![Preface
This book is devoted to the problem of shape identification. Problems of this type occur
in a number of important fields which belong to the class of inverse problems. As the
first of these fields, we mention inverse scattering problems where one wants to detect –
and identify – unknown objects through the use of acoustic, electromagnetic, or elastic
waves. Complex models in scattering theory involve boundary value problems for partial
differential equations such as Maxwell’s equations in electromagnetics, and one of the
important problems in inverse scattering theory is to determine the shape of the obstacle
from field measurements. Applications of inverse scattering problems occur in such
diverse areas as medical imaging, material science, nondestructive testing, radar, remote
sensing, or seismic exploration. A survey on the state of the art of the mathematical
theory and numerical approaches for solving inverse time harmonic scattering problems
until 1998 can be found in the standard monograph [43] by David Colton and Rainer
Kress. We also refer to Chapter 6 of [106] and [155] for an introduction and survey on
inverse scattering problems.
The second important area where the identification of unknown shapes plays an
important role is tomography, in particular, electrical impedance tomography or optical
tomography. Electrical impedance tomography is a technique to recover spatial prop-
erties of the interior of a conducting object from electrostatic measurements taken on
its boundary. For example, a current through a homogeneous object will, in general,
induce a different potential than the same current through the same object containing
an enclosed cavity. The problem of impedance tomography, we are interested in, is to
determine the shape of the cavity from measurements of the potential on the boundary
of the object. For survey articles on this subject we refer to [16] by Liliana Borcea and
[82] by Martin Hanke and Martin Brühl.
Shape identification problems are intrinsically nonlinear, i.e., the measured quan-
tities do not depend linearly on the shape. Even the notion of linearity does not make
sense since, in general, the set of admissible shapes does not carry a linear structure.
Traditional (and still very successful) approaches describe the objects by appropriate
parameterizations and compute the parameters by iterative schemes as, e.g., Newton-
type methods. Besides the well-known advantages (fast convergence) and disadvantages
(only local convergence properties) of iterative methods for nonlinear problems, these
methods share the common drawback that important information on the unknown object
such as the number of connectivity components or the type of the boundary condition has
to be known in advance. Nevertheless, methods of this type are widely used – in particular
because the first or second order derivatives can be characterized by using techniques](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-10-320.jpg)
![vi Preface
from the shape optimization theory. We refer to [156, 173] for general references and
[112, 134, 135, 84, 83, 86] for applications in inverse scattering theory.
In particular, classical iterative algorithms using explicit parameterizations of the
objects are not able to change the number of connectivity components during the algo-
rithm. This observation has led to the development of level set methods which are based
on implicit representations of the unknown object involving an “evolution parameter” t.
Since the pioneering work [154] by S. Osher and J. Sethian, this method has been further
developed and applied in a huge number of papers. We refer to [22] for a recent survey.
Since around 1995 iterative methods for solving problems in shape optimization have
been developed which completely avoid the use of parameterizations and replace the
classical Fréchet derivative by a geometrically motivated topological derivative, see,
e.g., [171, 172]. These methods have also been applied to problems in inverse scattering
theory in [15, 78]. We refer also to [67].
While very successful in many cases, iterative methods for shape identification prob-
lems – may they use classical tools as the Fréchet derivative or more recent techniques
such as domain derivatives, level curves, or topological derivatives – are computationally
very expensive since they require the solution of a direct problem in every step. Further-
more, for many important cases the convergence theory is still missing. This is due to the
fact that these problems are not only nonlinear but also because their linearizations are
improperly posed.Although there exist many results on the convergence of (regularized)
iterative methods for solving nonlinear improperly posed problems (see, e.g., [62, 87]),
the assumptions for convergence are not met in the applications to shape identification
problems.1
Thesedifficultiesanddisadvantagesofiterativeschemesgaverisetothedevelopment
of different classes of non-iterative methods which avoid the solution of a sequence of
direct problems. We briefly mention decomposition methods (according to the notion of
[44]) which consist of an analytic continuation step (which is linear but highly improperly
posed) and a nonlinear step of finding the boundary of the unknown domain by forcing
the boundary condition to hold. We refer to [6, 3, 45, 46, 123, 50, 125, 132] for some
versions of this approach. In Section 7.2 we will briefly recall the Dual Space Method of
Colton and Monk (see below) which belongs to this class.There is also a close connection
to the Point-Source Method of Roland Potthast in [157, 158, 160].
In this monograph, we will focus on a different class of non-iterative methods. The
common idea is the construction of criteria on the known data to decide whether a given
point z (or a curve or a set) is inside or outside the unknown domain D. By choosing a
grid of “sampling” points z (or collection of curves or sets) in a region known to contain
the unknown domain D one is therefore able to compute the (approximate) characteristic
function of D. In the following, we will collect these approaches under the name sampling
methods. They differ in the way of defining the criterion and in the type of test objects.
One of the first methods which falls into this class has been developed by David
Colton and one of the authors of this book (A.K.) in 1996 ([39]), now known as the
Linear Sampling Method. Its origin goes back to the Dual Space Method developed by
1 Or, at least, it is unknown whether these assumptions are fulfilled or not.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-11-320.jpg)
![Preface vii
David Colton and Peter Monk during 1985 and 1990 (see, e.g., [45, 46, 49] or [43]).
We also mention the work [105] of Victor Isakov in which singular functions are used
to prove uniqueness of an inverse scattering problem. The numerical implementation of
the Linear Sampling Method is extremely simple and fast because sampling is done by
points z only. For every sampling point z one has to compute the field of a point source
in z with respect to the background medium2 (if this is constant the response is even
given analytically) and evaluate a series, i.e., a finite sum in practice.
In 1998 Masaru Ikehata published the paper [96] in which he presented a method
now known as Ikehata’s Probe Method. Instead of points, the region is probed by curves
(called needles in [96]), and points on these curves are identified which belong to the
boundary of the unknown domain D. In many subsequent papers, mainly by Ikehata
and his collaborators, the probe method has been applied to several inverse scattering
problems and inverse conductivity problems (see [95, 97, 98, 99, 100, 101, 102, 33,
103, 104]). We explicitly mention [33, 63] for numerical implementations of the probe
method.
In 2000 Roland Potthast presented a sampling method in [159] which he later in
[160], Chapter 6, called the Singular Sources Method. The idea, formulated here for the
inverse scattering problem, is to approximate (from the given data) the scattered field
vs(·, z) which belongs to the field of a point source in z as incident field. This scattered
field vs(z, z), evaluated at the same source point z, is unbounded if z approaches the
boundary ∂D from the exterior. Therefore, the unknown region D is found as the set
of points z where vs(z, z) becomes large. In this sense, we consider this method as a
sampling method with respect to point sampling.
Sampling by sets is done in the Range Test and the No Response Test, developed
by Roland Potthast, John Sylvester, and Steven Kusiak in [163] and Russell Luke and
Roland Potthast in [141]. We refer also to [88].
Aproblem with all of the mentioned methods (except of Ikehata’s probe method) from
the mathematical point of view is that the computable criterion provides only sufficient
conditions which are, in general, not necessary. The Factorization Method, developed
by the authors in [114, 115, 76, 74] overcomes this drawback and provides a criterion
for z which is both, necessary and sufficient. Therefore, this method provides a simple
formula for the characteristic function of D which can easily be used for numerical
computations. We emphasize, that these results hold in the resonance region, i.e., no
asymptotic forms such as the Born approximation or the geometric or physical optics
approximations are assumed. Compared to Ikehata’s Probe Method the Factorization
Method is much more direct, both from the theoretical point of view as well as with
respect to the computational implementation. From the numerical point of view, the
Linear Sampling Method and the Factorization Method are equally simple and fast.
A typical feature of these two methods is that they make no explicit use of boundary
conditions or topological properties of D. In other words, they determine the unknown
domain without knowing in advance the type of boundary condition or the number of
components.
2 Essentially, one has to compute the Green’s function.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-12-320.jpg)
![viii Preface
Since their first presentations, the Linear Sampling Method and the Factorization
Method have been developed for several problems in inverse scattering theory and
tomography. We refer to [24, 23, 26, 27, 28, 35, 37, 38, 80, 65, 178] for some recent
work on the Linear Sampling Method and [7, 116, 117, 129, 118, 77, 119, 120] for
papers related to the Factorization Method in inverse scattering theory. The interesting
papers [8] and [11] by Tilo Arens and Armin Lechleiter discovers a deeper relationship
between these two methods. Stimulated by the first paper [114] Martin Brühl and Martin
Hanke investigated the Factorization Method for problems in impedance tomography
(see [18, 19, 20, 82, 21] and later [90, 92, 94]) by Nuutti Hyvönen. For a more general
approach to elliptic equations we refer to [120] and [68]. There are several applications of
the Factorization Method which are not covered in this monograph. We mention scatter-
ing problems for periodic surfaces or arcs (cf. [10, 9] and [129], respectively), for elastic
media (see [7, 29]), for static problems (cf. [81, 133, 131]), or in optical tomography
(cf. [14, 89, 91, 93]).
However, it should also be mentioned that the range of problems for which the
Factorization Method has been justified from the mathematical point of view is con-
siderably smaller than the one for the Linear Sampling Method or the other sampling
methods.
While the main subject of this monograph is the Factorization Method, we will
report on the Linear Sampling Method, Ikehata’s Probe Method, and Potthast’s Singular
Sources Method in Chapter 7. We also refer to the survey articles [161, 162] by Roland
Potthast on Sampling and Probe Methods and to [88] for an interesting relationship
between sampling methods and iterative methods.
The monograph is organized as follows. Chapters 1–4 study the Factorization
Method for scattering problems where the wave propagation is described by the three-
dimensional scalar Helmholtz equation. In Chapters 1–3 impenetrable scatterers D are
considered where boundary conditions on the boundary ∂D of D of Dirichlet, Neumann,
impedance or mixed type are imposed. Chapter 4 is devoted to the penetrable case,
and we show that the Factorization Method can be considered as an extension of the
well-known MUSIC-algorithm from signal processing (cf. [60, 30]).
There exist several variants of the Factorization Method. The – in our opinion –
most satisfactory version holds for scattering problems with non-absorbing media, such
as the obstacle scattering case with Dirichlet or Neumann boundary conditions. The
Factorization Method for this situation is investigated in detail in Chapter 1. The math-
ematically most important feature of these problems is the normality of the far field
operator. This makes it possible to use the spectral theory for normal operators. The
authors think that the Factorization Method is a particularly interesting and useful appli-
cation of this theory. As an intermediate step we prove a characterization of the scatterer
D by an inf-condition which is, although not as elegant as the final characterization by
the solvability of an equation, the basis for characterizations of D for scattering problems
for absorbing media.
We consider Chapter 1 also as an introduction into our method and emphasize that
it can not be left out by the reader because it sets up the basis for all subsequent
chapters.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-13-320.jpg)
![Preface ix
A first example of a case where the far field operator fails to be normal is studied in
Chapter 2. The impedance boundary condition serves as a simple model for an absorbing
medium. Since the far field operator F is no longer normal the final characterization of
Chapter 1 does not hold and has to be modified. This is done by considering a suitable
combination F# of the self-adjoint parts (F +F∗)/2 and (F −F∗)/(2i) of F which finally
leads to a characterization of D by the solvability of an equation involving F# instead of F.
