deﬁnition 2. All k ∈ C|z|>α such that there exists a nontrivial solution u ∈ H0 (D), v ∈ L2(D) to (4.2) are called Born transmission eigenvalues.

74 Real Born transmission eigenvalues do not exist for (4.2) so long as either m1 or m2 are strictly positive. Indeed, we follow the argument from [38] and assume there are u, v, and k ∈ R which satisfy (4.2). Multiplying (4.2) by its complex conjugate, dividing by m, and integrating over D yields

R m 2 2 2 R 2 D |m|2 |(∆u + k u)| dV = −k D (∆u + k u)v dV (4.5) 2 R 2 2 R ∂v ∂u = −k D u (∆v + k v) dV − k ∂D u ∂ν − v ∂ν ds = 0.

The last line vanishes from (4.3) and (4.4). By the assumptions on m, u satisfies ∆u + k2u = 0. Along with vanishing Cauchy data on ∂D this gives that u ≡ 0 and hence by (4.2), v ≡ 0. Note that, since v ∈ L2(D) with ∆v ∈ L2(D) and u ∈ H2(D), the boundary terms in (4.5) are understood in the sense of the duality H3/2(∂D),H−3/2(∂D) and H1/2(∂D),H−1/2(∂D), respectively. Indeed if v ∈ L2(D) with ∆v ∈ L2(D) then its trace v ∈ H−1/2(∂D) is defined by duality using the identity Z hv, τiH−1/2(∂D),H1/2(∂D) = (v∆w − w∆v) dx D where w ∈ H2(D) is such that w = 0 and ∂w/∂ν = τ on ∂D. Similarly, the trace of ∂v/∂ν ∈ H−3/2(∂D) is defined by duality using the identity

∂v Z , τ = − (v∆w − w∆v) dx ∂ν H−3/2(∂D),H3/2(∂D) D where w ∈ H2(D) is such that w = τ and ∂w/∂ν = 0 on ∂D. With the exception of the above argument, the existence, discreteness, and distribution in the complex plane of Born transmission eigenvalues for (4.2) has not been studied. We study each of these properties in this chapter. In Section 4.2 we give results for spherically stratified media. We show that there are contrasts whose associated eigenvalues are real. We also show that, except for special cases, the complex parts of these eigenvalues are unbounded. In Section 4.4, we discuss the case when the media is no longer spherically stratified. We show that eigenvalues to (4.2) are discrete when they exist and have no finite accumulation points.

75 4.2 Spherically Stratiﬁed Media In this section we consider D to be a ball and m to be spherically symmetric. Under these assumptions, we are able to obtain considerable information about Born transmission eigenvalues. We begin our study by deriving a function whose zeros

1 coincide with Born transmission eigenvalues. By deﬁning w = k2 u + v, we can write (4.2) as

∆w + k2w = −mv in D (4.6)

∆v + k2v = 0 in D (4.7)

∂v ∂w v = w, ∂ν = ∂ν on ∂D. (4.8)

We will use this formulation in this section.

3 Theorem 5. Let R > 0 be ﬁxed and assume D = BR(0) ⊂ R and m = m(|x|). Let j`(r) be a spherical Bessel function of the ﬁrst kind of order `. Then the zeros of the function Z R k 2 2 d`(k) := − 2 ρ m(ρ)j`(kρ) dρ, ` = 0, 1, 2,... R 0 are Born transmission eigenvalues k ∈ C|z|>α.

Proof. Let r = |x| > 0 andx ˆ = x/|x| ∈ S2 := {x ∈ R3 : |x| = 1}. Then there exist m unique α` ∈ C, |m| ≤ ` such that (4.3) has solutions of the form [36]

∞ ` X X m m 2 v(x) = α` j`(kr)Y` (ˆx), 0 ≤ r ≤ R, xˆ ∈ S `=0 m=−`

m where Y` (ˆx) are spherical harmonics of degree ` and order m. Solutions to (4.2) take the form ∞ ` X X m m 2 w(x) = β` z`(r)Y` (ˆx), 0 ≤ r ≤ R, xˆ ∈ S `=0 m=−` m for unknown constants β` and functions z`(r) where lim z`(r) < ∞. The orthogonality r→0 of spherical harmonics reduces the deﬁning equation for each z`(r) to

m 00 2 0 2 `(` + 1) α` z` (r) + z`(r) + k − 2 z`(r) = − m m(r)j`(kr). (4.9) r r β`

76 We proceed by the method of variation of parameters. Let y`(r) be a spherical Bessel function of the second kind and of order `. Then since the set of linearly independent solutions to the homogeneous equation, {j`(kr), y`(kr)}, has Wronskian equal to k−1r−2,

m α` R r 2 z`(r) = c1 + m kρ m(ρ)j`(kρ)y`(kρ) dρ j`(kr) β` 0 m α` R r 2 2 + c2 − m kρ m(ρ)j` (kρ) dρ y`(kr). β` 0

We can ensure lim z`(r) < ∞ by taking c2 = 0. Deﬁne f` and g` by r→0 Z r Z r 2 2 2 f`(r) = kρ m(ρ)j`(kρ)y`(kρ) dρ and g`(r) = kρ m(ρ)j` (kρ) dρ. 0 0

0 0 Using the identity f`(R)j`(kR)−g`(R)y`(kR) = 0, the boundary conditions (4.8) imply that there is a nontrivial solution to (4.6) if and only if   j`(kR)(f`(R) − 1)j`(kR) − g`(R)y`(kR) d`(k) = det   = 0. 0 0 0 kj`(kR) k(f`(R) − 1)j`(kR) − kg`(R)y`(kR)

Expanding this determinant shows that Born transmission eigenvalues correspond to the zeros of Z R 2 2 d`(k) = −k ρ m(ρ)j` (kρ) dρ. 0

Two examples suggest that the behavior of transmission eigenvalues diﬀers when m changes sign compared to when it does not. In what follows, we always take R = 1 for simplicity.

Example 1.

i We ﬁrst consider an example where m1 is strictly positive. Let m = 1 + k . We claim that the complex part of the zeros of d0(k) are unbounded as |k| → ∞. In the case of the full interior transmission problem without the Born approximation, we know of no examples of transmission eigenvalues with unbounded complex part.

77 sin r Since j0(r) = r , Z 1 sin2(kρ) (k + i) sin(2k) d0(k) = − (k + i) 2 dρ = 2 − 1 . 0 k 2k 2k

Hence, there are no non-zero real solutions to d0(k) = 0. Let k = x + iy for x, y ∈ R.

Some manipulation of the equation d0(k) = 0 reveals that d0(x + iy) = 0 if and only if either k = −i or

x = sin(x) cosh(y) and y = cos(x) sinh(y).

Taking the modulus of the ﬁrst equation shows |x| ≤ | cosh(y)|, which is possible only if |y| → ∞ as |x| → ∞. We show below that there are inﬁnitely many solutions to d0(k) = 0 which form a discrete set in the complex plane. Hence, there are Born transmission eigenvalues with arbitrarily large imaginary part.

Example 2.

When m changes sign in D, the behavior of transmission eigenvalues also changes.

For instance, real eigenvalues can exist when m1 is no longer bounded away from zero. Let m(r) = 1 − 2r. Then,

sin(k)(sin(k) − k cos(k)) d (k) = . 0 2k3

Both sin(k) = 0 and sin(k) = k cos(k) have infinitely many solutions, all of which are real. As such there are infinitely many real transmission eigenvalues associated with m. Numerical experiments suggest that d`(k) = 0 has both real and complex solutions when ` > 0. We now turn to a general theory to better explain these examples. We begin by demonstrating that for spherically-stratified media there are always infinitely many transmission eigenvalues.

Theorem 6. If D = BR(0) and the contrast is radially-dependent then d0(k) has inﬁnitely many zeros.

78 Proof. We claim that d0(k) is an entire function of k of order at most one and ﬁnite type. Indeed, expanding into a Taylor series about k = 0,

∞ X (−1)n22n−1 Z 1 Z 1 d (k) = k2n−1 m (ρ)ρ2n dρ + ik2n−2 m (ρ)ρ2n dρ . 0 (2n)! 1 2 n=1 0 0

This series converges for all k since km1kL∞ , km2kL∞ are bounded and so d0(k) is entire. Moreover, it is easily veriﬁed that there exists a positive constant A such that

max |d0(z)| ≤ A cosh 2|z| |z|=r for r suﬃciently large. This corresponds to d0(k) being of at most order one and type two (see, e.g., [40]).

Assume now to the contrary that d0(k) has only 0 ≤ M < ∞ zeros (not necessarily distinct). Since d0(k) is an entire function of at most order one and type two, the Hadamard factorization theorem [40] gives

M Y k d (k) = kneak+b 1 − 0 k j=1 j where n is an integer, a, b are constants, and kj, j = 0, 1,...,M, are the zeros of d0(k) (if there are no zeros, the ﬁnite product is replaced by the constant one). This factorization implies that d0(k) does not tend to zero for both k → ∞ and k → −∞ on the real axis. However, Z 1 Z 1 2 2 1 2 d0(k) = − kρ m(ρ)j0 (kρ) dρ = − m(ρ) sin (kρ) dρ. 0 k 0

i Since m = m1 + k m2, d0(k) → 0 as k → ±∞ on the real axis. This contradiction implies that d0(k) has inﬁnitely many zeros.

As demonstrated by the above examples, the location in the complex plane of Born transmission eigenvalues appears to depend on whether the contrast m changes sign or not. Two results explain these examples more precisely.

1 Theorem 7. Assume m1 ∈ C ([0, 1]). If the zeros of d0(k), k ∈ C|z|>α, are contained R 1 in a horizontal strip of the complex plane then 0 m1(ρ) dρ = 0.

79 R 1 R 1 Proof. Let α = 0 m1(ρ) dρ, β = 0 m2(ρ) dρ, and k = x + iy for x, y ∈ R. From the

identity sin(x + iy) = sin(x) cosh(y) + i cos(x) sinh(y) we have that for k ∈ C|z|>α,

R 1 −2Re(kd0(k)) = α − 0 m1(ρ) cos(2xρ) cosh(2yρ) dρ − (4.10) 1 R 1 x2+y2 βy − 0 m2(ρ)(x sin(2xρ) sinh(2yρ) + y cos(2xρ) cosh(2yρ)) dρ .

Assume that the zeros of d0 lie in a horizontal strip of the complex plane so that

if d0(x + iy) = 0 then y ∈ [a, b] for some −∞ < a < b < ∞. From (4.10), if y ∈ [a, b] then there is a constant C > 0 depending only on a and b so that

C Z 1 Z 1 |2Re((x + iy)d0(x + iy)) + α| ≤ |m1(ρ)| dρ + |m1(1)| + |m2(ρ)| dρ . |x| 0 0

This demonstrates that as k grows on the real axis, Re(kd0(k)) becomes arbitrarily close to −α/2. However, since there exist inﬁnitely many transmission eigenvalues, there exist transmission eigenvalues with arbitrarily-large real part and complex part

in [a, b] so that Re(kd0(k)) = 0. Since Re(kd0(k)) is a continuous function, this is a contradiction unless α = 0.

2 R 1 Theorem 8. If m2 ≡ 0, m1 ∈ C ([0, 1]), m1(1) 6= 0, and 0 m1(r) dr = 0, then d0(k) has inﬁnitely many real zeros.

0 00 Proof. Integrating-by-parts twice and using the assumption that m1(ρ) and m1(ρ) are bounded on 0 ≤ ρ ≤ 1, we have that for k > 0 Z 1 2 1 4k d0(k) = 2k m1(ρ) cos(2kρ) dρ = m1(1) sin(2k) + O . 0 k 2 Therefore, for large-enough k > 0, 4k d0(k) – and hence d0(k) itself – has inﬁnitely many real zeros.

Note that the restriction m1(1) 6= 0 may not be necessary. For example, if

m(ρ) = sin(2πρ) then all the conditions for Theorem (8) hold, but m1(1) = 0. Nonethe- less, π sin(k)2 d (k) = , 0 2k π2 − k2

80 0.003

0.002

0.001

15 20 25 30 35

-0.001

Figure 4.1: Plot of the function d0(k) associated with m = 1 − 1.95r. When d0(k) crosses the real axis, k is a transmission eigenvalue.

so that d0(k) = 0 whenever sin(k) = 0. Numerical examples suggest that (possibly ﬁnitely many) real zeros exist when R 1 0 m1(ρ) dρ 6= 0 as well. As an example, Figure 4.1 shows d0(k) on the real axis for m(ρ) = 1 − 1.95ρ. The intersections of d0(k) with the real k-axis correspond to the location of real Born transmission eigenvalues.

4.3 Transmission Eigenvalue Free Regions Before considering the case of general geometries and contrasts, we demonstrate that real, constant contrasts have a large region in which Born transmission eigenvalues cannot appear. While these results are consistent with the fact that there are no real Born transmission eigenvalues for real constants which do not change sign, we show that they may become arbitrarily close to the real axis. A recent result [94] gives an optimal characterization of the location of transmission eigenvalues for the strongly- scattering problem with real n ∈ C∞(D) such that n(x) 6= 1 for x ∈ ∂D. In particular, [94] along with a related result from [93] gives that transmission eigenvalues all fall in a strip of the real axis, {k ∈ C , Re(k) > 0 : |Im(k)| < C} for some positive and ﬁnite C. Note that taking m(x) = 1 in the spherically symmetric case immediately suggests that this theorem does not apply to the case of Born transmission eigenvalues; a short calculation for this choice of m gives the same results as Example 1. However, the case

81 of constant contrast is an unusual one in that the location of transmission eigenvalues is unchanged by the value of that constant. Further investigation is neccessary to apply ideas from [94] to the Born transmission eigenvalue problem. To investigate the region where transmission eigenvalues can and cannot exist, denote k = x + iy for a, b ∈ R and k2 =: τ = a + ib. Note that under this notation, a = x2 − y2 and b = 2xy. For simplicity, we only consider x, y ≥ 0 in this section.