Chapter 3 is devoted to mixed boundary conditions. The obstacle is assumed to
consist of several parts, and on some them we impose Dirichlet boundary conditions,
on the others Neumann or, more general, impedance boundary conditions. Even for the
Dirichlet–Neumann case, where the far field operator F is still normal, it is an open
problem whether or not the Factorization Method (in any of its forms) can be justified.
Numerical experiments indicate that this is indeed the case but a rigorous proof is not
known. However, if we a priori know some domains which enclose the parts with the
Dirichlet boundary condition and the impedance boundary condition we can modify the
operator F# appropriately to treat this case as well.
In Chapter 4 we study the penetrable case, i.e., scattering by an inhomogeneous
medium where we allow the medium to be absorbing. The techniques developed in
Chapters 1 and 2 allow us to prove a characterization of the shape of the contrast (which
is the difference between the indices of refraction of the scattering medium and the
background medium) by the same operators used in Chapters 1 and 2. We note already
here that this implies in practice that one does not need to know the type of obstacle –
penetrable or impenetrable – in advance.
While Chapters 1–4 treat scattering problems for the scalar Helmholtz equation we
investigate the Factorization Method in Chapters 5 and 6 for the scattering of time har-
monic electromagnetic waves and the problem of impedance tomography, respectively.
One assumption for the validity of the Factorization Method is that the square k2 of
the wavenumber k is not an eigenvalue of a corresponding eigenvalue problem. In the
case of an impenetrable obstacle with Dirichlet or Neumann boundary conditions this
eigenvalue problem is just the classical eigenvalue problem for − in the domain D with
respect to the boundary conditions. For the scattering by an inhomogeneous medium,
however, a new type of eigenvalue problem (the “interior transmission eigenvalue prob-
lem”) occurs which fails to be self-adjoint (cf. [55]). In Sections 4.5 and 5.5 we show
under certain assumptions on the index of refraction that the eigenvalues form at most a
countable set. The question of existence of eigenvalues is only settled for the spherically
stratified case.
In Chapter 6 we investigate the problem of impedance tomography. In contrast to
the scattering problems this problem is set up as a boundary value problem in a bounded
domain B. The inverse problem we are interested in is to determine the shape D of
an inclusion with different electrical properties than the background medium. In this
application, the Factorization Method is set up for the difference of the Neumann–
Dirichlet operators for the cases with and without inclusion rather than for the far field
operator.
As mentioned above, the Factorization Method is only one of a class of new
approaches for solving “geometric” inverse problems. In Chapter 7 we introduce the](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-14-320.jpg)
![x Preface
reader to some related sampling methods. In contrast to the Factorization Method they
all use heavily the fact that every solution of the Helmholtz equation in some domain
G can be approximated arbitrarily well by solutions of the Helmholtz equation in larger
domains. We summarize two of such approximation theorems in Section 7.1. The Linear
Sampling Method can be considered as the precursor of the Factorization Method and
is closely related to the latter one. We present this method in Section 7.2 and show its
relationship to the Dual Space Method of Colton and Monk. In Section 7.3 we present
the basic ideas of the Singular Sources Method of Roland Potthast. Finally, in Sub-
section 7.4.1 of Section 7.4 we explain the Probe Method of Masaru Ikehata for the
impedance tomography problem and extend it in Subsection 7.4.2 to the inverse scatter-
ing problem with boundary conditions of mixed type. Here we follow the presentation
of [74].
In this monograph we use several spaces of functions on domains G or their bound-
aries ∂G. We try to follow the standard notations for these spaces. The boundary value
problems in bounded domains are set up in the Sobolev space H1(G) of (Lebesgue) mea-
surable functions such that their derivatives (in the sense of distributions) are regular
and belong to L2(G). This space is equipped with the inner product
(u, v)H1(G) = (u, v)L2(G) +
∇u, ∇v
L2(G)
where we denote by (u, v)L2(G) =
G u(x) v(x) dx the inner product in L2(G). By v(x)
we denote the complex conjugate of v(x). If u and v are vector fields then u(x)v(x) has
to be understood as the scalar product
3
j=1 uj(x)vj(x). The corresponding norms are
denoted by · L2(G) and · H1(G)
For k ∈ N we denote by Ck(G) the space of functions for which all partial derivatives
up to order k exist in G and are continuously extendable to the closure G of G. The
norms in Ck(G) are denoted by · Ck (G). We set C∞(G) =
k∈N Ck(G). Then we can
equivalently define H1(G) as the completion of C1(G) with respect to the inner product
(·, ·)H1(G). Spaces of vector fields are denoted by H1(G, C3) or Ck(G, C3).
We assume that G is a Lipschitz domain. Sometimes we assume that even ∂G ∈ C1,α
or ∂G ∈ C2. For definitions of these notations as well as of spaces Cj,α(G) or Cj,α(∂G)
of Hölder continuous functions we refer to [146] (Section 3.2), [144], Chapter 3, [43],
Section 2.2, or [167], Section 6.4. Then the spaces C(∂G) and L2(∂G) are defined in the
usual way using local coordinates. It can be shown (see, e.g., [17], Section 3) that the
trace operator γ : C1(G) → L2(∂G), γ u = u|∂G, has a bounded extension on H1(G).
Its range space
R(γ ) =
ψ ∈ L2
(∂G) : there exists u ∈ H1
(G) with γ u = ψ
is denoted by H1/2(∂G) and equipped with the norm
ψH1/2(∂G) = inf
uH1(G) : u ∈ H1
(G) with γ u = ψ
.
We define H−1/2(∂G) as the dual space of H1/2(∂G). We denote by ·, · the dual form
in H−1/2(∂G), H1/2(∂G), which is the extension of the inner product (·, ·)L2(∂G) :](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-15-320.jpg)
![Preface xi
L2(∂G)×L2(∂G) → C to ·, · : H−1/2(∂G)×H1/2(∂G)) → C. We note that both, the
inner product (·, ·)L2(∂G) and the dual form ·, · are sesqui-linear forms. Furthermore,
H1
0 (G) =
u ∈ H1(G) : γ u = 0 on ∂G
can be constructed as the closure in H1(G) of
the space C1
0 (G) of C1-functions with compact support in G. We denote by R(F) and
N(F) the range space and the null space, respectively, of an operator F.
Finally, we come to the pleasant task of thanking those who have supported and
encouraged us for starting – and finishing – this project. Here we want to mention three
working groups who have influenced the research on sampling methods in a very essen-
tial way. First of all, we would like to thank the research group of Fioralba Cakoni,
David Colton, Russell Luke, and Peter Monk from the University of Delaware who not
only made important contributions to the field but also provided a warm and hospitable
environment for one of the authors (A.K.) to spend several weeks in Newark during
the past years. Second, the working group at the University of Göttingen of Rainer
Kress, Thorsten Hohage, and Roland Potthast (now at the University of Reading) are
one of the most active groups in Germany in the field of inverse scattering problems
and encouraged us to present the ideas of the Factorization Method to a broader audi-
ence of interested mathematicians, physicists, and engineers.Third, the working group of
Martin Hanke-Bourgeois at the University of Mainz developed the Factorization Method
for problems of electrical impedance tomography which started with the PhD thesis of
Martin Brühl [18]. During a common BMBF-project funded by the German Federal
Ministry of Education and Research many discussions resulted in new ideas, new gener-
alizations,andnewapplicationsoftheFactorizationMethod(see,e.g.,thejointpaper[69]
and also [122]).
Last but not least particular thanks are given to the members of our own research
group at the Department of Mathematics, in particular to Tilo Arens, Frank Hettlich,
Armin Lechleiter, and Sebastian Ritterbusch for many fruitful discussions and argu-
ments – and also for their willingness to shoulder “daily” duties during the preparation
of this monograph.
The colored versions of the plots are available on my homepage
www.mathematik.uni-karlsruhe.de/iag1/∼kirsch/en
Karlsruhe, Germany
May 2007
Andreas Kirsch
Natalia Grinberg](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-16-320.jpg)
![1
The simplest cases: Dirichlet and
Neumann boundary conditions
As pointed out in the introduction this chapter is devoted to the analysis of the
factorization methods for the most simplest case in scattering theory. We consider
the scattering of time-harmonic plane waves by an impenetrable obstacle D which
we model by assuming Dirichlet boundary conditions on the boundary = ∂D of D.
With respect to the factorization methods we will carry out all proofs in detail. We
point out that the title of this chapter should not lead to the wrong conclusion that this
“simplest case” could be left out by those readers interested in the factorization method
for more complicated models. In this chapter we formulate the basic functional analytic
results which form the basis of the method and will be referred to several times in later
chapters.
After a short derivation of the Helmholtz equation from the basic equations in con-
tinuum mechanics we will repeat in Section 1.2 some well-known results on the direct
scattering problem. We will omit the proofs but refer to the existing literature such
as [43]. However, we will emphasize the important “ingredients”: Rellich’s Lemma
and unique continuation for the problem of uniqueness of the scattering problem and
Green’s theorem for the derivation of important properties such as the reciprocity
principles.
Section 1.3 will collect analytical results on the inverse scattering problem such as
uniqueness of the inverse problem and properties of the far field operator. Here we will
present the proofs of those results which are necessary for the factorization method.
We start Section 1.4 with the basic factorization of the far field operator. Then in
Subsection 1.4.2 a quite general approach is discussed in which the domain D is char-
acterized by those points z for which the infimum of a certain function (depending on z)
is positive. For the special case where the scattering operator is unitary – which is the
case for the scattering by an obstacle under Dirichlet boundary conditions – this repre-
sentation can be transformed into the characterization of D by those points z for which
a certain equation of the first kind (where the right-hand side depends on z) is solvable
or not.
In Section 1.6 we will briefly treat the case of Neumann boundary conditions. The
analysis is quite analogous to the Dirichlet case.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-20-320.jpg)
![The direct scattering problem 5
Figure 1.1 Incident, scattered, and total field
We illustrate this situation in Figure 1.1 which shows the real parts of the incident
plane wave ui (left picture), the scattered wave us (middle picture), and the total field u
(right picture), respectively, in two dimensions1.
Observing that the incident field ui satisfies the Helmholtz equation (1.15) in all of R3
we note that the scattered field us solves the following exterior boundary value problem
for f = −ui:
Given f ∈ H1/2() find v ∈ H1
loc(R3 D) such that
v + k2
v = 0 outside D, (1.18)
v = f on . (1.19)
and
∂v
∂r
− ik v = O
r−2
for r = |x| → ∞ (1.20)
uniformly with respect to x̂ = x/|x|.
We note that the solution of (1.18) is understood in the variational sense. Indeed,
using the Helmholtz equation in Green’s first formula for the region DR =
x ∈ R3 D :
|x| R
DR
ϕ v + ∇ϕ · ∇v
dx =
R
ϕ
∂v
∂ν
ds (1.21)
yields
R3D
∇ϕ · ∇v − k2
ϕv
dx = 0 (1.22)
for any test function ϕ with compact support in R3 D (choose R such that that the ball
of radius R contains the support of ϕ). Here and throughout this monograph, ν = ν(x)
denotes the unit normal vector at x ∈ ∂D directed into the exterior of D. It follows from
1 The setting of the two-dimensional scattering problem, i.e., where x, θ ∈ R2 and D ⊂ R2, differs from the
one in three dimensions only in the radiations condition which has now the form ∂us/∂r − ik us = O
r−3/2
.