Assume k ∈ C|z|>α is a Born transmission eigenvalue for m so that (4.2) is satisﬁed. Multiplying (4.2) by ∆u + τu, integrating over D, and dividing the result by m > 0 yields 1 Z Z − |∆u + τu|2 dV = τ v∆u + τu dV. m D D Adding and subtracting the quantity τu to the right-hand-side and integrating-by-parts yields

1 Z Z |∆u + τu|2 dV = −τ u (∆v + τv) + (τ − τ)uv dV m D D Z = −τ(τ − τ) uv dV D 1 Z = (τ − τ) u (∆u + τu) dV. m D Simplifying this equality yields

2 2 2 2 2 2 k∆u + τukL2(D) − 2b kukL2(D) + 2abkukL2(D) − 2bk∇ukL2(D)kL2(D) i = 0.

Assume that there is a positive constant C1 so that b > C1 > 0. Then, the imaginary component of the previous equality simplifies to k∇uk2 2 2 L2(D) x − y = a = 2 ≥ Λ(D) > 0 kukL2(D) where Λ(D) is the first Dirichlet eigenvalue of the Laplacian on the domain D. Here we have used the Poincaréinequality. As such, transmission eigenvalues associated with m

2 2 can fall between two hyperbola, {k = x+iy ∈ C|z|>α : xy > C1 and x −y > C2}. For an example of the shape of this set, see Figure 4.2. Note that while we have assumed xy > C1 > 0, which excludes the case of k ∈ R, Born transmission eigenvalues can

82 25

0 0 5 10 15 20

Figure 4.2: Sample of the region in which transmission eigenvalues can appear for a real constant contrast. become arbitrarily close to the real axes as x → ∞. Their imaginary part also can grow without bound.

4.4 Transmission Eigenvalues for General Shapes and Contrasts In this section, we return to the case of more general contrasts such that D =

1 1 supp(n − 1) is a C domain. Make the change of variables u 7→ k2 u in (4.2). Then we 2 2 look for nontrivial solutions (u, v) ∈ H0 (D) × L (D) to

∆u + k2u = −mv in D (4.11)

∆v + k2v = 0 in D (4.12)

i ∞ where k ∈ C|z|>α and m(x) = m1(x) + k m2 for real-valued m1, m2 ∈ L (D). If there exists a nontrivial solution to (4.11) then k is a Born transmission eigenvalue as deﬁned in Deﬁnition2 and conversely. We now demonstrate that Born transmission eigenvalues form at most a discrete set in C|z|>α. Here we adapt to the case of absorbing weak scattering media the approach developed by Kirsch in [67] which revisits [90]. To this end, we must impose

83 restrictions on the contrast in a neighborhood of the boundary ∂D. In what follows, let N ⊂ D be a neighborhood of ∂D so that ∂D ⊂ N . We are interested in m(x) which have the same sign for all x ∈ N . Deﬁne

∗ m∗ = inf m1(x) and m = sup m1(x). x∈N x∈N

In the following theorems, we always take 0 < m∗ < 1. However, under the change of variables u 7→ −u, all results also hold for −1 < m∗ < 0.

2 2 Deﬁne the Hilbert space X(D) = H0 (D) × L (D) equipped with the norm

k(u, v)kX(D) = kukH2(D) + kvkL2(D) and corresponding inner product h·, ·iX(D). The variational form of (4.11) is to ﬁnd (u, v) ∈ X(D) such that Z Z ∆ϕ + k2ϕ v dx + ∆u + k2u ψ + mvψ dx = 0 for all (ϕ, ψ) ∈ X(D). D D

For each k ∈ C|z|>α, deﬁne the sesquilinear form Bk : X(D) × X(D) → C by Z Z 2 2 Bk(u, v; ϕ, ψ) := ∆ϕ + k ϕ v dx + ∆u + k u ψ + mvψ dx. D D

From the Riesz representation theorem, there exists a bounded linear operator Bk : X(D) → X(D) which satisﬁes

Bk(u, v; ϕ, ψ) = hBk(u, v), (ϕ, ψ)iX(D).

Born transmission eigenvalues are the values k ∈ C|z|>α such that Bk(u, v) = 0 does not have a unique solution (u, v) ∈ X(D).

We decompose the above sesquilinear form as Bk(u, v; ϕ, ψ) = Ak(u, v; ϕ, ψ) + Ck(v, ψ) where Z Z 2 2 Ak(u, v; ϕ, ψ) = ∆ϕ + k ϕ v dx + ∆u + k u ψ + m1vψ dx D D and i Z Ck(v; ψ) = m2vψ dx. k D

Again by the Riesz representation theorem, there are bounded linear operators Ak : 2 2 X(D) → X(D) and Ck : L (D) → L (D) so that

Ak(u, v; ϕ, ψ) = hAk(u, v), (ϕ, ψ)iX(D) and Ck(v; ψ) = hCk(v), ψiL2(D).

84 We will make use of the analytic Fredholm theory to determine that if transmission eigenvalues exist then they are discrete in the complex plane.

Theorem 9. For any k1, k2 ∈ C, the operator Ak1 − Ak2 is compact.

Proof. We will show that Ak1 − Ak2 maps weakly convergent sequences in X(D) to strongly convergent sequences in X(D). Assume that (uj, vj) converges weakly in X(D) to (0, 0) and let (ϕ, ψ) ∈ X(D). Then, Z 2 2 (Ak1 − Ak2 )(uj, vj; ϕ, ψ) = k1 − k2 vjϕ + ujψ dx dx. D We bound each of these terms separately. First, by the Cauchy-Schwarz inequality, Z

ujψ dx ≤ kujkL2(D)kψkL2(D). D 1 Next, deﬁne zj ∈ H (D) to satisfy ∆zj = vj in D and zj = 0 on ∂D. By Green’s second identity and the Cauchy-Schwarz inequality, Z Z Z

vjϕ dx = ∆zjϕ dx = zj∆ϕ dx ≤ kzjkL2(D)kϕkH2(D). D D D Therefore,

kAk1 − Ak2 kX(D) = sup06=(ϕ,ψ)∈X(D) |(Ak1 − Ak2 )(u, v; ϕ, ψ)| ≤ C kujkL2(D) + kzjkL2(D)

2 where C depends on k1 and k2. Since uj * 0 in H0 (D), Rellich’s compact embedding 2 1 2 theorem gives that uj → 0 in L (D). Moreover, zj * 0 in H (D) so zj → 0 in L (D).

Hence, (Ak1 − Ak2 )(u, v) converges strongly to zero in X(D) proving compactness of the operator.

Before continuing, we need a technical lemma from [67].

∞ Lemma 5. Assume m ∈ L (D) is such that either 0 < m∗ < 1 and m2(x) = 0 for all x ∈ N . Then there exist constants γ > 0 and c > 0 so that for all k = iκ0, κ0 > 0, Z Z 2 −γκ0 2 |v| dx ≤ ce m1|v| dx D\N N

2 2 for all solutions v ∈ L (D) of ∆v − κ0v = 0 in D.

85 We now prove a Babuˇska-Brezzi inf-sup condition which will be used to establish that there is a κ0 ∈ C so that (Aκ0 + Ck) is invertible with bounded inverse for all k ∈ C|z|>α (c.f. [67]). We say that a sesquilinear form L : X(D) × X(D) → C satisﬁes the inf-sup condition if there exists a c > 0 so that for all (u, v) ∈ X(D), |L(u, v; ϕ, ψ)| sup ≥ ck(u, v)kX(D). (4.13) 06=(ϕ,ψ)∈X(D) k(ϕ, ψ)kX(D)

∞ Theorem 10. Assume that m ∈ L (D) and 0 < m∗ < 1. Then for all k ∈ C|z|>α,

1. if m2(x) ≡ 0 for x ∈ N then there is a κ∗ > 0 so that for all κ0 > κ∗, Aiκ0 + Ck satisﬁes the inf-sup condition (4.13).

2. if Im(k) > 0 and there is some β > 0 so that m2(x) > β > 0 for x ∈ D then p for every 0 < κ0 < λ0(D), Aκ0 + Ck satisﬁes the inf-sup condition (4.13) where

λ0(D) > 0 the smallest Dirichlet eigenvalue for the negative Laplacian on the domain D.

Proof. In both cases, we assume by contradiction that there does not exist a constant c > 0 such that (4.13) holds. Then there is a sequence {(uj, vj)} ∈ X(D) with k(uj, vj)kX(D) = 1 such that |(A + C )(u, v; ϕ, ψ)| sup κ k → 0 as j → ∞ (4.14) 06=(ϕ,ψ)∈X(D) k(ϕ, ψ)kX(D) p where κ = iκ0 in Case 1 and κ = κ0 for 0 < κ0 < λ0(D) in Case 2. There is a weakly- convergent subsequence, still denoted by {(uj, vj)}, such that uj * u and vj * v for some (u, v) ∈ X(D). By (4.14), we have that ∆u + κ2u = −mv and ∆v + κ2v = 0 in D. We ﬁrst prove Case 1. We claim that (u, v) = (0, 0) in D. Indeed, Z Im(k) 2 0 = Re((Aiκ0 + Ck)(u, v; −u, v)) = m1 + 2 m2 |v| dx. (4.15) D |k|

Since 0 < m∗ < 1 and m2 ≡ 0 in N , Lemma5 and equation (4.15) imply that there is a suﬃciently-large κ∗ ∈ R so that if κ0 > κ∗ and k ∈ C|z|>α then Z m |v|2 dx = R m (x) + Im(k) m |v|2 dx 1 D\N 1 |k|2 2 N R 2 1 R 2 ≤ kmkL∞(D) D\N |v| dx ≤ 2 N m1|v| dx.

86 Thus, v ≡ 0 in N and hence by unique continuation v ≡ 0 in D. Since 0 = −(Aiκ0 + R 2 2 2 Ck)(u, 0; 0, u) = D |∇u| + κ0|u| dx, we have that u ≡ 0 in D as well.

We continue by proving a contradiction with k(u, v)kX(D) = 1. To this end, deﬁne N 0 to be a neighborhood of ∂D such that N 0 ⊂ N ∪ ∂D. Choose a nonnegative

∞ 0 cutoﬀ function η ∈ C (D) so that η = 0 in D\N and η = 1 in N . Since {(ηuj, −ηvj)} is bounded in X(D) and m2 ≡ 0 in N , assumption (4.14) yields

(Aiκ0 + Ck)(uj, vj; ηuj, −ηvj)

R 2 2 2 = N (∆(ηuj) − κ0ηuj) vj − η (∆uj − κ0uj) vj − m1η|vj| dx → 0 as j → ∞. Taking the real part of the above, Z 2 Re ujvj∆η + 2vj∇η · ∇uj − m1η|vj| dx → 0. N

2 1 2 From the compact embedding of H (D) in H (D), uj * 0 in H0 (D) implies that kujkH1(D) → 0 as well. Applying the Cauchy-Schwarz inequality then yields Z 2 m1η|vj| dx → 0, as j → ∞. (4.16) N

0 Since m1 is strictly positive in N and m1η ≥ δ > 0 in N for some δ > 0, we have that 2 0 vj → 0 in L (N ). Now pick another neighborhood of ∂D called N 00 with N 00 ⊂ N 0 ∪ ∂D. Deﬁne a new nonnegative cutoﬀ function with the same name, η ∈ C∞(D) with η = 0 in N 00

0 2 2 and η = 1 in D\N . There is a zj ∈ H (D) which is a solution to ∆zj − κ0zj = vj in

D and zj = 0 on ∂D. Since {(ηzj, 0)} is bounded in X(D), (4.14) gives

R 2 (Aiκ0 + Ck)(uj, vj; ηzj, 0) = D\N 00 (∆(ηzj) − κ0ηzj) vj dx R 2 = D\N 00 η|vj| + 2vj∇η · ∇zj + vjzj∆η dx → 0

2 2 1 as j → ∞. Since vj * 0 in L (D), zj * 0 in H (D) and so zj → 0 in H (D). By applying the Cauchy-Schwarz inequality and using the deﬁnition of η, we conclude that

2 00 2 vj → 0 on L (D\N ). Along with the previous argument, vj → 0 in L (D).

87 2 Finally, letting ϕ = 0 and ψ = ∆uj + k0uj in (4.14) yields

1 R 2 2 2 2 |uj − κ0uj| + m1vj (∆uj − κ0uj) dx k∆uj −κ0uj kL2(D) D 2 2 R ∆uj −κ0uj = k∆uj − κ0ujkL2(D) + m1vj 2 dx → 0. D k∆uj −κ0uj kL2(D)

2 2 2 Since vj → 0 in L (D), we have from Cauchy-Schwarz that ∆uj − κ0uj → 0 in L (D) 2 2 2 as well. Since uj * 0 in H0 (D), u → 0 in L (D). Hence, ∆uj → 0 in L (D). As 2 2 k∆ujkL2(D) is equivalent to kujkH2(D) for uj ∈ H0 (D), we have that uj → 0 in H (D).

Therefore, (uj, vj) → (u, v) = (0, 0) in X(D), contradicting the claim that k(u, v)kX(D) = 1.

Now consider Case 2. We again ﬁrst show that (u, v) = (0, 0). Since κ0 ∈ R, Z 1 2 0 = Im((Aκ0 + Ck))(u, v; u, v) = m2|v| dx. κ0

Since m2(x) > 0 for all x ∈ D, we conclude that v ≡ 0 in D. Next, integrating-by-parts 2 and using the Poincar´einequality for functions in H0 (D),

R 2 R 2 2 2 0 = (Aκ + Ck)(u, 0; 0, u) = D(∆u + κ0u)u dx = D −|∇u| + κ0|u| dx

2 2 ≤ (κ0 − λ0(D))kukL2(D). p The assumption that 0 < κ0 < λ0(D) gives u = 0 in D. After replacing iκ0 by κ0, nearly the same arguments as in the ﬁrst part of this proof give that (uj, vj) → (u, v) =

(0, 0) in X(D) which contradicts the assumption k(u, v)kX(D) = 1. In particular, (4.16) is replaced by Z Im(k) 2 m1 + 2 m2 η|vj| dx → 0 N |k| 2 0 which gives that vj → 0 in L (N ) since for all Im(k) > 0 there is an α ∈ R so that Im(k) m1(x) + |k|2 m2(x) ≥ α > 0 for all x ∈ N . Moreover, since κ0 is not a Dirichlet 2 eigenvalue, the equation ∆zj + κ0zj = vj in D with zj = 0 on ∂D has a solution 2 zj ∈ H (D). Following the proof from Case 1 gives the desired result.

To ﬁnish, we must show that there are k ∈ C|z|>α which are not Born transmission eigenvalues.

88 ∞ Theorem 11. Assume that m ∈ L (D), 0 < m∗ < 1, and m2(x) ≡ 0 for x ∈ N .

Then for suﬃciently large κ0 > 0 the operator Biκ0 : X(D) → X(D) is invertible with bounded inverse.

Proof. Let κ0 > 0. If m2 ≡ 0, Biκ0 = Aiκ0 . By Theorem 10, Aiκ0 is invertible with bounded inverse (see, e.g., [67]) which gives the result. Assume then that m2 is not identically zero in D. For 0 < κ1 6= κ0,

Biκ0 = (Aiκ1 + Ciκ0 ) + (Aiκ0 − Aiκ1 ) which, by Theorems9 and 10, is an invertible operator plus a compact operator when

κ1 is suﬃciently large. By Fredholm theory, it is suﬃcient to show that Biκ0 is injective.

Assume by contradiction that there is a sequence κ0,j → ∞ and functions (uj, vj) ∈

X(D) with k(uj, vj)kX(D) = 1 so that Biκ0,j (uj, vj) = 0.

−γκ0,j Let αj = ckmkL∞(D)e where c, γ > 0 are the constants in Lemma5. By this Lemma, Z 2 R 2 R 2 m1|vj| dx ≥ N m1|vj| dx − D\N |m1||vj| dx D R 2 ≥ (1 − αj) N m1|vj| dx > 0 since 0 < m∗ < 1.

On the other hand, since Biκ0,j (uj, vj) = 0, Z 1 2 0 = Re(Biκ0,j (uj, vj; −uj, vj)) = m1 + m2 |vj| dx D κ0,j R 2 Since m2 is not identically zero, this implies that D m1|vj| dx < 0 which is a contradiction.

With this in hand, the analytic Fredholm theory gives the following.

i ∞ Theorem 12. Assume that m = m1 + k m2 ∈ L (D) for k ∈ C and real-valued ∞ m1, m2 ∈ L (D) with m2(x) ≥ 0 for all x ∈ D. Assume further that there is a neighborhood N ⊂ D of ∂D with ∂D ⊂ N so that 0 < inf m1(x) < 1. If: x∈N

1. m2(x) ≡ 0 for x ∈ N then there is at most a discrete set of Born transmission eigenvalues k ∈ C|z|>α which do not have an accumulation point with ﬁnite modulus.

89 2. there is some β > 0 so that m2(x) > β > 0 for x ∈ D then there is at most a discrete set of Born transmission eigenvalues k ∈ C|z|>α with Im(k) > 0 which do not have an accumulation point with ﬁnite modulus.

Proof. By Theorem 11, if Hypothesis 1 holds and κ1 > 0 is suﬃciently large then Biκ1 is invertible. Moreover, from the arguments in the Introduction, if Hypothesis 2 holds then there are no real Born transmission eigenvalues, so if k ∈ R then Bk is invertible.

As such, there exist κ ∈ C|z|>α which are not transmission eigenvalues.

Under either hypothesis, Theorem 10 gives that for some κ0 ∈ C and k ∈ C,

Aκ0 + Ck is invertible with bounded inverse. In particular, under Hypothesis 1, this is true for any k ∈ C|z|>α and for κ0 > κ∗ > 0 with κ∗ suﬃciently large. Under p Hypothesis 2, it is true for k ∈ C|z|>α with Im(k) > 0 and 0 < κ0 < λ0(D), where

λ0(D) is the smallest Dirichlet eigenvalue for the negative Laplacian on D. Then for all (u, v) ∈ X(D) and for appropriate k, κ0,

−1 −1 (Aκ0 + Ck) Bk(u, v) = (u, v) + (Aκ0 + Ck) (Ak − Ak0 )(u, v).

Applying Theorem9 shows that this is of the form of an identity operator plus a compact operator. Since there are values of k which are not transmission eigenvalues, the analytic Fredholm theory then gives the result.

Remark 1. In this chapter we do not discuss the existence of transmission eigenvalues. The approach taken in [26] provides monotonicity of transmission eigenvalues in terms of the refractive index, but only works for real eigenvalues and one-sign contrast. This is precisely the case when real Born transmission eigenvalues do not exist. In principle, for real contrasts, the approach of Robbiano [85] can be applied to the Born transmission eigenvalue problem in order to obtain a general spectral theorem for a C∞ domain D and contrast m. For constant contrast m (possibly complex valued) the Born transmission eigenvalue problem becomes

2 2 (∆ + λ) u = 0, u ∈ H0 (D),

90 a quadratic eigenvalue problem independent of the contrast where we let k2 := λ. The investigation of this interesting eigenvalue problem is the subject of a forthcoming study.

Remark 2. It is rather surprising that there is little correlation between the transmission eigenvalues with and without the Born approximation. For example, without the Born approximation there are an inﬁnite number of real transmission eigenvalues whereas with the Born approximation there are no real eigenvalues. In the case when n > 1, the ﬁrst real transmission eigenvalue is proportional to the L∞-norm of

−1 (n − 1) , and hence it goes to +∞ as the contrast kn − 1kL∞ approaches 0. In the case of complex eigenvalues, there also appears to be minimal correlation, although in our opinion the information is too limited to make any quantitative statement.

91 Chapter 5

FAST METHODS FOR HELMHOLTZ EQUATION SIMULATION WITH APPLICATIONS TO TIME DOMAIN SCATTERING AND BAYESIAN INVERSE PROBLEMS

In this chapter, we use volume integral equation techniques for simulating the Helmoltz equation on unbounded domains. Volume integral equation methods have been used for many decades to simulate the solution to PDEs on unbounded and unstructured domains [46, 92]. However, in their basic form, they are extremely slow and require huge amounts of memory. More recently, fast methods have been developed which exploit the convolutional nature of volume integral equations with fast Fourier transforms (FFT) [92]. Unfortunately, such methods often require the domain of interest to be easily represented on a structured grid. In the ﬁrst section of this chapter, we combine the geometric ﬂexibility and theoretical foundation of classical volume integral equation techniques with the speed of FFT-based methods to simulate time harmonic scattering using the adaptive integral method (AIM) introduced in 1996 [15]. In the second section of this chapter, we discuss convolution quadrature (CQ), a technique which can be used for simulation of time dependent PDEs. We combine CQ with the AIM-type method for time harmonic scattering to derive a fast time dependent wave scattering algorithm. Finally, we use the fast Helmholtz equation simulation to solve a Bayesian inverse scattering problem.

92 5.1 Volume Integral Equations with Galerkin and FFT-Based Techniques We begin by presenting two methods for solving the time harmonic scattering problem introduced in Chapter1,

∆û + k2n(x)û = 0 in Rd, (5.1a) uˆ(x) =u î(x, dˆ) +u ˆs(x), (5.1b)

∂uˆs s limr→∞ r ∂r − ikuˆ = 0. (5.1c)

Recall that the Lippmann-Schwinger equation provides a volume integral equation representation foru ˆs,

s 2 s 2 i d uˆ (x) + k (Vˆ [muˆ ])(x) = −k (Vˆ [muˆ ])(x), x ∈ R , (5.2) where m(x) = 1 − n(x), Z ˆ ˆ (V f)(x) := Φk(x, z)f(z) dV (z), D ˆ and Φk(x, z) is the time harmonic free space fundamental solution of the Helmholtz equation. While the equivalence of (5.1) and (5.2) have been discussed previously, we state formally here that for real-valued n ∈ L∞(Rd),

Theorem 13 ([23]). If uˆs ∈ H1 is a solution of (5.1), then uˆs is a solution of loc(Rd) 2 s 2 s 1 d (5.2) in L (D). Conversely, if uˆ ∈ L (D) is a solution of (5.2), then uˆ ∈ Hloc(R ) and uˆs is a solution of (5.1).

∞ In this chapter we will always take real-valued m ∈ L (D) with m(x) ≥ m0 > 0 for some m0 ∈ R, though many of these results can be extended to complex m with Im(m) ≥ 0 [69].

5.1.1 Galerkin Approach A classical technique for calculating numerical approximations tou ˆs from the Lippmann-Schwinger equation is a Galerkin approach, sometimes called the Method of Moments (MoM) approach. We will proceed by deriving and discussing the well- posedness of a variational formulation of the Lippmann-Schwinger equation. We will

93 then use this variational formulation as a basis for a finite element approach to the problem, projecting the solution onto the space of P0 piecewise constants on a finite element mesh. Note that, because it is an integral operator with a weakly-singular kernel, Vˆ is compact on L2(D). As such, the Lippmann-Schwinger equation is of Fredholm type, and we can expect different behavior of the solution depending on the wavenumber. Indeed, as is well-known by an application of the unique continuation principle from the theory of elliptic PDEs, the Lippmann-Schwinger equation has a unique solution for every k > 0 (see, e.g., [23] Theorem 1.11). We will focus in this section on k > 0. However, when extending these results to time domain scattering in Section 5.2 below, we will take k in a subset of the complex plane. As will be discussed there, all the results presented in this section will carry over to that case as well. Continuing under these restrictions on k, we derive a weak form of the Lippmann- Schwinger equation. For the sake of generality, consider the problem of findingw ˆ ∈ L2(D) such that for all f ∈ L2(D),w ˆ satisfies

wˆ(x) + k2Vˆ (m(x)w ˆ)(x) = −k2Vˆ (f)(x), x ∈ D. (5.3)

Deﬁnev ˆ(x) = m(x)w ˆ(x) for x ∈ D. Then, (5.3) can be written as 1 vˆ(x) + k2Vˆ (ˆv)(x) = −k2Vˆ (muˆi)(x), x ∈ D. (5.4) m(x)

1 After solving this equation,w ˆ(x) = m(x) vˆ(x) can be used to ﬁndw ˆ in D. Moreover, in the case ofw ˆ =u ˆs and f = muˆi, the solution can be extended to the exterior of D via

s s i d uˆ (x) = −Vˆ m(z)(ˆu (z) +u ˆ (z)) (x), x ∈ R .

Hence, we will demonstrate the well-posedness of (5.4) before explaining the numerical method for its solution. First, however, we prove a boundedness property for Vˆ .

∞ Lemma 6. Assume m ∈ L (D) with m(x) ≥ m0 > 0 for some m0 ∈ R. Then, if vˆ ∈ L2(D),

ˆ ˆ kV (ˆv)k 2 ≤ C1(k)kvˆk 2 and kV (ˆv)k 1 d ≤ C2(k)kvˆk 2 L (D) L (D) Hk (R ) L (D)

94 2 where kfk 1 d = kfk 2 d + k k∇fk 2 d and C1 and C2 are constants depending Hk (R ) L (R ) L (R ) on k.

Proof. This theorem is given in [73] for a similar problem with imaginary wavenumber. The argument is similar for k > 0 and we give it here. Letw ˆ = Vˆ (ˆv) so that

∆w ˆ + k2wˆ =v. ˆ

Recall thatw ˆ ∈ H1(Rd) sincev ˆ ∈ L2(D). Multiplying by wˆ and integrating over D yields Z 2 2 2 k∇wˆk 2 d + k kwˆk 2 = vˆwˆ dV. L (R ) L (D) D From the triangle-inequality, we have

2 2 2 k∇wˆk 2 d + k kwˆk 2 ≤ kvˆk 2 kwˆk 2 L (R ) L (D) L (D) L (D)

Both results follow immediately from the facts that k > 0 and that kwˆkL2(D) ≤ kwˆk 1 d . Hk (R )

To derive a weak formulation, multiply (5.4) by ϕ ∈ L2(D) and integrate over D, yielding Z 1 Z vˆ + Vˆ (ˆv) ϕ dV = − Vˆ (fˆ)ϕ dV. D m(x) D Associating the left- and right-hand-sides with linear forms, we obtain the weak form of ﬁndingv ˆ ∈ L2(D) so that

a(ˆv, ϕ) = b(ϕ) ∀ϕ ∈ L2(D), where Z 1 a(ˆv, ϕ) = vˆ + Vˆ (ˆv) ϕ dV D m and Z b(ϕ) = − Vˆ (fˆ)ϕ dV. D

Note that by making the assumption m(x) ≥ m0 > 0 for all x ∈ D we lose some ﬂexibility but avoid the weighted L2 norms required in [73]. Although a is not a coercive operator for every k > 0, well-posedness of the variational problem can be derived from a G˚arding-type inequality.

95 2 ∞ Theorem 14. Let k > 0, f ∈ L (D), and m ∈ L (D) such that m(x) > m0 > 0 2 for some m0 ∈ R. Then a is continuous and injective on L (D) and satisﬁes the G˚arding-type inequality

−1 2 Re(a(ˆv, vˆ))) ≥ (kmk∞) kvˆkL2(D) + Re (Kv,ˆ vˆ)L2(D) . where K : L2(D) → L2(D) is a compact operator. Moreover, the variational formulation of ﬁnding vˆ ∈ L2(D) so that

a(ˆv, ϕ) = b(ϕ) for all ϕ ∈ L2(D) has a unique solution satisfying

2 kvˆkL2(D) ≤ C(k, m)kfkL(D) for a constant C > 0.

Proof.