We refer to [43] and Section 1.7.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-24-320.jpg)
![6 Dirichlet and Neumann boundary conditions
interior regularity results for elliptic differential equations (cf. [70]) that any solution of
(1.22) is a classical solution of the Helmholtz equation in R3D.We call v ∈ H1
loc(R3D)
a variational solution of (1.18), (1.19), and (1.20) if v solves (1.22) for all ϕ ∈ H1
0 (R3D)
which vanish outside of some ball, and v = f on in the sense of the trace theorem and
v satisfies Sommerfeld’s radiation condition (1.20).
Thefundamentalresultsonuniquenessandexistencearesummarizedinthefollowing
theorem.
Theorem 1.1 For any f ∈ H1/2() there exists a unique (variational) solution v ∈
H1
loc(R3 D) of (1.18), (1.19), and (1.20).
Furthermore, if the boundary data f is continuous on then v ∈ C2(R3 D) ∩
C(R3 D). If f is even continuously differentiable on then the normal derivative
∂v/∂ν exists and is continuous on .
For a proof we refer to [150] and [43] (see also Chapter 2, Section 2.1 where we show
existence and uniqueness for the Robin boundary conditions).
Solutions of the Helmholtz equation (1.18) in some exterior domain which satisfy also
Sommerfeld’s radiation condition (1.20) will be referred to as radiating solutions of
(1.18).
The uniqueness part in the proof of the previous theorem makes essential use of the
following result which is due to Rellich [166] (cf. [43]).
Lemma 1.2 Let v be a solution of the Helmholtz equation (1.18) in some region of the
form {x ∈ R3 : |x| R} satisfying
lim
r→∞
|x|=r
|v(x)|2
ds = 0 .
Then v vanishes for |x| R.
For a proof we refer to [43].
Green’s formula is the essential tool also in the proof of the following representation
formula for radiating solutions of the Helmholtz equation which sometimes is called
“Green’s representation formula”.
Theorem 1.3 Let v ∈ C2(R3 D) ∩ C(R3 D) be a radiating solution of (1.18) such
that v posseses a normal derivative on the boundary in the sense that the limit
∂v(x)
∂ν
= lim
h→+0
ν(x) · ∇v
x + hν(x)
, x ∈ ,
exists uniformly with respect to x ∈ . Then Green’s formula holds in the form
v(x) =
v(y)
∂(x, y)
∂ν(y)
−
∂v(y)
∂ν
(x, y) ds(y) , x /
∈ D . (1.23)](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-25-320.jpg)
![The far field patterns and the inverse problem 7
Again, ν(y) denotes the exterior unit normal vector at y ∈ and the fundamental
solution of the Helmholtz equation in R3 given by
(x, y) =
exp(ik|x − y|)
4π|x − y|
, x, y ∈ R3
, x = y . (1.24)
For a proof we refer to [43, Theorem 2.4].
As a direct consequence of this theorem one has the following result. Its proof can again
be found in [43], Theorems 2.5 and 2.6.
Theorem 1.4 Let v ∈ C2(R3 D) ∩ C(R3 D) be a radiating solution of (1.18) such
that v posses a normal derivative on the boundary in the sense of Theorem 1.3.
Then v is analytic in R3 D and has the asymptotic behavior
v(x) =
exp(ik|x|)
4π|x|
v∞
(x̂) + O(|x|−2
) , |x| → ∞ , (1.25)
uniformly with respect to x̂ = x/|x| ∈ S2. The function v∞ : S2 → C is analytic and is
called the far field pattern of v. It has the form
v∞
(x̂) =
v(y)
∂
∂ν(y)
e−ikx̂·y
−
∂v(y)
∂ν
e−ikx̂·y
ds(y) , x̂ ∈ S2
. (1.26)
As a consequence of the analyticity of v and Rellich’s Lemma 1.2 we have
Corollary 1.5
(a) If v vanishes on some open subset of R3 D then v vanishes everywhere in R3 D.
(Note that we always assume that the exterior of D is connected.)
(b) If v∞ vanishes on an open part of S2 (open relative to S2) then v vanishes in the
exterior of D.
Application of these results to the scattering problem (1.15), (1.16), and (1.17) assures
existence of a unique solution u for any incident field ui. Its dependence on the incident
direction is indicated by writing u = u(·, θ). Analogously, the far field pattern u∞ of us
dependsonthetwoangularvaluesx̂ andθ.Weindicatethisbywritingu∞ = u∞(x̂, θ)and
note that u∞ depends analytically on both variables.This follows, e.g., fromTheorem 1.6
below.
1.3 The far field patterns and the inverse problem
First, we will prove a reciprocity principle for u∞. It states the (physically obvious) fact
that it is the same if we illuminate an object from the direction θ and observe it in the
direction −x̂ or the other way around: illumination from x̂ and observation in −θ.
Theorem 1.6 (First reciprocity principle) Let u∞
x̂, θ
be the far field pattern corre-
sponding to the direction x̂ of observation and the direction θ of the incident plane wave.
Then
u∞
−x̂, θ
= u∞
−θ, x̂
for all x̂, θ ∈ S2
. (1.27)](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-26-320.jpg)
![8 Dirichlet and Neumann boundary conditions
Proof: Application of Green’s second theorem to ui and us in the interior and exterior
of D, respectively, yields
0 =
ui
(y, θ)
∂
∂ν
ui
(y, x̂) − ui
(y, x̂)
∂
∂ν
ui
(y, θ) ds(y) ,
0 =
us
(y, θ)
∂
∂ν
us
(y, x̂) − us
(y, x̂)
∂
∂ν
us
(y, θ) ds(y) .
(More precisely, to prove the second equation, one applies Green’s second theorem to
us in the region {x ∈ R3 D : |x| R} with R large enough and lets R tend to infinity.)
Now we use the representation (1.26) for the far field patterns u∞
−x̂, θ
and u∞
−θ, x̂
:
u∞
(−x̂, θ) =
us
(y, θ)
∂
∂ν
ui
(y, x̂) − ui
(y, x̂)
∂
∂ν
us
(y, θ) ds(y) ,
−u∞
(−θ, x̂) =
ui
(y, θ)
∂
∂ν
us
(y, x̂) − us
(y, x̂)
∂
∂ν
ui
(y, θ) ds(y) .
Adding these four equations yields
u∞
(−x̂, θ) − u∞
(−θ, x̂) =
u(y, θ)
∂
∂ν
u(y, x̂) − u(y, x̂)
∂
∂ν
u(y, θ) ds(y) . (1.28)
So far, we have not used the boundary condition on . With u(y, x̂) = 0 and u(y, θ) = 0
on the assertion follows.
There exists an interesting second reciprocity principle which relates the scattered field
us = us(x, θ) corresponding to the plane wave of direction θ as incident field to the
far field pattern v∞ = v∞(x̂, z) which corresponds to the point source (·, z) as inci-
dent field. Indeed, by the same arguments as in Theorem 1.6 one can show (cf. [160],
Section 2.1):
Theorem 1.7 (Second or mixed reciprocity principle) Let us(z, −θ) be the scattered
field at z ∈ R3 D which corresponds to the incident field ui(x, −θ) = exp(−ikx · θ),
x ∈ R3. Furthermore, let v∞(θ, z) be the far field pattern of the scattered field vs at θ
which corresponds to the incident field vi(x, z) = (x, z), x ∈ R3. Then
us
(z, −θ) = v∞
(θ, z) for all θ ∈ S2 and z /
∈ D . (1.29)
The far field patterns u∞(x̂, θ), x̂, θ ∈ S2, define the integral operator F : L2(S2) →
L2(S2) by
(Fg)(x̂) =
S2
u∞
(x̂, θ) g(θ) ds(θ) for x̂ ∈ S2
, (1.30)](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-27-320.jpg)
![The far field patterns and the inverse problem 11
(c) This follows now by comparing the forms of S∗S and SS∗.
(d) Let g ∈ L2(S2) be such that Fg = 0 on S2. From the definition of the far field
operator we note that Fg = v∞ where v∞ is the far field pattern which corresponds
to the incident field vi(x) =
S2 exp(ikx · θ) g(θ) ds(θ), x ∈ R3. Rellich’s Lemma 1.2
and analytic continuation imply that the scattered field vs vanishes outside of D.3 From
the boundary condition we conclude that the incident field vi vanishes on , i.e., vi is
a Dirichlet eigenfunction of − in D. From the assumption on the wavenumber vi has
to vanish in D and thus everywhere by analytic continuation. Expansion into spherical
wave functions by using the Jacobi–Anger expansion (see (1.82) below) yields that g
vanishes on S2. Finally, we show that the adjoint F∗ of F is one-to-one as well which
proves denseness of the range of F. Indeed, F∗g = 0 yields by using the reciprocity
relation (1.27) that
0 = (F∗
g)(x̂) =
S2
u∞(θ, x̂) g(θ) ds(θ) =
S2
u∞(−x̂, −θ) g(θ) ds(θ) ,
i.e., Fg̃ = 0 with g̃(θ) = g(−θ). Injectivity of F yields that g̃ = 0 and
thus also g = 0.
Now we turn to the formulation of the inverse scattering problem for which we will
introduce the factorization methods in Section 1.4.
Inverse Scattering Problem: Given the wavenumber k 0 and the far field patterns
u∞(x̂, θ) for all x̂, θ ∈ S2 determine the shape of the scattering obstacle D!
The following uniqueness theorem, taken from [43], assures that in principle the data
set
u∞(x̂, θ) : x̂, θ ∈ S2
is sufficient to determine D.
Theorem 1.9 For fixed wavenumber k 0 the far field patterns u∞(x̂, θ) for all
x̂, θ ∈ S2 uniquely determine the shape of the scattering obstacle D, i.e., if there are
two obstacles D1 and D2 with corresponding far field patterns u∞
1 (x̂, θ) and u∞
2 (x̂, θ),
respectively, then u∞
1 (x̂, θ) = u∞
2 (x̂, θ) for all x̂, θ ∈ S2 implies that D1 = D2.
As in Figure 1.1 we want to illustrate also the inverse scattering problem with two
examples, again in two dimensions. In this case, the far field patterns u∞ depend on
the two variables x̂ and θ from the unit circle in R2 which we identify with the interval
[0, 2π]. Furthermore, we identify x̂ = (cos φ, sin φ) with φ ∈ [0, 2π] and the unit
vector θ with the angle θ ∈ [0, 2π]. Figures 1.2 and 1.3 show contour plots of the real
and imaginary parts, respectively, of the far field patterns for two examples.
The inverse scattering problem is to identify the obstacle D from these plots.