As mentioned above, injectivity is well-known for the Lippmann-Schwinger equation with k > 0 (e.g., [23]). Continuity of a follows from the triangle-inequality and the bound on Vˆ given in Lemma6:

−1 2 ˆ |a(ˆv, ϕ)| ≤ (kmk∞) kvˆkL2(D) + k kV (ˆv)kL2(D) kϕkL2(D) ≤ CkvˆkL2(D)kϕkL2(D) where C = C(m, k) is a positive constant. To prove the G˚arding-type inequality, note that

Z 1 2 2 ˆ −1 2 ˆ Re(a(ˆv, vˆ)) = Re |vˆ| + k V (ˆv)vˆ dV ≥ (kmk∞) kvˆkL2(D)+Re k V (ˆv), vˆ . 2 D m L (D)

Since Vˆ is a compact operator on L2(D), the result follows immediately for K = k2Vˆ . These three conditions imply well-posedness of the variational problem (see, e.g.,

Theorem 4.2.9 of [88]). In particular, continuity of solutions with respect to kfkL2(D) ˆ ˆ follows from kV (f)kL2(D) ≤ CkfkL2(D) which is true due to Lemma6.

96 Remark: While the above theorem is unnecessary for the well-posedness of the problem with real wavenumbers due to the Fredholm alternative, in order to prove quasi-optimal error estimates of the ﬁnite element solution, we need a stronger result than is given by the Fredholm alternative.

5.1.2 Piecewise Constant Finite Element Discretization With the well-posedness of the continuous problem in hand, we deﬁne a spatial discretization using ﬁnite elements. In particular, introduce a triangularization of D

2 denoted by {Th}h>0 where h indicates the size of the mesh. We will approximate L (D) with piecewise constants ϕh ∈ P0 deﬁned on the support of each triangle K ∈ Th,

ϕh = 1K . Since the use of P0 elements in ﬁnite element approximations to scattering problems is not common, we explain the discretization scheme explicitly here.

The ﬁnite element space Vh is spanned by P0 deﬁned on each K ∈ Th. In particular, any w ∈ Vh can be written

N X w(x) = cj1Kj (x), x ∈ D j=1 where Th = {K1,K2,...,KN } and cj are are the constants 1 Z 1 Z c = w(x)1 (x) dx = w(x) dx. j R Kj |K | D 1Kj D j Kj

Note that cj are the average of w on the jth triangle. These elements can be used to interpolate L2(D) functions, as the following result demonstrates.

s Lemma 7 (Lemma A.5, [48]). Let v ∈ H (D) for some real s ∈ [0, 1] and let Ih be a piecewise-constant interpolation operator deﬁned so that for all K ∈ Th, Z

Ihv K ∈ P0, (Ihv − v)f dV = 0 ∀f ∈ P0. K

2 Then the L projection Ih satisﬁes the error estimate

s kv − IhvkL2(D) ≤ Ch |v|Hs(D),

s for a constant C > 0 independent of h and v where | · |Hs(D) is the H (D)-seminorm.

97 Notice, in particular, that the interpolation operator Ih which maps a function 2 w ∈ L (D) to its average on a triangle Kj meets the criteria required by the above

Lemma. In particular, it is clear that Ihw ∈ P0. Moreover, if f ∈ P0 and K ∈ Th, Z Z 1 Z Z (Ihw − w)f dV = f w dV dV − w dV K K |K| K K Z Z = f w dV − w dV = 0. K K Hence, coupled with the solvability theory for the continuous problem, we have solvability for the discrete problem.

Theorem 15. Let Vh, Ih, and Th be deﬁned as above. Then for h small enough, there exists a unique solution vˆh ∈ Vh satisfying

a(ˆvh, ϕh) = b(ϕh) ∀ϕh ∈ Vh which satisﬁes

kvˆ − vˆhkL2(D) ≤ Ch|vˆ|H1(D where C = C˜(|s|) > 0 is independent of h and v.

Proof. Existence, uniqueness, and quasi-optimality of the discrete problem follow be-

2 cause Vh ⊂ L (D) and because a is continuous, injective, and satisﬁes a G˚arding inequality on L2(D) ([88] Theorem 4.2.9). In particular,

kvˆ − vˆhkL2(D) ≤ C min kvˆ − wˆkL2(D) wˆ∈Vh for some constant C > 0. Note that h must be suﬃciently small for this quasi- optimality condition to hold. Hence, from Lemma7,

kvˆ − vˆhkL2(D) ≤ Ckvˆ − IhvˆkL2(D) ≤ Ch|vˆ|H1(D). (5.5)

1 1 d Note that the H (D) semi-norm is bounded becausev ˆ is in Hloc(R ) and |vˆ|H1(D) ≤ kvˆkH1(B) for any ball D ⊂ B.

98 To ﬁnish this theory, we return to the somewhat delicate question of quadrature. PN At the discrete level, forv ˆ(x) = j=1 cj1Kj (x), we solve

N N 1 Z X Z Z X Z c |K |+ c Φˆ (x, y) dV (y) dV (x) = Φˆ (x, y)f(y) dV (y) dV (x) m(x) i i j k k Ki j=1 Kj Ki j=1 Kj for each i = 1,...,N. At the matrix level, this corresponds to an equation of the form

(M + V)c = b where c is the N × 1 vector containing each cj and 1 [ ] = |K |δ , M ij m(x) i ij Z Z ˆ [V]ij = Φk(x, y) dV (y) dV (x), and Ki Kj N Z Z X ˆ [b]i = Φk(x, y)f(y) dV (y) dV (x). j=1 Ki Kj

Calculating the entries of V and b requires particularly special attention when i = j ˆ due to the singularity of Φk(x, y) at x = y. At this singularity, following [81], we p approximate Ki by equal-area circles (of radius Ri = |Ki|/π) and use the analytic expression

Z Z Z 2π Z Ri (1) (1) [V]ij = H0 (k|x − y|) dV (y) dV (x) ≈ |Ki| ρH0 (kρ) dρθ Ki Kj 0 0 4i 2πR = |K | + j H(1)(kR ) . i k2 k 1 j

Away from the singularity, we use

ˆ [V]ij = |Ki||Kj|Φ(xci , yci )

where xci is the centroid of triangle Ki. Numerical experiments demonstrate that this set of quadrature rules works well for small h. The same quadrature rules are used in calculating [b]i.

5.1.3 Numerical Results for Galerkin Approximation Before continuing to fast approximations of solutions to the Lippmann-Schwinger equation, we investigate the behavior of this Galerkin solution. In particular, we let

99 Convergence Rate for various meshes 10 -3

10 -4 error 2 L

L2 Error O(h) 10 -5 10 -2 10 -1 Step size h

Figure 5.1: Convergence rate for time harmonic scattering from a ball with constant contrast simulated with P0 ﬁnite elements.

D be a ball of radius 0.275 with contrast m = p(2). Under these assumptions we can use separation of variables to construct an exact solution (see AppendixA). To

∗ approximate error, we calculate kehkL2(D) = ehMeh where M is the mass matrix for the finite element mesh and eh is the difference between the approximate and series solution evaluated at the centroid of each triangle in the finite element mesh. As Figure 5.1 demonstrates, we see a convergence rate of O(h), as expected. We will discuss these results in more detail below, in comparison to the same simulation done with fast techniques.

5.1.4 The Adaptive Integral Method (AIM) While the Galerkin method introduced above is an accurate and ﬂexible method for computing solutions to the Lippmann-Schwinger equation, it requires a the assembly and inversion of large dense matrices. Naive implementations of this method take O(N 3) CPU time to solve and O(N 2) memory usage, where N is the number of elements in the ﬁnite element grid. Iterative solution methods, on the other hand, require

2 O(kiterN ) solution time, where kiter is a constant related to the iterative method, and the same O(N 2) memory. In recent decades, a number of methods have been developed

100 for overcoming these potentially-crushing memory and computation time properties by splitting the computation into “near” field and “far” field components. Typically in these approaches, the near field components (which correspond to operations on the sparse near-by elements of the finite element grid) are calculated accurately, while the far field elements (which correspond to the remaining elements) are calculated quickly and up to a user-specified tolerance. One class of these methods is typified by the fast multipole method (FMM), introduced in the 1980s [49, 33]. In the FMM, interactions between far field elements are approximated using analytical expansions of solutions to the Helmholtz equation and a tree-based data structure is used to reduce solution time from O(N 2) to O(N log N). Of more interest to this section are fast solving techniques which make use of the fast Fourier transform (FFT) to speed up solution times to O(N log N). In particular, far field elements are approximated by auxiliary elements on a Cartesian grid. The Lippmann-Schwinger integral can then be calculated quickly on this auxiliary grid using the FFT. While this idea is straightforward to apply to approximating functions defined on a Cartesian grid, more care must be taken in order to apply such a technique to solutions on an unstructured mesh. We describe in detail now the adaptive integral method (AIM), which is one way of doing just this. We also mention the precorrected- FFT (p-FFT) technique [82, 83], which is similar in spirit to the AIM, and was invented around the same time, but which has slightly different properties. The AIM was introduced for electromagnetic scattering in a series of papers in the early 1990s [15] and has been further developed in the succeeding years [16, 17]. For more detail about these developments, as well as a description of a time-domain version of the AIM, see [12]. The key to FFT-based fast solvers is that the integral operator Vˆ described above has a translation-invariant kernel so that the integral operator is a spatial convolution. At the continuous level, for sufficiently-smooth f,

ˆ −1 ˆ (V f)(x) = Fω→x Fx→ωΦk (ω)(Fx→ωf)(ω) (x),

101 where Fx→ω is the spatial Fourier transform. At the discrete level, Fourier transforms can be computed more quickly than matrix-vector multiplication (order O(N log N) rather than O(N d) for x ∈ Rd and f ∈ CN ). Hence, the previous equation allows for the multiplication of the discrete M + Vˆ matrix by a vector in order O(N log N) time and coupled with an iterative solver, then, FFT-based solvers take only O(kiterN log N) ˆ time. Unfortunately, the discrete convolution theorem requires that Φk and f be evaluated on a uniform Cartesian grid, so it cannot be applied immediately to the Galerkin approximation to Vˆ on a ﬁnite element triangulation. This is the problem alleviated in the AIM and p-FFT algorithms.

Specifically, we aim to write Vˆ ≈ Vˆ near + Vˆ far where Vˆ near is computed using a finite element approximation and Vˆ far is computed with a fast Fourier transform on an auxiliary grid. Loosely speaking, Vˆ near corresponds to a the elements of Vˆ coming from the interactions of near-by elements and Vˆ far those coming from far-away elements. Intuitively, the contributions from far-away elements can be approximated by auxiliary elements on a near-by Cartesian mesh without significantly affecting V because the distance between far-away elements is much larger than the distance between points on the finite element and auxiliary meshes.

To this end, denote by G ∈ Rd a set of Cartesian grid points surrounding D. d i i For each element Ki in the ﬁnite element mesh, pick (MG + 1) points, C := {Cα, d α = 1,..., (MG + 1) }, which cover Ki in a speciﬁed way. Here, MG is related to the accuracy of our method, always chosen to be MG = 2 in this chapter. Moreover, one of i the elements of Cα is be chosen as the nearest element in G to the centroid of Ki and i the other elements of Cα chosen as that elements nearest neighbors. We then make the approximation Z Z ˆ X X ˆ i j [V]ij = Φk(x, y) dV (y) dV (x) ≈ PiαPjβΦk(Cα,Cβ) K K i j α∈Ci β∈Cj

The matrix P contains elements which map from the Cartesian grid to the ﬁnite element grid. For example, in [15], P is chosen as a polynomial interpolation matrix, which interpolates exactly up to order MG + 1. We discuss this choice and two others below.

102 With this mapping from Cartesian to ﬁnite element grids in place, we are able to decompose the action of Vˆ into its near- and far-ﬁeld components. In particular, set

Vˆ ≈ Vˆ near + Vˆ far where  ˆ far [ ]ij − [ ]ij dist(i, j) < dh ˆ near V V [V ]ij = 0 o.w.

ˆ far PMG +1 ˆ i j and [V ]ij = α6=β=1 PiαPjβΦk(Cα,Cβ). Here, dist(i, j) defines the distance between two elements i and j and can be defined in different ways, to be discussed below. The parameter d indicates, in terms of finite element mesh size h, the maximum distance between two elements which are in each other’s near fields.

We remark two things here: first, the matrix Vˆ near is sparse, leading to easy storage and fast multiplication with vectors. Second, we will later be computing Vˆ far over all of G. As such, to avoid double counting the near field contributions, we need to subtract away the elements of Vˆ far corresponding to the near field calculations. Indeed, because we will solve the Lippmann-Schwinger equation using an iterative scheme, we are interested in quickly calculating Vˆ farf for some vector f. Because ˆ Φk(x, y) is a Toeplitz block-block toeplitz matrix for x, y ∈ G, we are able to do this quickly and with minimal memory usage using the FFT. Through a well-known process (in R2) of embedding each Toeplitz block into a circulant block, and each of those blocks into a block circulant matrix, multiplication by a toeplitz matrix reduces to a discrete Fourier transform. Moreover, this process only requires N distinct evaluations ˆ 3 of Φk(x, y), saving considerable memory. The process is similar in R .

5.1.5 Interpolation Between Finite Element and Cartesian Grid Key to the success of the AIM is the construction of the interpolation operator

P which maps between the ﬁnite element mesh and the auxiliary Cartesian grid. There have been a number of ways suggested for this task in the literature. In this section, we introduce two techniques for construction P which were previously-introduced as well as one which is new to the author’s knowledge. While the two previously-introduced

103 techniques balance accuracy and speed, the new technique sacriﬁces accuracy in favor of speed and ease of implementation.

The ﬁrst technique we introduce is the method for computing P from the original article describing the AIM [15]. It is, in essence, polynomial interpolation of the ﬁnite element basis functions onto a Cartesian grid. In the case of P0 elements discussed here, P is chosen as the solution to ! Z X p(x, y; m) 1Kj − Pαj dV = 0 D α∈Cj

mx my where p(x, y; m) = x y is a polynomial in x and y of degree m = (mx, my) for 0 ≤ 2 mx, my ≤ (MG +1) . By picking P in this way, each ﬁnite element basis function is inter- 2 polated to the auxiliary grid exactly for polynomials of degree up to (MG +1) . This dis- i j tance between two elements is deﬁned by dist(i, j) = minα∈Ci,β∈Cj max Cα − Cβ where the max is in reference to the larger x- or y-coordinate.

We call this choice of parameters AIMd where d is the near-field distance, as discussed above. Note that separating near-field elements from far-field elements can be slow, O(N 2), if not implemented in an efficient manner. However, an a quadtree (or octtree in 3D) approach would significantly speed up this calculation.

The second technique we introduce is based on the calculation of P used in the p-FFT method, and also introduced in some implementations of AIM [18]. In this method, we minimize the L2 diﬀerence between PVˆφ and Vˆφ, for some basis function φ, both evaluated on some surface outside the (M +1)2 grid points nearest the centroid

j of supp(φ). In particular, let Γj be a surface containing both Kj and C . Then, P is chosen as Z ˆ X ˆ i P = arg min Φk(x, y) dV (y) − PΦ(x, Cα) . K j α∈Cj 2 L (Γj )

Experimentally, we have found that choosing Γj ⊃ D for all j and deﬁning distance by i j dist(i, j) = |C − C |2 works well so long as the ball dist(i, j) < dh is smaller than Γj j and larger than Kj and C . Note that P = P(k).

104 Finally, we introduce a method for calculating P assuming that each element is in its own near ﬁeld and no other element is. For this reason, we term it the AIM0 set of parameters, since we are in essence setting d = 0 in the AIMd method. However, unlike the polynomial interpolation scheme for matching grid points and ﬁnite element points proposed in AIMd, we select P in a less-accurate but easier-to-compute way: we

|Kj | 2 take jα = 2 for each j = 1,...,N and α = 1,... (MG + 1) . Notice that P (MG +1)

Z X 1Kj dV − Piα = 0 D α∈Ci when P is chosen this way, so we are interpolating the ﬁnite element basis functions onto the auxiliary grid using constants. Despite this large approximation, we see surprisingly accurate results and very fast solution times.

5.1.6 Numerical Results for the Time Harmonic Problem Before we apply AIM to problems which require repeated solutions of the

Helmholtz equation, we show numerical results that suggest that both the P0 Galerkin scheme and its approximation with AIM super-converge at O(h2). While this is supris- ing in comparison with the expected O(h) convergence rate, it seems to be due to computing error integrals on the barycenters of each triangle. This is unlike in Fig- ure 5.1, for which the error integral was computed with a Gaussian quadrature, and where we see the expected O(h) error rates. We first simulate scattering from a ball with constant index of refraction. Next we compute scattering simulations from more complicated objects with non-constant indices of refraction. We again see that both the Galerkin solution and its AIM approximation super-converge at the higher-than- expected rates. Along with convergence rates, we demonstrate in both instances that the AIM solution is computed in significantly less time, particularly on fine meshes.

All results are in R2 and we always takeu ˆi(x; d) = exp (ikx · d) for d = [0, 1]. All numerical schemes were implemented in Matlab and simulations were computed on a laptop with a 2.4 Ghz AMD Radeon R5 CPU and 6GB of memory. All schemes are solved iteratively with Matlab’s implementation of the conjugate gradient

105 squared method. For each set of AIM parameters, the auxiliary mesh has spatial step

1 hG = 6 h where h is the ﬁnite element mesh size.

Scattering from a ball with constant index of refraction

To begin, we simulate scattering from a ball with radius of 0.275 with constant √ contrast m(x) = 2 for x ∈ D. This conﬁguration is the same as was used in [73], which we will compare against in later time domain simulations. In Table 5.1 we show three diﬀerent parameter choices for AIM, corresponding to the three choices described above. Note that convergence rates for both the Galerkin scheme and both AIM with

2 both AIM12 and p-FFT parameters converge at a rate of O(h ). Conversely, the AIM schemes with parameter chosen to minimize time, AIM0 converges at a rate between approximately O(h) and O(h2). Note that in the tables below, solve time refers to the amount of time required to solve the appropriate system, while total time includes both solve time and any assembly of required matrices. Notably, total time includes the separation of near- and far-ﬁeld elements. For the AIM12 method, this is the costliest component of the solution, though this is not an inherent complication of the method; a tree code could be used to perform this calculation in O(N log N) rather than the O(N 2) which has been implemented. Despite this, AIM12 is faster than the standard Galerkin scheme for large N.

Scattering from ball with non-constant index of refraction

In this section, we consider scattering from the same ball of radius 0.275 discussed above, but now the index of refraction is non-constant. In particular, we set the con- tract m(x, y) = (2 + 2 sin(x)) which satisﬁes the properties required in the convergence theorems given above. We now take the wavenumber as k = 5. Since no exact solution is available for this problem, we compute the solution using P0 elements on a ﬁne mesh (N = 2048) and compare how the solution on coarser meshes converges to this solution. As Table 5.2 shows, we see the same super-convergence rate as before for both

106 Simulation method N = 128 N = 512 N = 2048 N = 8192 L2-error×105 9.99 2.51 0.627 0.157 Galerkin scheme Convergence rate – 2.0 2.2 2.2 Solve time (s) 0.005 0.048 1.034 98.45 Total time (s) 0.252 2.474 41.32 1205.30 L2-error×105 9.13 1.81 0.45 0.11 AIM12 Convergence rate – 2.4 2.0 2.0 Solve time (s) 0.460 2.531 11.25 60.31 Total time (s) 0.672 5.268 54.81 814.43 L2-error×105 7.50 4.28 1.43 0.76 AIM0 Convergence rate – 1.6 0.8 2.7 Solve time (s) 0.134 0.752 1.66 11.61 Total time (s) 0.161 0.909 3.20 27.54 L2-error×105 9.97 2.51 0.62 0.15 p-FFT Convergence rate – 2.0 2.0 2.2 Solve time (s) 0.099 0.718 1.744 14.06 Total time (s) 0.573 3.346 28.22 350.78

Table 5.1: Simulation results for time-harmonic scattering from a ball with constant index of refraction.

the Galerkin scheme and the AIM scheme with p-FFT and AIM12 parameters. Also as before, the AIM with AIM0 parameters converges at a sub-optimal rate of O(h), but is significantly faster than the other parameter choices. Somewhat surprisingly, the AIM with AIM12 parameters is slower than a standard Galerkin scheme for the parameters tested in this example. While we expect the AIM12 solution to become faster than the Galerkin solution as N → ∞, this example demonstrates that, absent an efficient implementation of the AIM with AIM12 parameters (e.g., using a quadtree data structure to separate near and far field elements), N may need to be quite large for any speed gains to occur. As in the previous example, the p-FFT parameters seem to balance accuracy and speed.

Scattering from non-convex shape with non-constant index of refraction

Finally we consider scattering from a non-convex shape with non-constant index of refraction. The shape is a circle with a wedge cut out of it, as shown in Figure 5.2. We have the same analysis as in the previous example: the AIM with parameters AIM12 is

107 Simulation method N = 128 N = 512 N = 2048 N = 8192 L2-error×103 0.895 0.214 0.0429 – Galerkin scheme Convergence rate – 2.3 2.4 – Solve time (s) 0.004 0.082 1.372 82.73 Total time (s) 0.348 2.834 41.46 843.2 L2-error×103 1.323 0.418 0.107 – AIM12 Convergence rate – 1.8 2.0 – Solve time (s) 0.557 2.971 11.80 61.48 Total time (s) 0.759 5.719 56.51 858.3 L2-error×103 1.481 1.232 0.487 – AIM0 Convergence rate – 0.3 1.4 – Solve time (s) 0.143 0.820 2.302 16.37 Total time (s) 0.169 0.980 3.706 31.19 L2-error×103 0.887 0.212 0.0425 – p-FFT Convergence rate – 2.3 2.4 – Solve time (s) 1.360 2.634 4.555 18.50 Total time (s) 1.879 5.335 28.13 356.9

Table 5.2: Simulation results for time-harmonic scattering from a ball with non- constant index of refraction.

slow but accurate, the AIM with parameters AIM0 is less accurate but extremely fast, and the AIM with parameters p-FFT strikes a balance between them. The Galerkin scheme converges as expected.

5.2 Convolution Quadrature for Time Domain Equations Recall from Chapter3 that the time domain acoustic wave equation can be expressed in a Lippmann-Schwinger-type integral equation as well. That is, the acoustic wave equation for the total ﬁeld

∂u − c2∆us = 0, (x, t) ∈ d × + ∂t2 R R with casual scattered component is equivalent to the integral equation Z Z u(x, t)+ m(z)Φ(x, z, t − τ)u(z, τ; y) dV (z) dτ = R D i d + u (x, t; y), x 6= y ∈ R , t ∈ R .

Here, m = c−2 − 1.

108 0.1

-0.1

-0.2

-0.3

-0.4

-0.5 -0.6 -0.4 -0.2 0 0.2 0.4

Figure 5.2: The shape of the non-convex object oﬀ which we simulate scattering. The ﬁnite element triangulation with N = 192 used is shown as well.

Simulation method N = 48 N = 192 N = 768 N = 3072 L2-error×103 1.232 0.3019 0.0608 – Galerkin scheme Convergence rate – 2.0 2.3 – Solve time (s) 0.003 0.005 0.108 1.87 Total time (s) 0.191 0.443 6.837 112.0 L2-error×103 1.951 0.8778 0.1123 – AIM12 Convergence rate – 1.2 3.0 – Solve time (s) 0.311 1.391 5.314 19.52 Total time (s) 0.356 1.875 13.89 137.8 L2-error×103 2.313 1.159 0.5486 – AIM0 Convergence rate – 1.0 1.0 – Solve time (s) 0.117 0.557 2.047 8.952 Total time (s) 0.138 0.682 2.742 19.91 L2-error×103 1.301 0.320 0.0663 – p-FFT Convergence rate – 2.0 2.3 – Solve time (s) 0.311 1.391 5.314 19.52 Total time (s) 0.356 1.875 13.90 137.8

Table 5.3: Simulation results for scattering from a non-convex shape with non-constant index of refraction.

109 In this section, we explain a method for solving this time domain Lippmann- Schwinger equation using the Galerkin spatial discretization described above and a time-stepping method known as convolution quadrature (CQ). Although CQ can be seen as a method for accurately calculating convolution operators, it is often used as a numerically stable and simple way to solve the convolution-type integral equations which often model wave phenomena. First introduced in the late 1980s [74, 75], CQ has become a fast-growing research topic over the past decade, and one of the primary methods for computing time domain approximations to wave-like equations. For much more information, see, e.g., [61, 87]. One way to view CQ is as a method for computing semi-discrete approximations to time domain PDEs. Following [73], we re-write the wave equation into ﬁrst-order form, setting w = ut. This yields

d + ut = w, wt = c∆u, (x, t) ∈ (R , R ).

Assume we have a multistep method to solve the diﬀerential equation y0 = g(y, t), t > 0, y(0) = 0 such that

k k X X αjyn−j = ∆t βjg(tn−j, yn−j), n = 0, 1,... j=0 j=0 where yn ≈ y(tn) for tn = n∆t, n = 0, 1,... and ∆t > 0 is the chosen time step.

Moreover, {αj, βj} describe the multistep method and are deﬁned such that α0/β0 > 0. For example, the backward Euler method is deﬁned by

yn − yn−1 = ∆tg(tn, yn) so α0 = −α1 = 1 and β0 = 1 while β1 = 0. Many popular time-discretization schemes ﬁt into this pattern. After this discretization, the ﬁrst-order system above becomes

k k X X αjun−j = ∆t βjwn−j j=0 j=0 k k X X αjwn−j = ∆tc βjun−j. j=0 j=0

110 P∞ j For some ξ ∈ C, deﬁne the formal power series U(x, ξ) = j=0 uj(x)ξ and W(x, ξ) = P∞ j n j=0 wj(x)ξ . Multiplying the above system by ξ , summing over all n, and applying

Cauchys product formula with the fact that the discrete ﬁelds are causal so that uj(x) = wj(x) = 0 for j ≤ 0, we obtain

k ! k ! X j X j αjξ U = ∆t βjξ W j=0 j=0 k ! k ! X j X j αjξ W = c∆t βjξ ∆U. j=0 j=0

Pk j Pk j Eliminating W and deﬁning δ(ξ) := j=0 αjξ / j=0 βjξ yields

δ(ξ)2 U = c(x)∆U, x ∈ d. (5.6) ∆t R

Note that (5.6) is similar in form to the Helmholtz equation with “wavenumber” s = δ(ξ) ∆t , but with the incorrect sign in the term with the wavenumber. Using the same arguments as above, we can still derive a Lippmann-Schwinger integral equation for U s = U − U i, s 2 ˆ s 2 ˆ i 2 U + s V (qcU ) = −s V qcU in L (D).

Here, slightly abusing notation, we have used Z ˆ ˆ d V (f)(x) := Φis(x, z)f(z) dV (z), x ∈ R (5.7) D ˆ where Φis is the typical fundamental solution for the Helmholtz equation with wavenumber is. All of the results given in Section 5.1 apply to the Lippmann-Schwinger equation as well. In particular, it was proven in [73] that

Theorem 16. Let s ∈ C with Re(s) > 0 and deﬁne the sesquilinear form a and anti-linear form b as in Section 5.1, now with V corresponding to (5.7). Then, a is continuous and coercive on L2(D) and b is bounded on L2(D). Hence, the variational problem of ﬁnding vˆ ∈ L2(D) such that

a(ˆv, ϕ) = b(ϕ) ∀ϕ ∈ L2(D)

111 2 has a unique solution vˆ ∈ L (D) such that kvˆkL2(D) ≤ CkfkL2(D). Moreover, the discrete problem of ﬁnding vˆh ∈ Vh so that

a(ˆvh, ϕh) = b(ϕh) ∀ϕh ∈ Vh

where Vh is as in Section 5.1 has a unique solution which satisﬁes

kvˆ − vˆhkL2(D) ≤ Ch|vˆ|H1(D).

Proof. The well-posedness of the continuous variational formulation is shown in the proof of Theorem 4.1 in [73] via the Lax-Milgram lemma. The well-posedness and convergence rate of the discrete problem follows immediately from the fact that Vh ⊂ L2(D), Lemma7, and C´ea’slemma [19].