In the first example (Figure 1.2) the contour lines are straight lines, i.e., u∞ is constant
along lines of the form φ − θ = const. In terms of the unit vectors x̂ and θ this can be
written in the form x̂·θ = const. By the following result of Karp [108] (which is also true
for the two-dimensional case) we conclude that this first example corresponds to D being
a disk. The second example corresponds to a domain D which is certainly not a disk.4
3 Here we make use of the assumption that the exterior of D is connected.
4 This belongs to the domain D of Figure 1.7.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-30-320.jpg)
![12 Dirichlet and Neumann boundary conditions
300
200
100
0 0 100 200 300
0
50
100
150
200
250
300
350
350
300
250
200
150
100
50
0
Figure 1.2 Real (left) and imaginary (right) parts of u∞ = u∞(φ, θ) for φ, θ ∈ [0, 2π]
350
300
250
200
150
100
50
0
0 100 200 300
300
200
100
0
0
50
100
150
200
250
300
350
Figure 1.3 Real (left) and imaginary (right) parts of u∞ = u∞(φ, θ) for φ, θ ∈ [0, 2π]
Theorem 1.10 Let k 0 and u∞ = u∞(x̂, θ) for x̂, θ ∈ S2 be the far field patterns
corresponding to some domain D ⊂ R3. If
u∞
(x̂, θ) = u∞
(Qx̂, Qθ)
for all x̂, θ ∈ S2 and all rotations, i.e., all real orthogonal matrices Q ∈ R3×3 with
det Q = 1, then D is a ball with center zero.
The proof is a simple consequence of the uniqueness result of Theorem 1.9, see
Section 5.1 of [43].](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-31-320.jpg)
![Factorization methods 13
Remarks:
(a) We note that by the analyticity of u∞ with respect to both variables it is sufficient to
require that u∞
1 (x̂, θ) and u∞
2 (x̂, θ) coincide for all x̂ and θ from open subsets of S2
or even from an infinite number of pairs (x̂, θm). We refer to [130] for more details.
(b) The historically first uniqueness result for this inverse scattering problem is due to
Schiffer (see remark in [138], Section V.5). The proof depends crucially on the com-
pact imbedding property of H1
0 (G) in L2(G) for any bounded open set G to ensure
that the spectrum of − in G with respect to Dirichlet boundary conditions is dis-
crete. The analogous argument for other boundary conditions such as the Neumann
boundary condition requires the compact imbedding property of H1(G) in L2(G)
which holds only under smoothness assumptions on the boundary of G. Since the
argument is applied to G being the (set-)difference of two obstacles D1 and D2
smoothness of this difference cannot be assured. Therefore, Schiffer’s proof cannot
be transfered to other boundary conditions. Based on results of Isakov [105] for
penetrable obstacles Kirsch and Kress [124] obtained uniqueness results for several
kinds of boundary conditions.
(c) To the authors knowledge it is an open problem whether the far field patterns u∞(x̂, θ)
for all x̂ ∈ S2 but only one incident wave ui(x) = exp(ikx · θ) uniquely determines
D if no a priori information on D is available. Partial results are known if a priori
information on the obstacle D is available. It has been shown by Colton and Sleeman
in [57] (cf. [43], Theorem 5.2) that the scatterer is uniquely determined by the far
field patterns of a finite number of incident plane waves provided a priori information
on the size of the obstacle D is available. In recent papers by Elschner, Yamamoto,
Liu, and others (see [61, 140]) uniqueness of the inverse scattering problem for one
incident plane wave and polyhedral scatterers is shown.
(d) A similar theorem shows that the scatterer is uniquely determined by the far field
patterns for an infinite number of incident plane waves with distinct wavenumbers
from a bounded interval in R0. We refer again to [130] for more details.
1.4 Factorization methods
We recall from (1.30) that the far field patterns u∞ define the far field operator F :
L2(S2) → L2(S2) by
(Fg)(x̂) =
S2
u∞
x̂, θ
g(θ) ds(θ) , x̂ ∈ S2
. (1.36)
With respect to the inverse problem, this operator contains the known data. It is the aim
to give explicit characterizations of the unknown domain D by this “data operator” F.
This section is organized as follows. In Subsection 1.4.1 we will derive a factorization
of the operator F in the form
F = G T G∗
(1.37)](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-32-320.jpg)
![16 Dirichlet and Neumann boundary conditions
To prove denseness of the range of G we recall the definition of the spherical
harmonics Ym
n defined by
Ym
n (x̂) = Ym
n (φ, ϕ) =
2n + 1
4π
(n − |m|)!
(n + |m|)!
P|m|
n (cos φ) eimϕ
, (1.42)
for −n ≤ m ≤ n and n ∈ N. Here (φ, ϕ) are the spherical polar coordinates of x̂ ∈ S2
and Pm
n the associated Legendre functions. The spherical harmonics Ym
n are normalized
such that they form a complete orthonormal system in L2(S2). Furthermore, we denote
by hn the spherical Hankel functions of the first kind and order n ∈ N. We refer to [43],
Sections 2.3 and 2.4, for a brief introduction to these functions. Now we make use of the
fact that every element f in L2(S2) can be approximated by a finite linear combination of
the spherical harmonics, i.e., for every ε 0 there exists N ∈ N and constants cm
n ∈ C
with
f −
N
n=0
|m|≤n
cm
n Ym
n
L2(S2)
≤ ε .
We choose the origin inside of D and define the function v by
v(x) =
k
4π
N
n=0
|m|≤n
cm
n ei(n+1)π/2
hn(k|x|)Ym
n (x̂) , x = 0 .
Thefarfieldpatternofhn(k|x|)Ym
n (x̂)isdeterminedbytheasymptoticbehaviorofhn(t)as
t tends to infinity and is given by (4π/k) exp
−i(n+1)π/2
Ym
n (x̂) (cf. [43]). Therefore,
v∞
=
N
n=0
|m|≤n
cm
n Ym
n
and thus f − v∞L2(S2) ≤ ε. Observing that v∞ = Gv| yields the assertion.
The operator T in (1.37) will be the adjoint of the single layer boundary operator S :
H−1/2() → H1/2(), defined by
(Sϕ)(x) :=
(x, y) ϕ(y) ds(y) , x ∈ , (1.43)
where denotes again the fundamental solution of the Helmholtz equation in three
dimensions as defined in (1.24). Note that we write this as an ordinary integral although,
strictly speaking, this has to be understood as the bounded extension of the classical
operator S defined on L2().
In the following lemma we summarize some of the well-known properties of S which
will imply, in particular, that T in (1.37) is an isomorphism. By H−1/2(), H1/2()
we denote the dual form which, in our setting, is the extension of the inner product of
L2(). In particular, this dual form is sesqui-linear, i.e., the mappings ϕ → ϕ, ψ and
ψ → ϕ, ψ are linear where the bar denotes the complex conjugation.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-35-320.jpg)
![Factorization methods 17
Lemma 1.14 Assume that k2 is not a Dirichlet eigenvalue of − in D. Then the following
holds.
(a) S is an isomorphism from the Sobolev space H−1/2() onto H1/2().
(b) Imϕ, Sϕ 0 for all ϕ ∈ H−1/2() with ϕ = 0. Again, ·, · denotes the duality
pairing in H−1/2(), H1/2().
(c) Let Si be the single layer boundary operator (1.43) corresponding to the wavenumber
k = i. The operator Si is self-adjoint and coercive as an operator from H−1/2()
onto H1/2(), i.e., there exists c0 0 with
ϕ, Siϕ ≥ c0ϕ2
H−1/2()
for all ϕ ∈ H−1/2
() . (1.44)
(d) The difference S − Si is compact from H−1/2() into H1/2().
Proof: (a) The mapping properties of S in Sobolev spaces are intensively studied
in [144].
(b) Define, for any ϕ ∈ H−1/2(), the single layer potential v by
v(x) =
ϕ(y) (x, y) ds(y) , x ∈ R3
. (1.45)
Then v is a solution of the Helmholtz equation in R3 . From the theory of potentials
with H−1/2()−densities it is known (see [111, 144]) that v ∈ H1(D) ∩ H1
loc(R3 D),
that the traces v± and ∂v±/∂ν exist in the variational sense with v± = Sϕ and ϕ =
∂v−/∂ν − ∂v+/∂ν. Here, v± denotes the limit from the exterior (+) and interior (−),
respectively. Therefore, using Green’s formula in D and in DR :=
x ∈ R3 D : |x| R
we conclude that
ϕ, Sϕ =
∂v−
∂ν
−
∂v+
∂ν
, v
(1.46)
=
D∪DR
|∇v|2
− k2
|v|2
dx −
|x|=R
v
∂v
∂r
ds (1.47)
=
D∪DR
|∇v|2
− k2
|v|2
dx − ik
|x|=R
|v|2
ds + O
1
R
(1.48)
as R tends to infinity. Taking the imaginary part yields
Imϕ, Sϕ = −k lim
R→∞
|x|=R
|v|2
ds = −
k
(4π)2
S2
|v∞
|2
ds ≤ 0 . (1.49)
Let now Imϕ, Sϕ = 0 for some ϕ ∈ H−1/2(). Then v∞ = 0. From (1.49), Rellich’s
Lemma 1.2, and unique continuation we conclude that v vanishes outside of D.Therefore,
Sϕ = 0 on by the trace theorem. Since S is an isomorphism ϕ has to vanish.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-36-320.jpg)
![20 Dirichlet and Neumann boundary conditions
Then, for any φ ∈ Y, φ = 0,
φ ∈ R(B) if and only if inf
ψ, Fψ
: ψ ∈ Y∗
, ψ, φ = 1
0 . (1.58)
Here again, R(B) denotes the range of the operator B : X → Y.
Furthermore, if φ = Bϕ0 ∈ R(B) for some ϕ0 ∈ X then
inf
ψ, Fψ
: ψ ∈ Y∗
, ψ, φ = 1
≥
c
ϕ02
X
. (1.59)
Proof: First, we observe that
ψ, Fψ
=
B∗
ψ, AB∗
ψ
≥ cB∗
ψ2
X ∗ for all ψ ∈ Y∗
. (1.60)
Let now φ = Bϕ0 for some ϕ0 ∈ X . For ψ ∈ Y∗ with ψ, φ = 1 we have
ψ, Fψ
≥ cB∗
ψ2
X ∗ =
c
ϕ02
X
B∗
ψ2
X ∗ ϕ02
X
≥
c
ϕ02
X
B∗
ψ, ϕ0
2
=
c
ϕ02
X
ψ, Bϕ0
=φ
2
=
c
ϕ02
X
.
This provides the lower bound of (1.59).
Second, assume that φ /
∈ R(B). Define the closed subspace V :=
ψ ∈ Y∗ :
ψ, φ = 0
. We show that B∗(V) is dense in R(B∗) ⊂ X ∗. This is equivalent to the
statement that the annihilators B∗(V)
⊥
and R(B∗)
⊥
= N(B) coincide. Therefore,
let ϕ ∈ B∗(V)
⊥
, i.e., B∗ψ, ϕ = 0 for all ψ ∈ V, i.e., ψ, Bϕ = 0 for all ψ ∈ V,
i.e., Bϕ ∈ V⊥ = span {φ}. Since φ /
∈ R(B) this implies Bϕ = 0, i.e., ϕ ∈ N(B).
By a consequence of the Hahn–Banach Theorem one can find φ̂ ∈ Y∗ with
φ̂, φ = 1. Choose a sequence {ψ̂n} in V such that
B∗
ψ̂n −→ −B∗
φ̂ as n → ∞ .
We set ψn = ψ̂n +φ̂. Then ψn, φ = 1 and B∗ψn → 0. From the first equation of (1.60)
we conclude that
ψn, Fψn
≤ A B∗
ψn2
X ∗
and thus ψn, Fψn −→ 0, n → ∞, which proves that
inf
ψ, Fψ
: ψ ∈ Y∗
, ψ, φ = 1
= 0 .
In our applications we will often prove the coercivity condition (1.57) with the help of
the following lemma.
Lemma 1.17 Let X be a reflexive Banach space and A, A0 : X ∗ → X be linear and
bounded operators such that
(i) ϕ, Aϕ ∈ C (−∞, 0] for all ϕ ∈ closure R(B∗) with ϕ = 0,
(ii) ϕ, A0ϕ is real-valued, and there exists c0 0 with
ϕ, A0ϕ ≥ c0ϕ2
X ∗ for all ϕ ∈ R(B∗
), (1.61)
(iii) A − A0 is compact.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-39-320.jpg)
![22 Dirichlet and Neumann boundary conditions
where again G : H1/2() → L2(S2) is defined by Gf = v∞ and v solves the boundary
value problem (1.38), (1.39), and (1.40).
Furthermore, if φ = Gf for some f ∈ H1/2() then
inf
(ψ, Fψ)L2(S2)
: ψ ∈ L2
(S2
), (ψ, φ)L2(S2) = 1
≥
c
f 2
H1/2()
(1.65)
for some c 0 independent of φ.
Combining this with the characterization of Theorem 1.12 of D by the range of G yields
immediately the main result of this subsection.
Theorem 1.20 Assume that k2 is not a Dirichlet eigenvalue of − in D. For any z ∈ R3
define again φz ∈ L2(S2) by (1.41), i.e.,
φz(x̂) := e−ikx̂·z
, x̂ ∈ S2
.
Then z ∈ D if, and only if,
inf
(ψ, Fψ)L2(S2)
: ψ ∈ L2
(S2
), (ψ, φz)L2(S2) = 1
0 . (1.66)
Therefore, the characteristic function of D is given by
χD(z) = sign inf
(ψ, Fψ)L2(S2)
: ψ ∈ L2
(S2
), (ψ, φz)L2(S2) = 1
, z ∈ R3
.
Furthermore, for z ∈ D we have the estimate:
inf
(ψ, Fψ)L2(S2)
: ψ ∈ L2
(S2
), (ψ, φz)L2(S2) = 1
≥
c
(·, z)2
H1/2()
(1.67)
for some constant c 0 which is independent of z.
Proof: It remains to prove the estimate (1.67). It follows directly from (1.65) and the
observation that φz = G(·, z)| for z ∈ D (see proof of Theorem 1.12).
We note again that the evaluation of the form of χD(z) uses only known information on
the far field operator F. Although satisfactory from the theoretical point of view there is
a major drawback with respect to the computationally point of view since it is very time
consuming to solve a minimization problem for every sampling point z.
We conclude this subsection with the remark that Theorem 1.20 provides an alter-
native – and very explicit – proof of uniqueness of the inverse scattering problem under
the assumption that k2 is not a Dirichlet eigenvalue of − in D.
1.4.3 The (F ∗F )1/4-method
As an important observation from Theorem 1.16 we note that the inf-criterion in the char-
acterization (1.58) depends only on F and not on the factorization itself. This observation
leads directly to the first part of the following result (see [71]).
Theorem 1.21 Let H be a Hilbert space and let F : H → H have two factorizations of
the form
F = B1 A1 B∗
1 = B2 A2 B∗
2 (1.68)](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-41-320.jpg)
![24 Dirichlet and Neumann boundary conditions
Remark: The operator F1/2 can be defined by using an eigensystem of F. Indeed,
if λj ≥ 0 are the eigenvalues of the non-negative (!) operator F with corresponding
normalized eigenfunctions ψj ∈ H then F has the form
Fψ =
j
λj (ψ, ψj)H ψj , ψ ∈ H , (1.70)
and thus
F1/2
ψ =
j
λj (ψ, ψj)H ψj , ψ ∈ H .
Proof of the corollary: The operator F admits a second factorization in the form F =
F1/2F1/2. The assertion follows directly from Theorem 1.21 because the operators A
and the identity satisfy both the coercivity condition (1.69).
The far field operator F : L2(S2) → L2(S2) for the Dirichlet boundary condition fails
to be self-adjoint and this corollary is not applicable. However, it is normal and – even
more – the operator I + ik
8π2 F is unitary. For this situation there exists a corresponding
result which we first formulate and prove in the general setting before we apply it to the
factorization of the far field operator.
Theorem 1.23 Let H be a Hilbert space, X a reflexive Banach space and let the compact
operator F : H → H have a factorization of the form
F = B A B∗
with operators B : X → H and A : X ∗ → X such that Imϕ, Aϕ = 0 for all
ϕ ∈ closure R(B∗) with ϕ = 0. Let furthermore A be of the form A = A0 + C for some
compact operator C and some self-adjoint operator A0 which is coercive on R(B∗)
in the sense of (1.61). Finally, assume that F is one-to-one and I + irF is unitary for
some r 0. Then the ranges of B and (F∗F)1/4 coincide. Furthermore, the operators
(F∗F)−1/4B and B−1(F∗F)1/4 are isomorphisms from X onto H and from H onto X ,
respectively.
Proof: First we note that by Lemma 1.17 the operator A satisfies the coercivity condition
(1.69), i.e., there exists c 0 with
ϕ, Aϕ
≥ cϕ2
X ∗ for all ϕ ∈ R(B∗
) . (1.71)
The unitarity of I + irF implies that F is normal. Therefore, there exists a complete
set of orthonormal eigenfunctions ψj ∈ H with corresponding eigenvalues λj ∈ C,
j = 1, 2, 3, . . . (see, e.g., [168]). Furthermore, since the operator I + irF is unitary the
eigenvalues λj of F lie on the circle of radius 1/r and center i/r. The spectral theorem
for normal operators yields that F has the form (1.70), i.e.,
Fψ =
∞
j=1
λj(ψ, ψj)H ψj , ψ ∈ H . (1.72)](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-43-320.jpg)
![28 Dirichlet and Neumann boundary conditions
is solvable in L2(S2). We write φz in spectral form as
φz =
j
(φz, ψj)L2(S2) ψj .
From (1.74) we observe that (1.81) is solvable if, and only if, the series
j
(φz, ψj)L2(S2)
2
|λj|
converges,7 and in this case
g =
j
(φz, ψj)L2(S2)
|λj|
ψj
is the solution of (1.81). Therefore, a point z ∈ R3 belongs to D if, and only if, the series
j
(φz, ψj)L2(S2)
2
|λj|
converges which proves the characterization (1.80).
The essential assumption under which the characterization (1.80) had been derived was
the normality of the far field operator F and the unitarity of the scattering operator S. In
many cases the operator F fails to be normal. Examples of theses cases are “absorbing”
media D or limited angle data u∞(x̂, θ). If the far field operator fails to be normal not
very much is known about eigenvalues. We refer to [43] for some results. In particular,
a complete set of eigenfunctions usually does not exist. Therefore, the technique of this
subsection does not work. As we will see later, the minimization approach is still appli-
cable but, as we mentioned already, it is very time consuming from the computational
point of view.
Before we modify the approach of this subsection in the next chapter appropriately
we observe that the convergence of the series in (1.80) depends only on the rates of decay
of the eigenvalues |λj| and the expansion coefficients. From the obvious estimate
1
√
2
| Re λj| + | Im λj|
≤ |λj| ≤ | Re λj| + | Im λj|
and the observation that Im λj 0 we note that we can replace |λj| in (1.80) by | Re λj|+
Im λj. Furthermore, we observe that | Re λj|+Im λj are the eigenvalues of the self-adjoint
and positive operator
F# = | Re F| + Im F
where the self-adjoint parts Re F and Im F are defined by
Re F =
1
2
F + F∗
and Im F =
1
2i
F − F∗
.
7 This is just Picard’s criterion, see [113].](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-47-320.jpg)
![An explicit example 29
Therefore, we can replace the operator (F∗F)1/4 in (1.81) by F
1/2
# . We will see in the
next chapters that the characterization (1.81) by F
1/2
# has a much wider applicability.
We continue with the example of D being the unit ball in R3.
1.5 An explicit example
Let D be the unit ball in R3 centered at the origin. We will compute the quantities which
appear in the series (1.80).
First, we expand the incident and scattered fields into spherical wave functions. Let
jn and hn be the, respectively, spherical Bessel functions and spherical Hankel functions
of the first kind and order n ∈ N and let Ym
n (x̂) be the spherical harmonics of order n
normalized such that they form a complete orthonormal system in L2(S2) (see (1.42)).
The Jacobi–Anger expansion (cf. [43]) has the form
ui
(x) = eikx·θ
= 4π
∞
n=0
|m|≤n
in
jn(k|x|) Ym
n (x̂) Ym
n (θ) , x ∈ R3
. (1.82)
Again, the unit vector θ ∈ S2 denotes the direction of incidence and x̂ = x/|x|. It is
immediately seen (at least formally) that the scattered field is given by
us
(x) = −4π
∞
n=0
|m|≤n
in jn(k)
hn(k)
hn(k|x|) Ym
n (x̂) Ym
n (θ) , |x| ≥ 1 . (1.83)
The far field pattern of hn(k|x|) Ym
n (x̂) is again given by (4π/k) exp
−i(n+1)π/2
Ym
n (x̂)
(compare proof of Lemma 1.13). As shown rigorously in [43] the far field pattern of us
is derived by the term-by-term asymptotics of hn(k|x|) Ym
n (x̂), i.e.,
u∞
(x̂, θ) = −
(4π)2
k
∞
n=0
|m|≤n
in jn(k)
hn(k)
e−i(n+1)π/2
Ym
n (x̂) Ym
n (θ)
=
(4π)2 i
k
∞
n=0
|m|≤n
jn(k)
hn(k)
Ym
n (x̂) Ym
n (θ) , x̂, θ ∈ S2
. (1.84)
From this we observe that the far field operator F : L2(S2) → L2(S2) from (1.36) is
given by
(Fg)(x̂) =
S2
u∞
x̂, θ
g(θ) ds(θ) =
(4π)2 i
k
∞
n=0
|m|≤n
jn(k)
hn(k)
gm
n Ym
n (x̂) , x̂ ∈ S2
,
(1.85)
where
gm
n =
S2
g(θ) Ym
n (θ) ds(θ) , |m| ≤ n , n ∈ N ,](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-48-320.jpg)
![30 Dirichlet and Neumann boundary conditions
are the expansion coefficients of g ∈ L2(S2). From (1.85) we observe that
λn =
(4π)2 i
k
jn(k)
hn(k)
, n ∈ N , (1.86)
are the eigenvalues of F of multiplicity 2n + 1. The asymptotic behavior
jn(k) =
kn
1 · 3 · · · (2n + 1)
1 + O
1
n
, (1.87)
hn(k) =
1 · 3 · · · (2n − 1)
i kn+1
1 + O
1
n
yields that
λn = −
(4π)2 k2n
(2n − 1)!! (2n + 1)!!