Note in the theorem above that we require Re(s) > 0. This is for exactly the same reason that we took wavenumbers ω with Im(ω) > 0 in Chapter3. This difference in the restrictinos on wavenumber is the result of a difference in sign convention for the Fourier-Laplace transform between the results presented in this chapter and in Chapter 3. This change does not affect the analytic properties of Fourier-Laplace transforms, and we employ these conflicting definitions simply because the one used here is typical in literature on CQ while the one used in Chapter3 is typical when discussing the time domain LSM. This derivation suggests how we can use CQ, tied into the Galerkin method for solving the time-harmonic Lippmann-Schwinger equation given above, to simulate acoustic wave propagation: by solving the time-harmonic Lippmann-Schwinger equation at various wavenumbers s ∈ C, we will be able to calculate an approximation to us. Indeed, it was shown in [73] that the semi-discrete problem derived above with CQ is optimally convergent and does not restrict ∆t. In particular, for some final time

M T > 0 discretized with M time steps {tm}m=0 such that ∆t = T/M, we have

Theorem 17 ([73], 4.2). Suppose the multistep method deﬁned by {αj, βj} is A-stable of order p, and δ(ξ) has no poles on the unit circle in the complex plane. Let ui ∈

112 p+3 + 2 s,∆t s Hσ (R ,L (D)) and let un be the approximate solution to u (·, tn) corresponding to step-size ∆t. Then,

M !1/2 X s s,∆t p i ∆t ku (·, t ) − u (·)k 2 ≤ C(∆t) ku k p3 + 2 . m m L (D) Hσ (R ,L (D)) m=0 Here, C = C(p, σ).

Indeed, due to the Galerkin-in-space discretization shown above, we have the following stronger result:

Theorem 18. Assume the same requirements as in Theorem 17 and that spatial discretization is computed using the Galerkin method with convergence rate O(hq), q ≥ 1. Then, the Galerkin-in-space, convolution-quadrature-in-time method is unconditionally convergent of optimal order,

s s,∆t p q ku (·, m∆t) − um kL2(D) = O(∆t ) + O(h ) uniformly over bounded intervals t ∈ [0,T ].

Proof. Apply Theorem 5.4 of [76].

As we explain in the next section, this method can be approximately decomposed into M + 1 independent Helmholtz-like equation solutions, with diﬀerent wavenumbers s ∈ C. We will show that using the Galerkin scheme for solving the Lippmann- Schwinger equation proposed above with CQ gives a stable and optimally-convergent numerical scheme for simulating the wave equation.

5.2.1 Approximation by Independent Helmholtz Equations and Error Anal- ysis We begin by describing the CQ process in more detail, then moving on to describe how to approximate the solution to the wave equation by a sequence of independent Helmholtz equations. We then give results on the convergence rates related to this approximation. Note that these results are an application to volume integral

113 equations of ideas presented for boundary integral equations previously in, e.g., [14], which we are following below. The results to not change signiﬁcantly from there and are only included for completeness. In principle, CQ is a method for estimating integrals of the form Z t ˆ (A(∂t)f)(t) := a(t − τ)f(τ) dτ, τ > 0 0 where A = L(a) is the Laplace transform of an operator a. Though we are speaking very loosely here, more information on the theory behind the method is available in the monograph [87]. In the Lippmann-Schwinger example, a is Z (a(t − τ)f(τ))(x) = m(z)Φ(x − z, t − τ)f(x, τ) dV (z). D ˆ R ˆ ˆ The Laplace transform is then (V f)(x) = D m(z)Φs(x, z)f(z) dV (z) for complex s ˆ such that Re(s) ≥ σ0 > 0. The CQ process proceeds through a discretization of V (∂t) at discrete times tm via m ˆ ∆t X ∆t ˆ V (∂t )f (tm) := ωm−j(V )f(tj). j=0

∆t ˆ Here, the convolution weights ωj (V ) are deﬁned as I ˆ ∆t ˆ 1 V (δ(ζ)/∆t) ωj (V ) := j+1 ds(ζ) 2πi C ζ where C is a circle of radius 0 < λ < 1. In the numerical implementation of CQ, these weights are approximated using the trapezoid rule sampled on M + 1 equispaced grid

2πi ` M+1 points λξM+1 where ζM+1 = e . This leads to the approximate weights

M λ−j X ωλ,∆t(Vˆ ) := Vˆ (s )ζ`j . j M + 1 ` M+1 `=0 −` ˆ Here, s` := δ(λζM+1)/∆t. This leads to the CQ approximation to V (∂t),

M ˆ λ,∆t X λ,∆t ˆ V (∂t )f (tm) := ωm−j(V )f(tj) m=0 M λ−m X = Vˆ (s )fˆ (x)ζm` , m = 0,...,M. M + 1 ` ` M+1 `

114 ˆ Here, f` is the scaled discrete Fourier transform

M ˆ X j −`j f` := λ f(tj)ζM+1. j=0 Up to this point, we have simply been talking about approximating the integral op- ˆ ˆ λ,∆t erator V (∂t). However, the above representation of V (∂t ) in terms of the scaled discrete Fourier transform leads to an approximation to the solution of the time domain Lippmann-Schwinger equation via a system of decoupled Helmholtz problems. In particular, applying CQ to the time domain Lippmann-Schwinger equation leads to the semi-discretization s ˆ λ,∆t s u (x, tm) + V (∂t )u (x) (tm) = f(x, tm), m = 0,...,M.

ˆ λ,∆t m Substituting in the deﬁnition of V (∂t ), multiplying by λ , and applying the scaled discrete Fourier transform with respect to m yields s ˆ s ˆ uˆ`(x) + V (s`)ˆu` (x) = f`(x), ` = 0,...,M

s s whereu ˆ` is the scaled discrete Fourier transform of u evaluated at `. The inverse of the scaled discrete Fourier transform is given by

M λ−` X f := fˆ ζ`j . ` M + 1 j M+1 j=0

s As such, the approximation to u (x, tm) given by CQ is

M M λ−m X λ−m X −1 us(x, t ) = uˆsζm` = I + Vˆ (s )fˆ (x)ζm` , m M + 1 ` M+1 M + 1 ` ` M+1 `=0 `=0

ˆ s ˆ and so solving the equation (I + V )ˆu = f for each of the frequencies s` leads to an approximate solution to the Lippmann-Schwinger equation. Note moreover, that by by reversing the process which led from the semi-discrete approximation of the Lippmann-Schwinger equation to the decoupled system of Helmholtz equations yields

M −1 s X ∆t,λ ˆ ˆ u (x, tm) = ωm−j I − V fj j=0

115 −1 s ˆ ∆t ∆t demonstrating that u (x, tm) = I + V (∂t )f(x, tm). Introduce the notation A(∂t ) = ˆ ∆t (I + V )(∂t ) in the following. λ,∆t Error estimates for the solution associated to the approximate weights ωj were derived in [14]. They are based on a lemma of the following form.

Lemma 8 ([73],4.1). Let Re(s) ≥ σ0 > 0. Then,

−1 |s| 2 2 kAh (s)kLm(D)→Lm(D) ≤ . σ0 Indeed, following the same proof as Theorem 5.5 in [14] gives

Theorem 19. Let 0 < λ < 1. Assume the same requirements as in Theorem 17 and that spatial discretization is computed using an order hq, q ≥ 1 Galerkin method. Then, the solution associated to decoupled Helmholtz equations

s,λ,h −1 ∆t,λ um = (A) (∂t fh) (tm) satisﬁes

M+1 s s,λ,h λ 2 −1 p q ku (·, t ) − u k 2 ≤ C T ∆t + ∆t + h m m Lm(D) 1 − λM+1 where C is independent of T, λ, M, h, and ∆t.

Theorem 19 allows us to choose λ to achieve optimal convergence order; for q = 1 and p = 2, taking h ∼ ∆t2 and λ ∼ ∆t3/(M+1) yields order O(∆t2) convergence.

Here, the notation A ∼ B indicates that there is are constants C1,C2 independent of

∆t and h so that A ≤ C1B and B ≤ C2A. Note for below that A . B indicates A ≤ CB for a C which is again independent of ∆t and h.

5.2.2 Implementation and Numerical Results We are interested in comparing results with [73]. As such, in all our numerical experiments, we scatter against a circle of radius 0.275 with constant speed of sound √ c(x) = 2 for x ∈ D. We simulate over [0, 8] and use the incident ﬁeld

i 2 u (x, t) = sin(α(t − x1/c0)) exp(−β(t − x1/c0 − γ) ), α = 4, β = 1.4, γ = 4.

116 Convergence rate of CQ-Galerkin scheme 10 -1

10 -2 error 2

10 -3

-4 log Space-Time L 10 h=0.0958 h=0.0512 h=0.0261 h=0.0131 O( t 2) 10 -5 10 -3 10 -2 10 -1 log t Figure 5.3: Convergence rate of CQ-Galerkin scheme for multiple meshes.

Following the discussion above, we solve M +1 independent Helmholtz problems, corresponding to M + 1 time steps, with wavenumbers

δ(λξM ) s = , m = 0,...,M ` ∆t where λ < 1 is chosen to stabilize computations. As discussed above, we take, λ = max(∆t3/(M+1), 1/2M ). We use the BDF2 time stepper corresponding to δ(ξ) = (ξ2 −

4ξ + 3)/2. We use the P0 Galerkin approach for the spatial discretization of these Helmholtz problems. In Figure 5.3, we see an initial convergence rate of order O(∆t2) for each ﬁxed h. The curves then level out at a point where spatial errors begin to dominate temporal errors. This is suggests that, in the example considered, the convergence error estimates given above are sharp. Note that choosing ui such that ui(x, t) ≈ 0 for x ∈ D, t < 0 is crucial for accurate convergence; in numerical experiments with the same ui as above but with

i −3 γ = 2, we see a loss of stability as ∆t → 0. For γ = 2, ku (·, 0)kL2(D) ≈ 10 whereas i −9 for γ = 4, ku (·, 0)kL2(D) ≈ 10 . We have noticed that increasing λ in the case of γ = 2 restores some stability, though further investigation is needed to better understand this

117 phenomena.

5.3 Application I: CQ-AIM In this section, we apply the AIM algorithm to speeding up the calculation of time domain wave equation simulations with CQ. In particular, we solve each of the Helmholtz-type equations associated with the CQ time stepping with the AIM algorithm. Before showing numerical results, we analyze the error in each convolution weight generated by using AIM versus using MoM. These results will allow us to select algorithm parameters in a successful manner. Note that the following results are very similar to those presented for CQ-BEM algorithms presented in [14, 30] and are included simply to verify that those results apply to the volume integral approach as well. In this section, we ﬁx a BDF2 time stepper, though the results can be modiﬁed for other choices.

5.3.1 Error Analysis ˆ AIM Introduce the notation Ah := I + Vh to deﬁne the AIM approximation such that

ˆ ˆ 2 2 kVh (s`) − Vh(s`)kLm(D)→Lm(D) < . (5.8)

The numerical results in Section 5.1 suggest that is a function of wavenumber, the original ﬁnite element mesh, the auxiliary Cartesian mesh, and the AIM parameters related to the mapping between those meshes and the parameters which split near and far ﬁeld points, though we do not have an exact relationship between these variables and . Similar to before, we use the notation

s,λ,h, −1 ∆t,λ h un = (Ah) (∂t )g (tn)

∆t,λ to indicate the solution (5.2) using perturbed ωj and using the AIM to solve the relevant Helmholtz equations. Then,

118 1 σ0 Theorem 20. Let min Re(s`) > σ0 > 0 and assume (5.8) holds with < . 2 max |s`| `=0,...,M `=0,...,M Then,

s,λ,h, s,λ,h 2 −M −2 2 kun − u kLm(D) ≤ CT λ ∆t where C = C(g).

Proof. The result follows from Lemma8 and following the steps of the proof of Propo- sition 5.8 in [14].

Hence, we establish a relationship between λ and and the overall convergence error of the CQ-AIM method.

Theorem 21. Assume the same requirements as in Theorems 17 and (20). Then,

M+1 s,λ,h, s 2 −M −2 λ 2 −1 2 q ku − u (·, t )k 2 ≤ C T λ ∆t + T ∆t + ∆t + h . n n Lm(D) 1 − λM+1

Here, C = C(g, σ0).

Proof. The result immediately from Theorems 17 and 20.

Hence, optimal convergence of O(∆t2) follows from

M+1 3 2 4 M 7/2 λ ∼ ∆t , h . ∆t , . ∆t λ . h . (5.9)

5.3.2 Numerical Results We examine the same example as in Section 5.2.2. In particular, we solve the same problems using the AIM approximation to Vˆ (s). We focus here on the speed increase resulting from approximating the spatial solution with AIM and ﬁx a spatial mesh with h = 0.0131 and four diﬀerent number of time steps, M = 20, 40, 80, and 160. Note that, since T = 8, we have that h < ∆t for each of these choices of M. Further, as the numerical experiments given in Section 5.2.2 demonstrate, we can expect

ˆ ˆ 2 2 kVh (s`) − Vh(s`)kLm(D)→Lm(D) <

119 Simulation method M = 20 M = 40 M = 80 M = 160 L2-error 0.0906 0.0335 0.0090 0.0022 Galerkin scheme Convergence rate – 1.4 1.9 2.0 Solve time (s) 10.89 24.25 51.49 115.1 Total time (s) 538.0 1049.1 2058.7 4066.9 L2-error 0.0905 0.0335 0.0088 0.0022 AIM0 Convergence rate – 1.4 2.0 2.0 Solve time (s) 17.61 36.82 83.44 208.3 Total time (s) 22.93 46.91 101.0 242.3 L2-error 0.0906 0.336 0.0089 0.0022 p-FFT Convergence rate – 1.4 1.9 2.0 Solve time (s) 17.81 38.67 91.76 198.46 Total time (s) 124.4 229.4 455.3 878.0

Table 5.4: Time domain simulation results for scattering from a ball with constant index of refraction. for a small value of , as required in the previous error analysis. Although we do not have a precise indicator of , choosing parameters in accordance with (5.9) should yield a convergence rate of O(∆t2). As 5.4 shows, this is indeed the case. Moreover, as table 5.4 shows, the CQ-AIM algorithm is significantly faster than CQ with a standard Galerkin scheme. The primary reason for this is that Vˆ is a function of wavenumber and, hence, must be recomputed at each time step. Thus, even though the overall time spent solving each system by the Galerkin scheme is less than each AIM scheme, assembly time for the AIM is so much faster than for the Galerkin scheme that the AIM schemes are as much as 20 times faster. Note that the timing figures reported in Table 5.4 were computed in serial. Because each of the Helmholtz problems we solve is independent, parallel computing can be used to significantly reduce the numbers given below. Nonetheless, the AIM0 parameters result in faster solutions than the p-FFT parameters, which result in faster solutions than the Galerkin scheme.