1 + O
1
n
, (1.88)
where we used the convenient notation p!! = 1 · 3 · 5 · · · p for any odd number p. Next,
we compute the expansion coefficients of the functions (1.41) by the Jacobi–Anger
expansion (1.82), i.e.,
φz(x̂) = e−ikz·x̂
= 4π
∞
n=0
|m|≤n
(−i)n
jn(k|z|) Ym
n (x̂) Ym
n (ẑ) , x̂ ∈ S2
,
where ẑ = z/|z|. From this we conclude that
(φz, Ym
n )L2(S2) = 4π (−i)n
jn(k|z|) Ym
n (ẑ) .
Using the formula
|m|≤n
Ym
n (ẑ)
2
=
2n + 1
4π
which is a special form of the addition theorem (see [43]) and the asymptotic form (1.87)
of jn(k|z|) we conclude that
|m|≤n
(φz, Ym
n )L2(S2)
2
= 4π(2n + 1)
(k|z|)2n
[(2n + 1)!!]2
1 + O(1/n)
.
Combining this with (1.88) yields
|m|≤n
(φz, Ym
n )L2(S2)
2
|λn|
=
2n + 1
4π k2n
(2n − 1)!!
(2n + 1)!!
(k|z|)2n
1 + O
1
n
=
|z|2n
4π
1 + O
1
n
.](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-49-320.jpg)
![32 Dirichlet and Neumann boundary conditions
The solution is again understood in the variational sense, i.e., v ∈ H1
loc(R3 D) is a
variational solution of (1.92) and (1.93) if it satisfies
R3D
∇u · ∇φ − k2
u φ
dx = f , φ
for all φ ∈ H1(R3 D) with compact support. Here, ·, · denotes again the dual form
in H−1/2(), H1/2(). Existence and uniqueness of this exterior Neumann boundary
value problem is well established, see [183], Lecture 4, [150], Section 2.6, or Chapter 1,
Theorem 2.2.
The far field patterns u∞ = u∞(x̂, θ) of the scattered fields us define again the far
field operator F : L2(S2) → L2(S2) by (compare (1.36))
(Fg)
x̂
=
S2
u∞
x̂, θ
g
θ
ds(θ) for x̂ ∈ S2
. (1.95)
By literally the same proofs as inTheorems 1.6 and 1.8 one shows reciprocity (1.27) of the
far field patterns, normality of F and unitarity of the scattering operator S = I + ik
8π2 F.
Furthermore, F is one-to-one if k2 is not a Neumann eigenvalue of − in D.8 Also, the
uniqueness result of Theorem 1.9 holds.
The analogous results of Lemmas 1.13 and 1.14 and Theorem 1.15 are formulated
in the following theorem.
Theorem 1.26 (a) The far field operator F : L2(S2) → L2(S2) from (1.95) has a
factorization in the form
F = −G N∗
G∗
(1.96)
where G : H−1/2() → L2(S2) maps f ∈ H−1/2() into the far field pattern
v∞ = Gf of the exterior Neumann boundary value problem (1.92), (1.93), and
(1.94), and N : H1/2() → H−1/2() is the normal derivative of the double layer
potential, defined by
(Nϕ)(x) =
∂
∂ν
ϕ(y)
∂
∂ν(y)
(x, y) ds(y) , x ∈ , (1.97)
for ϕ ∈ H1/2().
(b) G is compact, one-to-one with dense range in L2(S2).
(c) N is an isomorphism from H1/2() onto H−1/2() if k2 is not a Neumann eigenvalue
of − in D.
(d) ImNϕ, ϕ 0 for all ϕ ∈ H1/2() with ϕ = 0 if k2 is not a Neumann eigenvalue
of − in D. Again, ·, · denotes the duality pairing in H−1/2(), H1/2().
8 k2 is called a Neumann eigenvalue of − in D if there exists a non-trivial solutionu ∈ C2(D) ∩ C1(D)
of the Helmholtz equation in D such that ∂u/∂ν vanishes on .](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-51-320.jpg)
![The Neumann boundary condition 33
(e) Let Ni be the boundary operator (1.97) corresponding to the wavenumber k = i.
The operator −Ni is self-adjoint and coercive as an operator from H1/2() onto
H−1/2(), i.e.,
−Niϕ, ϕ ≥ c0ϕ2
H1/2()
for all ϕ ∈ H1/2
() . (1.98)
(f) The difference N − Ni is compact from H1/2() into H−1/2().
Remark: We note that the classical definition of Nϕ by (1.97) makes only sense for
sufficiently smooth densities (i.e., for Hölder continuously differentiable functions on ,
see [43]). It can be shown that N has a bounded extension to an operator from H1/2()
into H−1/2() which we also denote by N (see [144]).
Proof: (a) Analogously to H from (1.51) we define the operator ∂H : L2(S2) →
H−1/2() by
(∂H)g(x) :=
∂
∂ν
S2
g(θ) eikx·θ
ds(θ) = ik
S2
g(θ) (ν(x) · θ) eikx·θ
ds(θ) , x ∈ .
(1.99)
Its adjoint (∂H)∗ : H1/2() → L2(S2) is now given by
(∂H)∗
ϕ(x̂) = −ik
(ν(y) · x̂) ϕ(y) e−ikx̂·y
ds(y) =
ϕ(y)
∂
∂ν(y)
e−ikx̂·y
ds(y)
(1.100)
for x̂ ∈ S2. We note that (∂H)∗ϕ is now the far field pattern of the double layer potential
v(x) =
ϕ(y)
∂
∂ν(y)
(x, y) ds(y) , x ∈ R3
D , (1.101)
and ∂v/∂ν = Nϕ provided ϕ ∈ C1,α() (see [42]). This yields (∂H)∗ϕ = GNϕ and
thus ∂H = N∗ G∗ by a density argument. Furthermore, we note that F = −G(∂H) and
therefore F = −G N∗G∗.
(b) This follows from similar the arguments as in the proof of Lemma 1.13.
(c) For this property we refer again to [144].
(d) Here we proceed exactly as in the proof of Lemma 1.14. Define, for any ϕ ∈
H1/2() the double layer potential v by
v(x) =
ϕ(y)
∂(x, y)
∂ν(y)
ds(y) , x ∈ R3
. (1.102)
Then v ∈ H1(D)∩H1
loc(R3 D) is again a solution of the Helmholtz equation in R3 .
The traces v± and ∂v±/∂ν exist in the variational sense with ϕ = v+ − v− and ∂v−
∂ν =
∂v+
∂ν = Nϕ. Therefore, using Green’s formula in D and in DR :=
x ∈ R3 D : |x| R](https://image.slidesharecdn.com/505080-250529202829-4126e0f1/85/The-Factorization-Method-For-Inverse-Problems-1st-Edition-Andreas-Kirsch-52-320.jpg)
![This ebook is for the use of anyone anywhere in the United States
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you are located before using this eBook.
Title: An Annapolis First Classman
Author: Edward L. Beach
Illustrator: Frank T. Merrill
Release date: July 29, 2012 [eBook #40368]
Most recently updated: October 23, 2024
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The Factorization Method For Inverse Problems 1st Edition Andreas Kirsch
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- 6. Oxford Lecture Series in Mathematics and its Applications Series Editors John Ball Dominic Welsh
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- 8. The Factorization Method for Inverse Problems Andreas Kirsch and Natalia Grinberg Institute of Algebra and Geometry University of Karlsruhe (TH) Karlsruhe, Germany 1
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- 10. Preface This book is devoted to the problem of shape identification. Problems of this type occur in a number of important fields which belong to the class of inverse problems. As the first of these fields, we mention inverse scattering problems where one wants to detect – and identify – unknown objects through the use of acoustic, electromagnetic, or elastic waves. Complex models in scattering theory involve boundary value problems for partial differential equations such as Maxwell’s equations in electromagnetics, and one of the important problems in inverse scattering theory is to determine the shape of the obstacle from field measurements. Applications of inverse scattering problems occur in such diverse areas as medical imaging, material science, nondestructive testing, radar, remote sensing, or seismic exploration. A survey on the state of the art of the mathematical theory and numerical approaches for solving inverse time harmonic scattering problems until 1998 can be found in the standard monograph [43] by David Colton and Rainer Kress. We also refer to Chapter 6 of [106] and [155] for an introduction and survey on inverse scattering problems. The second important area where the identification of unknown shapes plays an important role is tomography, in particular, electrical impedance tomography or optical tomography. Electrical impedance tomography is a technique to recover spatial prop- erties of the interior of a conducting object from electrostatic measurements taken on its boundary. For example, a current through a homogeneous object will, in general, induce a different potential than the same current through the same object containing an enclosed cavity. The problem of impedance tomography, we are interested in, is to determine the shape of the cavity from measurements of the potential on the boundary of the object. For survey articles on this subject we refer to [16] by Liliana Borcea and [82] by Martin Hanke and Martin Brühl. Shape identification problems are intrinsically nonlinear, i.e., the measured quan- tities do not depend linearly on the shape. Even the notion of linearity does not make sense since, in general, the set of admissible shapes does not carry a linear structure. Traditional (and still very successful) approaches describe the objects by appropriate parameterizations and compute the parameters by iterative schemes as, e.g., Newton- type methods. Besides the well-known advantages (fast convergence) and disadvantages (only local convergence properties) of iterative methods for nonlinear problems, these methods share the common drawback that important information on the unknown object such as the number of connectivity components or the type of the boundary condition has to be known in advance. Nevertheless, methods of this type are widely used – in particular because the first or second order derivatives can be characterized by using techniques
- 11. vi Preface from the shape optimization theory. We refer to [156, 173] for general references and [112, 134, 135, 84, 83, 86] for applications in inverse scattering theory. In particular, classical iterative algorithms using explicit parameterizations of the objects are not able to change the number of connectivity components during the algo- rithm. This observation has led to the development of level set methods which are based on implicit representations of the unknown object involving an “evolution parameter” t. Since the pioneering work [154] by S. Osher and J. Sethian, this method has been further developed and applied in a huge number of papers. We refer to [22] for a recent survey. Since around 1995 iterative methods for solving problems in shape optimization have been developed which completely avoid the use of parameterizations and replace the classical Fréchet derivative by a geometrically motivated topological derivative, see, e.g., [171, 172]. These methods have also been applied to problems in inverse scattering theory in [15, 78]. We refer also to [67]. While very successful in many cases, iterative methods for shape identification prob- lems – may they use classical tools as the Fréchet derivative or more recent techniques such as domain derivatives, level curves, or topological derivatives – are computationally very expensive since they require the solution of a direct problem in every step. Further- more, for many important cases the convergence theory is still missing. This is due to the fact that these problems are not only nonlinear but also because their linearizations are improperly posed.Although there exist many results on the convergence of (regularized) iterative methods for solving nonlinear improperly posed problems (see, e.g., [62, 87]), the assumptions for convergence are not met in the applications to shape identification problems.1 Thesedifficultiesanddisadvantagesofiterativeschemesgaverisetothedevelopment of different classes of non-iterative methods which avoid the solution of a sequence of direct problems. We briefly mention decomposition methods (according to the notion of [44]) which consist of an analytic continuation step (which is linear but highly improperly posed) and a nonlinear step of finding the boundary of the unknown domain by forcing the boundary condition to hold. We refer to [6, 3, 45, 46, 123, 50, 125, 132] for some versions of this approach. In Section 7.2 we will briefly recall the Dual Space Method of Colton and Monk (see below) which belongs to this class.There is also a close connection to the Point-Source Method of Roland Potthast in [157, 158, 160]. In this monograph, we will focus on a different class of non-iterative methods. The common idea is the construction of criteria on the known data to decide whether a given point z (or a curve or a set) is inside or outside the unknown domain D. By choosing a grid of “sampling” points z (or collection of curves or sets) in a region known to contain the unknown domain D one is therefore able to compute the (approximate) characteristic function of D. In the following, we will collect these approaches under the name sampling methods. They differ in the way of defining the criterion and in the type of test objects. One of the first methods which falls into this class has been developed by David Colton and one of the authors of this book (A.K.) in 1996 ([39]), now known as the Linear Sampling Method. Its origin goes back to the Dual Space Method developed by 1 Or, at least, it is unknown whether these assumptions are fulfilled or not.