120 5.4 Application II: Bayesian Inverse Scattering In this final application, we introduce a different type of inversion algorithm than has been discussed in the rest of the thesis. Rather than reconstructing a single estimate of the location of hidden obstacles, we construct a probability distribution which indicates the location and material properties of hidden obstacles along with a quantification of uncertainty in the reconstruction. This type of reconstruction is called a Bayesian solution to an inverse problem because it is based on Bayes’ theorem from probability. In particular, we will use the framework of infinite dimensional Bayesian inverse problems developed in [89] which explains, among other things, how to choose parameters correctly so as to give a solvable statistical inverse problem (even from a deterministically ill-posed problem in the sense discussed previously in this thesis). A numerical implementation of this framework was proposed in [20], on which we elaborate below. While we focus here on a rather basic application of the Bayesian framework, we include this section for three reasons: first, we wish to demonstrate the applicability of the AIM for multiple types of problems. As we will see, AIM can be used to significantly speed up the calculation of Bayesian inverse scattering problems as well as it did the CQ calculations. Second, we wish to introduce a large class of inverse problems, along with their upsides and downsides, which were not covered in this thesis prior to this. Finally, while we will only study a toy problem here, the idea of converting an optimization problem into a series of PDEs whose discretization does not need to be fully stored in a computer is an important one which can lead to powerful results [20]. To this end, we return to the problem of reconstructing the speed of sound in a medium. For the sake of simplicity, we will consider the problem of time-harmonic scattering by plane waves in R2 from a medium with refractive index n, though all of this could be done for, e.g., time domain scattering in R3 (indeed, a very similar problem to this was explored in the article [20]). Assume, then, that we collect the scattered

2 2 ﬁeld data coming from y ∈ Γi on a boundary Γm. Denote by f : L (D) → L (Γm) the function so that f(m) = us(·, y) . Under the assumption of Gaussian additive noise Γm

121 with zero mean and covariance operator Γnoise, η ∼ N (0, Γnoise), we seek to solve the equation f(m) = yobs + η where yobs are the noisy data corresponding to us(·, y) . We can write f as Γm −1 4 ˆ 2 ˆ ˆ i f(m) = k V I + k V V (u (·, y)) (x), x ∈ Γm, y ∈ Γi.

Although this formula is somewhat unwieldy, such an relationship is needed in the numerical approximation of the Bayesian inverse problem described below. Prior to this point in the thesis, we have only been interested in supp(m). Here, we are interested in more. Informally, we seek the probability distribution of m. We also assume we have some knowledge about the index of refraction which allows us to a priori assume a probability distribution for m, without yobs. This data could come from expert information about the material being examined (or could encode a complete lack of information). Still informally, these two probability distributions are related through Bayes’ formula,

obs obs πpost(m|y ) ∝ πprior(m)πlike(y |m), (5.10)

obs obs where πpost(m|y ) is the a posteriori probability distribution of m given y , πprior(m) is the a priori probability distribution of m without any additional information, and

obs obs πlike(y |m) is the likelihood probability density function of y given m. Note that by the assumption of Gaussian noise η, we have obs 1 obs 2 πlike(y |m) ∝ exp − f(m) − y Γ−1 2 noise

2 where k · kM is the L (D) norm weighted by an operator M. The proportionality constants in the likelihood deﬁnition and in (5.10) are independent of m, so we omit them here and below. The above discussion has been informal because the standard version of Bayes’ formula (5.10) does not hold in inﬁnite dimensions. While we do not detail the theoretical considerations, note that some care must be taken when discussion Bayes’ theorem

122 in infinite dimensions. In particular, these considerations show that a well-posed inverse problem problem follows from the choice of prior covariance operator as a solution operator to an elliptic PDE. Note that after a finite element discretization, the discrete solutions to the inverse problem will satisfy the classical Bayes’ theorem. To continue, we first formalize the Bayesian inverse problem which we will solve. For simplicity, we focus on the problem with Gaussian prior distributions and under the assumption of small noise. Next, we discuss the discrete approximation to this inverse problem using finite elements and the Lippmann-Schwinger solver introduced earlier in this chapter. In both of these components, we follow [20]. We will describe how to solve an optimization problem by converting it into a series of solutions of various Helmholtz-like equations. Finally, we show simulated solution to our inverse problem.

5.4.1 Infinite Dimensional Bayesian Inverse Problems We seek a discrete approximation to the infinite dimensional parameter m ∈ L∞(D). Due to theoretical aspects of this problem described in [20, 89], the prior distribution must have specific properties in order for the discretized inverse problem to be well-behaved. We will focus exclusively on Gaussian prior distributions, so assume

2 2 µ0 = N (m0, C0) is a prior Gaussian measure on L (D) with mean m0 ∈ L (D) and 2 2 2 covariance operator C0 : L (D) → L (D) and assume µ0(L (D)) = 1. Following [20], assume that C0 comes from a covariance function c(x, y) which is deﬁned so that if ξ is a Gaussian random ﬁeld with probability measure µ0 then for x, y ∈ D,

c(x, y) = E [(ξ(x) − m0(x))(ξ(y) − m0(y))] , where E is the expectation operator on L2(D). Then, for φ ∈ L2(D), Z (C0φ)(x) = c(x, y)φ(y) dy, x ∈ D. D

In order to produce a well-posed inverse problem, we take C0 to be a power of the solution operator of an elliptic, Laplacian-like PDE. In particular, for s ∈ L2(D), let

123 A : H1(D) → L2(D) be deﬁned so that m = A−1s satisﬁes

− ∆ · (a(x)∆)m + c(x)m = s in D (5.11) ∂m = 0 on ∂D. ∂ν

−2 We set C0 = A . For simplicity in calculating the posterior distribution, we make the approximation

obs y + η = f(m) ≈ f(m ˆ ) + Fmˆ (m − mˆ ) wherem ˆ is a small perturbation of m and Fmˆ is the Fr´echet derivative of f with respect to m evaluated atm ˆ . For suﬃciently smooth nonlinear f, whenm ˆ is very close to m in an appropriate norm, the approximation does not incur much error. For an appropriate choice ofm ˆ and under the assumption of small noise, it can be shown that the posterior distribution µy is a Gaussian measure N (m, ˆ Cpost) where

post ∗ −1 −1 C = Fmˆ ΓnoiseFmˆ + C0 .

The posterior mean here is the so-called maximum a posteriori (MAP) point, deﬁned as 1 obs 2 1 2 mMAP := arg min kf(m) − y kΓ−1 + kA(m − m0)kL2(D) . (5.12) m∈L2(D) 2 noise 2

In summation, under the small noise assumption that η ∼ N (0, γnoiseI) with

γnoise 1 and the assumption that prior measure is Gaussian with mean m0 and covariance operator A−2, Bayes’ theorem gives that the posterior measure is Gaussian

post with mean mMAP and covariance operator C . In order to calculate a discrete approximation to these quantities, we will project onto a space of P1 ﬁnite elements to solve the same variational formulation posed in Section 5.1. The MAP point can then be calculated with traditional nonlinear optimization schemes and Cpost through its deﬁnition.

124 5.4.2 Numerical Implementation Applications of large scale Bayesian inverse problems include mapping the internal structure of the Earth and the movement of enormous ice sheets over time. These applications potentially have billions of unknowns, corresponding to points on a ﬁnite element grid describing the geometry of these applications. As such, state-of-the-art methods for Bayesian inverse problems focus on methods which scale well in terms of the number of parameters to be reconstructed and number of parallel computing cores available. For example, [20] demonstrate that Bayesian solutions can be computed which scale the same way as highly parallelizable forward solvers. As such, we derive a method for solving the Bayesian inverse problem through a series of solutions to Helmholtz-like PDEs. The overall solution will then scale in the same way as the solution to the Helmholtz equation solver. In this context, we now describe an way to compute a discrete approximation

y to dµ without having to form any dense matrices. We will compute the application dµ0 of the Fr´echet derivative of f in the computation of mMAP by solving an appropriate Lippmann-Schwinger equation and its approximation with the AIM.

2 As before, denote by Vh a finite dimensional subspace of L (D) with basis func- N tions {φj}j=1. This section, let φj be continuous P1 elements on a triangularization of D. We will comment on the discretization of the Lippmann-Schwinger equation with these elements in the next section. We are attempting to calculate the posterior PN distribution of mh = j=1 mjφj. Define M to be the finite element mass matrix, as before. Then, we will replace L2(D) inner products in the infinite dimensional space

T −2 by weighted inner products (m1, m2)M = m1 Mm2. The covariance operator C0 = A can be discretized on this space via the stiﬀness matrix Z Kij = α (a(x)∇φi(x)) · ∇φj(x) dV (x) + Mij, i, j = 1,...,N. D

Then in the weighted inner product space, A = M−1K is the ﬁnite dimensional approximation to A.

125 obs These discretizations and the definition of πlike(y |m) lead to simple expressions for the prior and posterior distributions, 1 π (m) ∝ exp − k (m − m )k2 prior 2 A h 0,h M 1 obs 2 1 2 πpost(m) ∝ exp − kfh(mh) − y kΓ−1 − kA(mh − m0,h)k 2 noise 2 M where m0,h is the discrete approximation to the prior mean m0 and fh is the discrete approximation to f. These follow from the fact that the prior distribution has post probability law N (m0,C0) the posterior distribution is N (mh,MAP,Ch ), the finite dimensional approximation of the value discussed above. To generate samples of the posterior distribution, we first need to calculate mh,MAP. To do this, we use Newton’s (i) method, linearizing fh around an initial guess mh ,

obs (i) (i) y + η = fh(mh + δmh) ≈ fh(mh ) + F (i) δmh mh where F is the discrete approximation to the Fr´echet derivative of f. Writing the optimization problem for mMAP into an equation for δm yields

−1 ∗ ∗ ∗ −1 ∗ ∗ −1 obs (i) ∗ −1 (i) Fm(i) ΓnoiseM F (i) + K M K δmh = M F (i) Γnoise(y −fh(mh ))+K M K(m0,h−mh ). h mh mh

Note that the M∗ terms come from the weighted ﬁnite element norm. This equation is solved using the conjugate gradient method and a new estimate is generated as (i+1) (i) mh = mh +βδmh. We use Armijo backtracking to choose β and aid in globalization [79]. This is continued until a suﬃcient error tolerance has been reached. To do this process in a matrix-free way, we need to relate the application of

∗ fh, Fm, and Fm to solutions of Helmholtz-like equation which can be solved with the

Lippmann-Schwinger equations derived earlier in this chapter. Note that fh can be 2 2 written as two operators, fh = γΓm G where γΓm : L (D) → L (Γm) takes a scattered

field corresponding to m to the boundary Γm and where G is the solution operator mapping from (m, uî(·, y)) to the scattered field in D. Both of these can be done by solving Lippmann-Schwinger equations. In particular, ifv ˆ satisfies

∆ˆv + k2(1 + m)ˆv = −k2mg

126 for g ∈ L2(D) then

2 ˆ γΓm (x)v = −k V (m(ˆv + g))(x), x ∈ Γm and (Gm)(z) = −k2((I + k2Vˆ )−1V (mg))(z), z ∈ D.

(y) Similarly, the Fr´echet derivative Fm for one incident location y can be written in terms (y) of solutions to Helmholtz equations. We can write Fm = γΓm H where H is a discrete solution operator to a particular Helmholtz equation. Assume

∆û + k2(1 + m)û = 0 is the total field corresponding to m and incident fieldu î(·, y). Then,v ˆ = Hδm satisfies

∆ˆv + k2(1 + m)ˆv = −k2δmuˆ

(y1) (y2) (yM ) T ∗ and Fm = [Fm , Fm ,..., Fm ] . Finally, in a similar way, we compute that (γΓm vˆ)(x) = 2 R (y) ∗ −k Φˆ (x, z)ˆv(z) dV (z) for x ∈ D and that ( m ) g = uˆwˆ where Γm k F

2 2 ∗ ∆w ˆ + k (1 + m)w ˆ = −k γΓm g

∗ (y1) ∗ (y2) ∗ (yM ) ∗ andu ˆ is as before. Then, Fm = [(Fm ) , (Fm ) ,..., (Fm ) ]. Note that in the ﬁnite −1 (y) ∗ ∗ (y) ∗ dimensional approximation, we must replace γΓm by M γΓM and (Fm ) by M (Fm ) . To calculate Γ , we could solve the equation −1 ∗ Γ−1 + Γ wˆ = post M FmMAP noiseFmMAP prior I using the same matrix free calculations as above. However, this would create a dense matrix which would be diﬃcult to store in the case of of a large parameter space. In- stead, we approximate Γpost by a truncated singular value decomposition - which can be computed in a matrix-free way. This has the further advantage of regularizing the numerical calculation of Γprior.

5.4.3 Simulations In this section, we demonstrate the eﬀectiveness of the numerical scheme described above. We ﬁrst describe the discretization of the Lippmann-Schwinger equation

127 using P1 elements and the adaptations which need to be made to the AIM for these elements. Next, we temporarily leave the setting of Bayesian inverse problems in order to demonstrate that the matrix-free approach described in this section can be used to approximate the contrast from scattered ﬁeld data using Tikhonov regularization. In particular, by choosing the parameters appropriately, this problem is equivalent to calculating mMAP. We will show that the AIM for P1 elements signiﬁcantly outperforms the unaccelerated Galerkin scheme in solution time and memory usage. Finally, we will use AIM to approximate the solution to a Bayesian inverse scattering problem.