- 12. Preface vii David Colton and Peter Monk during 1985 and 1990 (see, e.g., [45, 46, 49] or [43]). We also mention the work [105] of Victor Isakov in which singular functions are used to prove uniqueness of an inverse scattering problem. The numerical implementation of the Linear Sampling Method is extremely simple and fast because sampling is done by points z only. For every sampling point z one has to compute the field of a point source in z with respect to the background medium2 (if this is constant the response is even given analytically) and evaluate a series, i.e., a finite sum in practice. In 1998 Masaru Ikehata published the paper [96] in which he presented a method now known as Ikehata’s Probe Method. Instead of points, the region is probed by curves (called needles in [96]), and points on these curves are identified which belong to the boundary of the unknown domain D. In many subsequent papers, mainly by Ikehata and his collaborators, the probe method has been applied to several inverse scattering problems and inverse conductivity problems (see [95, 97, 98, 99, 100, 101, 102, 33, 103, 104]). We explicitly mention [33, 63] for numerical implementations of the probe method. In 2000 Roland Potthast presented a sampling method in [159] which he later in [160], Chapter 6, called the Singular Sources Method. The idea, formulated here for the inverse scattering problem, is to approximate (from the given data) the scattered field vs(·, z) which belongs to the field of a point source in z as incident field. This scattered field vs(z, z), evaluated at the same source point z, is unbounded if z approaches the boundary ∂D from the exterior. Therefore, the unknown region D is found as the set of points z where vs(z, z) becomes large. In this sense, we consider this method as a sampling method with respect to point sampling. Sampling by sets is done in the Range Test and the No Response Test, developed by Roland Potthast, John Sylvester, and Steven Kusiak in [163] and Russell Luke and Roland Potthast in [141]. We refer also to [88]. Aproblem with all of the mentioned methods (except of Ikehata’s probe method) from the mathematical point of view is that the computable criterion provides only sufficient conditions which are, in general, not necessary. The Factorization Method, developed by the authors in [114, 115, 76, 74] overcomes this drawback and provides a criterion for z which is both, necessary and sufficient. Therefore, this method provides a simple formula for the characteristic function of D which can easily be used for numerical computations. We emphasize, that these results hold in the resonance region, i.e., no asymptotic forms such as the Born approximation or the geometric or physical optics approximations are assumed. Compared to Ikehata’s Probe Method the Factorization Method is much more direct, both from the theoretical point of view as well as with respect to the computational implementation. From the numerical point of view, the Linear Sampling Method and the Factorization Method are equally simple and fast. A typical feature of these two methods is that they make no explicit use of boundary conditions or topological properties of D. In other words, they determine the unknown domain without knowing in advance the type of boundary condition or the number of components. 2 Essentially, one has to compute the Green’s function.
- 13. viii Preface Since their first presentations, the Linear Sampling Method and the Factorization Method have been developed for several problems in inverse scattering theory and tomography. We refer to [24, 23, 26, 27, 28, 35, 37, 38, 80, 65, 178] for some recent work on the Linear Sampling Method and [7, 116, 117, 129, 118, 77, 119, 120] for papers related to the Factorization Method in inverse scattering theory. The interesting papers [8] and [11] by Tilo Arens and Armin Lechleiter discovers a deeper relationship between these two methods. Stimulated by the first paper [114] Martin Brühl and Martin Hanke investigated the Factorization Method for problems in impedance tomography (see [18, 19, 20, 82, 21] and later [90, 92, 94]) by Nuutti Hyvönen. For a more general approach to elliptic equations we refer to [120] and [68]. There are several applications of the Factorization Method which are not covered in this monograph. We mention scatter- ing problems for periodic surfaces or arcs (cf. [10, 9] and [129], respectively), for elastic media (see [7, 29]), for static problems (cf. [81, 133, 131]), or in optical tomography (cf. [14, 89, 91, 93]). However, it should also be mentioned that the range of problems for which the Factorization Method has been justified from the mathematical point of view is con- siderably smaller than the one for the Linear Sampling Method or the other sampling methods. While the main subject of this monograph is the Factorization Method, we will report on the Linear Sampling Method, Ikehata’s Probe Method, and Potthast’s Singular Sources Method in Chapter 7. We also refer to the survey articles [161, 162] by Roland Potthast on Sampling and Probe Methods and to [88] for an interesting relationship between sampling methods and iterative methods. The monograph is organized as follows. Chapters 1–4 study the Factorization Method for scattering problems where the wave propagation is described by the three- dimensional scalar Helmholtz equation. In Chapters 1–3 impenetrable scatterers D are considered where boundary conditions on the boundary ∂D of D of Dirichlet, Neumann, impedance or mixed type are imposed. Chapter 4 is devoted to the penetrable case, and we show that the Factorization Method can be considered as an extension of the well-known MUSIC-algorithm from signal processing (cf. [60, 30]). There exist several variants of the Factorization Method. The – in our opinion – most satisfactory version holds for scattering problems with non-absorbing media, such as the obstacle scattering case with Dirichlet or Neumann boundary conditions. The Factorization Method for this situation is investigated in detail in Chapter 1. The math- ematically most important feature of these problems is the normality of the far field operator. This makes it possible to use the spectral theory for normal operators. The authors think that the Factorization Method is a particularly interesting and useful appli- cation of this theory. As an intermediate step we prove a characterization of the scatterer D by an inf-condition which is, although not as elegant as the final characterization by the solvability of an equation, the basis for characterizations of D for scattering problems for absorbing media. We consider Chapter 1 also as an introduction into our method and emphasize that it can not be left out by the reader because it sets up the basis for all subsequent chapters.
- 14. Preface ix A first example of a case where the far field operator fails to be normal is studied in Chapter 2. The impedance boundary condition serves as a simple model for an absorbing medium. Since the far field operator F is no longer normal the final characterization of Chapter 1 does not hold and has to be modified. This is done by considering a suitable combination F# of the self-adjoint parts (F +F∗)/2 and (F −F∗)/(2i) of F which finally leads to a characterization of D by the solvability of an equation involving F# instead of F. Chapter 3 is devoted to mixed boundary conditions. The obstacle is assumed to consist of several parts, and on some them we impose Dirichlet boundary conditions, on the others Neumann or, more general, impedance boundary conditions. Even for the Dirichlet–Neumann case, where the far field operator F is still normal, it is an open problem whether or not the Factorization Method (in any of its forms) can be justified. Numerical experiments indicate that this is indeed the case but a rigorous proof is not known. However, if we a priori know some domains which enclose the parts with the Dirichlet boundary condition and the impedance boundary condition we can modify the operator F# appropriately to treat this case as well. In Chapter 4 we study the penetrable case, i.e., scattering by an inhomogeneous medium where we allow the medium to be absorbing. The techniques developed in Chapters 1 and 2 allow us to prove a characterization of the shape of the contrast (which is the difference between the indices of refraction of the scattering medium and the background medium) by the same operators used in Chapters 1 and 2. We note already here that this implies in practice that one does not need to know the type of obstacle – penetrable or impenetrable – in advance. While Chapters 1–4 treat scattering problems for the scalar Helmholtz equation we investigate the Factorization Method in Chapters 5 and 6 for the scattering of time har- monic electromagnetic waves and the problem of impedance tomography, respectively. One assumption for the validity of the Factorization Method is that the square k2 of the wavenumber k is not an eigenvalue of a corresponding eigenvalue problem. In the case of an impenetrable obstacle with Dirichlet or Neumann boundary conditions this eigenvalue problem is just the classical eigenvalue problem for − in the domain D with respect to the boundary conditions. For the scattering by an inhomogeneous medium, however, a new type of eigenvalue problem (the “interior transmission eigenvalue prob- lem”) occurs which fails to be self-adjoint (cf. [55]). In Sections 4.5 and 5.5 we show under certain assumptions on the index of refraction that the eigenvalues form at most a countable set. The question of existence of eigenvalues is only settled for the spherically stratified case. In Chapter 6 we investigate the problem of impedance tomography. In contrast to the scattering problems this problem is set up as a boundary value problem in a bounded domain B. The inverse problem we are interested in is to determine the shape D of an inclusion with different electrical properties than the background medium. In this application, the Factorization Method is set up for the difference of the Neumann– Dirichlet operators for the cases with and without inclusion rather than for the far field operator. As mentioned above, the Factorization Method is only one of a class of new approaches for solving “geometric” inverse problems. In Chapter 7 we introduce the