Continuing, note that we search for a P1 approximation to mh so that the 1 covariance operator, which maps between H (D) and itself, can be applied to mh with no difficulties. Adopting the same arguments made in Section 5.1, we see that there is a unique solution to the Lippmann-Schwinger equation in the standard linear P1 element space which converges at order O(h2). To calculate the entries of the integral operator V, we use standard Gaussian quadrature away from the singularity and use a so-called singularity subtraction technique near the singularity. For full details, see [71]. Though convergence studies are not included in this chapter, we find that the Galerkin scheme implemented in this way converges at the expected O(h2). To adopt the AIM to this new set of finite elements, we use the p-FFT parameters and expand the auxiliary grid around the nodes of the finite element mesh. We

2 set the spatial discretization of the auxiliary grid to be so that the set of (MG + 1) auxiliary points surrounding node i in the ﬁnite element mesh is roughly the same size as the support of ϕi, the basis function centered at node i. Choosing parameters in this way has led to a fast technique which approximates the Galerkin scheme well. To demonstrate these techniques yield reasonable results, we now approximate the solution to (5.12) with the parameter choices Γnoise = I, m0,h = 0, and A = αI for α 1 acting as a regularization parameter. In this way, we are computing a standard

obs Tikhonov regularized solution to the problem fh(mh) = y . In these experiments, we set mh correspond to a square scatterer centered at the origin with side lengths of size

0.1 and contrast mh = 0.25. We reconstruct mh on a larger domain, the ball centered

128 Simulation method h = 0.0512 h = 0.0261 h = 0.0131 Assembly time (s) 21.36 367.92 – Galerkin scheme Average iteration time (s) 21.61 184.01 – Size of V (megabytes) 0.105 1.336 – Assembly time (s) 3.00 11.25 52.38 p-FFT Average iteration time (s) 37.88 93.64 1739.8 Size of V (megabytes) 0.0671 0.346 1.51 Table 5.5: Time and memory usage for calculation Tikhonov regularized solution of contrast.

at the origin of radius 0.275. Note that mh ≡ 0 outside of the square scatterer, which is what we aim to reconstruct on the larger domain. To simulate forward data, we use uˆi(x; d) = exp (ikx · d) with d = [0, 1] and k = 15, and solve the Lippmann-Schwinger equation using P0 elements. We do not add noise for this set of experiments. Note that while we are possibly committing an inverse crime here, we are only including these experiments as a feasibility study for a toy problem and, hence, do not anticipate precisely similar results for real data. We send the incident wave from 60 transmitters equispaced on a ball of radius 1.5 centered at the origin and collect the scattered ﬁeld at the same points. In Figure 5.4, we see sample reconstructions using both the Galerkin scheme and the AIM. Note that reconstructions become more accurate, in comparison to the exact value of mh, as h → 0. Of more interest is Table 5.5, which shows the time and memory usage of both the Galerkin scheme and the AIM for various values of h. As expected, the AIM vastly outperforms the full Galerkin simulation in terms of time and memory usage. Indeed, the full Galerkin scheme for h = 0.0131 was too slow to be computed in a reasonable amount of time, so Table 5.5 does not include results from it. Note that average iteration time refers to the average amount of time spent on each (i) iteration to compute each mh . Finally, we return to the Bayesian inverse scattering problem. We use the same set-up as in the Tikhonov case, but now with a prior distribution coming from the considerations discussed above. In these experiments, we add normally distributed

129 0.3 0.25 0.3 0.25

0.2 0.2 0.2 0.2

0.1 0.1 0.15 0.15 0 0 0.1 0.1 -0.1 -0.1

0.05 0.05 -0.2 -0.2

-0.3 0 -0.3 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

0.3 0.25 0.3 0.25

0.2 0.2 0.2 0.2

0.1 0.1 0.15 0.15 0 0 0.1 0.1 -0.1 -0.1

0.05 0.05 -0.2 -0.2

-0.3 0 -0.3 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

0.3 0.25 0.3 0.25

0.2 0.2 0.2 0.2

0.1 0.1 0.15 0.15 0 0 0.1 0.1 -0.1 -0.1

0.05 0.05 -0.2 -0.2

-0.3 0 -0.3 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Figure 5.4: Tikhonov regularized solutions to f(m) = yobs. All images have the same color scale, and white lines indicate ﬁnite element mesh. Left column: Solutions corresponding to a Galerkin scheme h = 0.0512 (top) and h = 0.261 (middle), and the exact solution projected onto a mesh of size h = 0.0131 (bottom). Right column: Solutions corresponding to an AIM scheme h = 0.0512 (top) and h = 0.261 (middle), h = 0.0131 (bottom).

130 noise with zero mean and covariance matrix proportional to the identity with standard deviation 0.1. As this is a toy model, we chose our prior to qualitatively behave like the exact solution, though more sophisticated choices can of course be made if there is extra physical information in the problem [20]. In particular, we set a = 1/10 and c = 10 to construct the prior through (5.11). The PDE defined by (5.11) is simulated with the same P1 elements as are used to simulate the Lippmann-Schwinger equations. A sample selected from the prior distribution is given in Figure 5.5 (top left). Note that in amplitude it is of the same order of magnitude as the scatterer defined by mh and that, as is expected for our solution, the prior sample changes smoothly. In Figure 5.5 (top right), we show the mean and a sample of the solution to the Bayesian inverse problem with prior described above and in Figure 5.5 (bottom) we show a sample selected from the computed posterior distribution. Note that the prior variance has affected the reconstruction such that the square scatterer is no longer clearly resting in a homogeneous background. However, the scatterer is still the most significant object in the reconstructions. Moreover, influence from both the prior and MAP can be seen in the sample taken from the posterior distribution - in areas where the prior is large, the posterior is large too, but the MAP estimate dominates the reconstruction. Hence, even in this toy example, we see that the method described above can be used to solve Bayesian inverse problems in a way which incorporates prior information and forward modeling assumptions.

131 0.3 0.3 0.25 3

0.2 0.2 2 0.2

0.1 0.1 1 0.15 0 0 0 0.1 -0.1 -0.1 -1 0.05 -0.2 -0.2 -2 -0.3 -0.3 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

0.3 0.25

0.2 0.2

0.1 0.15 0 0.1 -0.1

0.05 -0.2

-0.3 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Figure 5.5: Bayesian solution to inverse scattering problem. (Top right) A sample generated by the prior distribution on a mesh of size h = 0.0512. (Top left) A sample generated by the posterior distribution on a mesh of size h = 0.0512. (Bottom) The MAP solution on a mesh of size h = 0.0512. Note that the posterior sample and MAP solution use the same color scheme, which is diﬀerent from the posterior color scheme. White lines indicate the mesh.

132 Chapter 6

OUTLOOK AND OPEN PROBLEMS

This thesis has been focused on improving the applicability of the theory of inverse scattering problems to real world problems. In particular, we have focused on decreasing the amount of spatial scattered field data which needs to be collected in order to apply non-iterative reconstruction techniques, and on speeding up wave equation simulations for the reconstructions of physical parameters using iterative techniques. Nonetheless, in many ways these types of questions are new and, in the case of data reduction for qualitative methods, have not been significantly studied. As such, this work is a starting point for a number of further questions, some of which we detail here. Perhaps the most significant open problem related to this work is related to the use of the Born approximation in theoretical justification. With respect to the results in this thesis, the most important question related to this is

1. What are the minimum assumptions on n and D so that a quasi-backscattering experimental set-up yields satisfactory reconstructions?

Indeed, even in the weak scattering case, more work must be done in general on optimal data collection requirements outside of the quasi-backscattering framework. An important application is the reconstruction of inhomogeneities with phase-free data. Whether such a scheme could be applied in a quasi-backscattering or other data-limited case is not clear and worth investigating. Beyond that, however, more work on the quasi-backscattering approach comes from adapting the it to models other than acoustic scattering. For example, scattering from sound-soft or sound-hard obstacles, anisotropic penetrable media, or even scattering electromagnetic or elastic waves

133 have not been considered. The behavior of scattered waves in each of these models is similar and there is no obvious reason why the results presented in Chapters2 or3 are not applicable. However, in particular for time domain problems, a careful theoretical analysis must be done before proceeding. Questions related to linear sampling in the time domain and the Born transmission eigenvalue problem are also worth exploring further. Of special importance are

2. Under what conditions do Born transmission eigenvalues all lie in a horizontal strip of the real axis; and

3. Under these conditions, can the Linear Sampling Method for the Born regime be fully justiﬁed in this case?

As a ﬁnal set of open problems, we remark again on the connections between Bayesian inverse problems and the other main topics of this thesis: qualitative inverse problems and data reduction techniques.

4. Given an a priori assumption about a scatterer, is it possible to place transmitters and receivers in a way so-as to maximize the probability of building a successful reconstruction?

5. Can the LSM, or other non-iterative methods, and Bayesian techniques be linked in a way which uses the positive aspects of both reconstruction methods? For example, in the case of a non-constant background index of refraction - say a quickly oscillating one - the LSM fails unless this index of refraction is known precisely. Similarly, a Bayesian technique would likely produce a probability distribution describing the index of refraction, but the underlying numerical discretization would have to be extremely ﬁne in order to capture the fast oscillations, leading to very long computational times. A faster and more accurate solution would be obtained by combining the strengths of non-iterative and iterative techniques.

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142 Appendix A

SEPARATION OF VARIABLES SOLUTIONS

In this Appendix, we derive a series solution for the Helmholtz equation with constant refractive index in two–dimensional balls. These solutions are, for example, be used to test the accuracy of numerical approximations. They are also used in Chapter 4 when ﬁnding solutions to a related problem. Hence, we now derive a series solution for time-harmonic acoustic scattering from a ball with constant index of refraction. As such, assume the index of refraction n(x) = 1 + (m − 1)χD where m ∈ (0, 1) is a constant and where the domain D = 2 BR(0) ⊂ R , for R > 0. We write the inhomogeneous Helmholtz equation as a system of equations for functions with support deﬁned inside D and outside of D,

2 ∆ˆuint + k muˆint = 0 in D

s i 2 s i 2 ∆(ûext +u ˆ ) + k (ûext +u ˆ ) = 0 in R \D s (1)/2 ∂uêxt s lim r + ikuêxt = 0, r = |x|. r→∞ ∂r

Note that this is equivalent to the Helmholtz equation, whereu înt is the total field i s inside the domain D,u ˆ is the incident field, andu êxt is the scattered field outside i ˆ ˆ of D. Assume thatu ˆ (x; y) = Φk(x, y) where Φk is the fundamental solution for the Helmholtz equation in R2 with wave number k > 0. We add an additional set of constraints to these equations which ensures the total field in all of R2 is a continuous function, ∂uˆ ∂uˆs ∂uî uˆ (R) =u ˆs (R) +u î(R) and int (R) = ext (R) + (R). (A.1) int ext ∂ν ∂ν ∂ν

s i As discussed above, we can expand each ofu ˆint, uˆext, andu ˆ as power series of special functions depending separately on the radial variable and angular variable(s).

143 The well-known addition theorems allow us to do this for the incident field. In partic- ˆ 2 ular, let Φk be the fundamental solution for the Helmholtz equation in R with wave number k > 0. Then, we have [36] ∞ i X Φˆ (x, y) = H(1)(k|x|)J (k|y|) exp(inθ)) k 4 n n n=−∞ (1) where |x| > |y| and θ is the polar angle between x and y. The functions Hn and Jn are Hankel functions of the first kind and Bessel functions, respectively. For significantly more detailed information on these functions see for example [21, Chapter 3.2], [36, Chapters 2.3 and 2.4], [72, Chapters 5-8], [77, Chapters 2 and 3], or [95, Chapters XV and XVII]. The scattered field can be written in terms of the same spatial and angular functions as the incident field. In particular, ∞ s X (1) uêxt(x; y) = anHn (k|x|) exp (inxˆ) n=0 and ∞ X uînt(x; y) = bnJn(km|x|) exp (inxˆ). n=0 Here an, bn ∈ C are unknown coefficients depending on R, k, and y. We use the addition theorem to writeu î in terms of |x|, |y|,x ˆ, andy ˆ. Using (A.1) allows us to solve for the unknown coefficients an and bn. In particular, multiplying the equation expressing continuity across ∂D by exp (−i`xˆ) and integrating over S1 yields i b J (kmR) = a H(1)(kR) + H(1)(k|y|)J (kR) exp (−i`yˆ), ` ∈ . ` ` ` ` 4 ` ` Z Similarly, the continuity of normal derivatives condition yields i kmb J 0(kmR) = ka H(1),0(kR) + kH(1)(k|y|)J 0(kR) exp (−i`yˆ), ` ∈ . ` ` ` ` 4 ` ` Z

To ﬁnd for an and bn, we solve the system of equations

 (1)      H` (kR) −J`(kmR) a` i (1) J`(kR)     = − H` (k|y|) exp (−i`yˆ)   . (1),0 0 4 0 H` (kR) −mJ`(kmR) b` J`(kR) for each ` ∈ Z. We do not explicitly solve for an or bn, as their forms are not particularly elucidating.

144 Appendix B

Permissions for “Quasi-backscattering in the frequency domain” This chapter was based on the article, ”A quasi-backscattering problem for inverse acoustic scattering in the Born regime,” published in Inverse Problems, volume 31. Copyright 2015 by Institue of Physics Publishing.

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Permissions for ”Direct imaging of small scatterers using reduced time dependent data” This chapter was based on the article, ”Direct imaging of small scatterers using reduced time dependent data,” published in the Journal of Computational Physics volume 338. Copyright 2017 by Elsevier.

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145 Permissions for “The Born transmission eigenvalues problem” This chapter was based on the paper, ”The Born transmission eigenvalue problem,” published in Inverse Problems, volume 32. Copyright 2016 by Institue of Physics Publishing.

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146