- 15. x Preface reader to some related sampling methods. In contrast to the Factorization Method they all use heavily the fact that every solution of the Helmholtz equation in some domain G can be approximated arbitrarily well by solutions of the Helmholtz equation in larger domains. We summarize two of such approximation theorems in Section 7.1. The Linear Sampling Method can be considered as the precursor of the Factorization Method and is closely related to the latter one. We present this method in Section 7.2 and show its relationship to the Dual Space Method of Colton and Monk. In Section 7.3 we present the basic ideas of the Singular Sources Method of Roland Potthast. Finally, in Sub- section 7.4.1 of Section 7.4 we explain the Probe Method of Masaru Ikehata for the impedance tomography problem and extend it in Subsection 7.4.2 to the inverse scatter- ing problem with boundary conditions of mixed type. Here we follow the presentation of [74]. In this monograph we use several spaces of functions on domains G or their bound- aries ∂G. We try to follow the standard notations for these spaces. The boundary value problems in bounded domains are set up in the Sobolev space H1(G) of (Lebesgue) mea- surable functions such that their derivatives (in the sense of distributions) are regular and belong to L2(G). This space is equipped with the inner product (u, v)H1(G) = (u, v)L2(G) + ∇u, ∇v L2(G) where we denote by (u, v)L2(G) = G u(x) v(x) dx the inner product in L2(G). By v(x) we denote the complex conjugate of v(x). If u and v are vector fields then u(x)v(x) has to be understood as the scalar product 3 j=1 uj(x)vj(x). The corresponding norms are denoted by · L2(G) and · H1(G) For k ∈ N we denote by Ck(G) the space of functions for which all partial derivatives up to order k exist in G and are continuously extendable to the closure G of G. The norms in Ck(G) are denoted by · Ck (G). We set C∞(G) = k∈N Ck(G). Then we can equivalently define H1(G) as the completion of C1(G) with respect to the inner product (·, ·)H1(G). Spaces of vector fields are denoted by H1(G, C3) or Ck(G, C3). We assume that G is a Lipschitz domain. Sometimes we assume that even ∂G ∈ C1,α or ∂G ∈ C2. For definitions of these notations as well as of spaces Cj,α(G) or Cj,α(∂G) of Hölder continuous functions we refer to [146] (Section 3.2), [144], Chapter 3, [43], Section 2.2, or [167], Section 6.4. Then the spaces C(∂G) and L2(∂G) are defined in the usual way using local coordinates. It can be shown (see, e.g., [17], Section 3) that the trace operator γ : C1(G) → L2(∂G), γ u = u|∂G, has a bounded extension on H1(G). Its range space R(γ ) = ψ ∈ L2 (∂G) : there exists u ∈ H1 (G) with γ u = ψ is denoted by H1/2(∂G) and equipped with the norm ψH1/2(∂G) = inf uH1(G) : u ∈ H1 (G) with γ u = ψ . We define H−1/2(∂G) as the dual space of H1/2(∂G). We denote by ·, · the dual form in H−1/2(∂G), H1/2(∂G), which is the extension of the inner product (·, ·)L2(∂G) :
- 16. Preface xi L2(∂G)×L2(∂G) → C to ·, · : H−1/2(∂G)×H1/2(∂G)) → C. We note that both, the inner product (·, ·)L2(∂G) and the dual form ·, · are sesqui-linear forms. Furthermore, H1 0 (G) = u ∈ H1(G) : γ u = 0 on ∂G can be constructed as the closure in H1(G) of the space C1 0 (G) of C1-functions with compact support in G. We denote by R(F) and N(F) the range space and the null space, respectively, of an operator F. Finally, we come to the pleasant task of thanking those who have supported and encouraged us for starting – and finishing – this project. Here we want to mention three working groups who have influenced the research on sampling methods in a very essen- tial way. First of all, we would like to thank the research group of Fioralba Cakoni, David Colton, Russell Luke, and Peter Monk from the University of Delaware who not only made important contributions to the field but also provided a warm and hospitable environment for one of the authors (A.K.) to spend several weeks in Newark during the past years. Second, the working group at the University of Göttingen of Rainer Kress, Thorsten Hohage, and Roland Potthast (now at the University of Reading) are one of the most active groups in Germany in the field of inverse scattering problems and encouraged us to present the ideas of the Factorization Method to a broader audi- ence of interested mathematicians, physicists, and engineers.Third, the working group of Martin Hanke-Bourgeois at the University of Mainz developed the Factorization Method for problems of electrical impedance tomography which started with the PhD thesis of Martin Brühl [18]. During a common BMBF-project funded by the German Federal Ministry of Education and Research many discussions resulted in new ideas, new gener- alizations,andnewapplicationsoftheFactorizationMethod(see,e.g.,thejointpaper[69] and also [122]). Last but not least particular thanks are given to the members of our own research group at the Department of Mathematics, in particular to Tilo Arens, Frank Hettlich, Armin Lechleiter, and Sebastian Ritterbusch for many fruitful discussions and argu- ments – and also for their willingness to shoulder “daily” duties during the preparation of this monograph. The colored versions of the plots are available on my homepage www.mathematik.uni-karlsruhe.de/iag1/∼kirsch/en Karlsruhe, Germany May 2007 Andreas Kirsch Natalia Grinberg
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- 18. Contents Preface v 1 The simplest cases: Dirichlet and Neumann boundary conditions 1 1.1 The Helmholtz equation in acoustics 2 1.2 The direct scattering problem 4 1.3 The far field patterns and the inverse problem 7 1.4 Factorization methods 13 1.4.1 Factorization of the far field operator 15 1.4.2 The inf-criterion 19 1.4.3 The (F∗F)1/4-method 22 1.5 An explicit example 29 1.6 The Neumann boundary condition 31 1.7 Additional remarks and numerical examples 35 2 The factorization method for other types of inverse obstacle scattering problems 40 2.1 The direct scattering problem with impedance boundary conditions 40 2.2 The obstacle reconstruction by the inf-criterion 49 2.3 Reconstruction from limited data 52 2.4 Reconstruction from near field data 54 2.5 The F# – factorization method 57 2.5.1 The functional analytic background 57 2.5.2 Applications to some inverse scattering problems 62 2.6 Obstacle scattering in a half-space 63 2.6.1 The direct scattering problem 65 2.6.2 The factorization method for the inverse problem 67 3 The mixed boundary value problem 70 3.1 The direct scattering problem 70 3.2 Factorization of the far field operator 76 3.3 Application of the F# – factorization method 79 4 The MUSIC algorithm and scattering by an inhomogeneous medium 86 4.1 The MUSIC algorithm 86 4.2 Scattering by an inhomogeneous medium 91 4.3 Factorization of the far field operators 95 4.4 Localization of the support of the contrast 97 4.5 The interior transmission eigenvalue problem 102
- 19. xiv Contents 5 The factorization method for Maxwell’s equations 109 5.1 Maxwell’s equations 109 5.2 The direct scattering problem 111 5.3 Factorization of the far field operator 123 5.4 Localization of the support of the contrast 125 5.5 The interior transmission eigenvalue problem 133 6 The factorization method in impedance tomography 141 6.1 Derivation of the models 141 6.2 The Neumann-to-Dirichlet operator and the inverse problem 142 6.3 Factorization of the Neumann-to-Dirichlet operator 148 6.4 Characterization of the inclusion 150 7 Alternative sampling and probe methods 159 7.1 Two approximation results 159 7.2 The dual space method and the linear sampling method 163 7.3 The singular sources method 171 7.4 The probe method 176 7.4.1 The probe method in impedance tomography 176 7.4.2 The probe method for the inverse scattering problem with mixed boundary conditions 183 Bibliography 189 Index 199
- 20. 1 The simplest cases: Dirichlet and Neumann boundary conditions As pointed out in the introduction this chapter is devoted to the analysis of the factorization methods for the most simplest case in scattering theory. We consider the scattering of time-harmonic plane waves by an impenetrable obstacle D which we model by assuming Dirichlet boundary conditions on the boundary = ∂D of D. With respect to the factorization methods we will carry out all proofs in detail. We point out that the title of this chapter should not lead to the wrong conclusion that this “simplest case” could be left out by those readers interested in the factorization method for more complicated models. In this chapter we formulate the basic functional analytic results which form the basis of the method and will be referred to several times in later chapters. After a short derivation of the Helmholtz equation from the basic equations in con- tinuum mechanics we will repeat in Section 1.2 some well-known results on the direct scattering problem. We will omit the proofs but refer to the existing literature such as [43]. However, we will emphasize the important “ingredients”: Rellich’s Lemma and unique continuation for the problem of uniqueness of the scattering problem and Green’s theorem for the derivation of important properties such as the reciprocity principles. Section 1.3 will collect analytical results on the inverse scattering problem such as uniqueness of the inverse problem and properties of the far field operator. Here we will present the proofs of those results which are necessary for the factorization method. We start Section 1.4 with the basic factorization of the far field operator. Then in Subsection 1.4.2 a quite general approach is discussed in which the domain D is char- acterized by those points z for which the infimum of a certain function (depending on z) is positive. For the special case where the scattering operator is unitary – which is the case for the scattering by an obstacle under Dirichlet boundary conditions – this repre- sentation can be transformed into the characterization of D by those points z for which a certain equation of the first kind (where the right-hand side depends on z) is solvable or not. In Section 1.6 we will briefly treat the case of Neumann boundary conditions. The analysis is quite analogous to the Dirichlet case.
- 21. 2 Dirichlet and Neumann boundary conditions 1.1 The Helmholtz equation in acoustics In the first part of this monograph we consider acoustic waves that travel in a medium, such as a fluid. Let v(x, t) be the velocity vector of a particle at x ∈ R3 and time t. Let p(x, t), ρ(x, t), and S(x, t) denote the pressure, density, and specific entropy, respectively, of the fluid. We assume that no exterior forces act on the fluid. Then the movement of the particle is described by the following equations: ∂v ∂t + (v · ∇)v + 1 ρ ∇p = 0 (Euler’s equation) , (1.1) ∂ρ ∂t + div(ρv) = 0 (continuity equation) , (1.2) f (ρ, S) = p (equation of state) , (1.3) ∂S ∂t + v · ∇S = 0 (adiabatic hypothesis) , (1.4) where the function f depends on the fluid. This system is nonlinear in the unknown functions v, ρ, p, and S. Let the stationary case be described by v0 = 0, time-independent distributions ρ = ρ0(x) and S = S0(x), and constant p0 such that p0 = f ρ0(x), S0(x) . The linearization of this nonlinear system is given by the (directional) derivative of this system at (v0, p0, ρ0, S0). For deriving the linearization, we set v(x, t) = ε v1(x, t) + O(ε2 ), p(x, t) = p0 + ε p1(x, t) + O(ε2 ), ρ(x, t) = ρ0(x) + ε ρ1(x, t) + O(ε2 ), S(x, t) = S0(x) + ε S1(x, t) + O(ε2 ), and we substitute this into (1.1), (1.2), (1.3), and (1.4). Ignoring terms with O(ε2) leads to the linear system ∂v1 ∂t + 1 ρ0 ∇p1 = 0, (1.5) ∂ρ1 ∂t + div(ρ0 v1) = 0, (1.6) ∂f (ρ0, S0) ∂ρ ρ1 + ∂f (ρ0, S0) ∂S S1 = p1, (1.7) ∂S1 ∂t + v1 · ∇S0 = 0. (1.8) First, we eliminate S1. Since 0 = ∇f ρ0(x), S0(x) = ∂f (ρ0, S0) ∂ρ ∇ρ0 + ∂f (ρ0, S0) ∂S ∇S0 ,
- 22. The Helmholtz equation in acoustics 3 we conclude by differentiating (1.7) with respect to t and using (1.8) ∂p1 ∂t = c(x)2 ∂ρ1 ∂t + v1 · ∇ρ0 , (1.9) where the speed of sound c is defined by c(x)2 := ∂ ∂ρ f ρ0(x), S0(x) . Now we eliminate v1 and ρ1 from the system. This can be achieved by differentiating (1.9) with respect to time and using equations (1.5) and (1.6). This leads to the wave equation for p1: ∂2p1(x, t) ∂t2 = c(x)2 ρ0(x) div 1 ρ0(x) ∇p1(x, t) . (1.10) Now we assume that all quantities are time-periodic. In particular, p1 is of the form p1(x, t) = Re u(x) e−iωt with frequency ω 0 and some complex-valued function u = u(x) depending only on the spatial variable. Substituting this into the wave equation (1.10) yields the three- dimensional reduced equation for u: ρ0(x) div 1 ρ0(x) ∇u(x) + ω2 c(x)2 u(x) = 0 . (1.11) If ∇ρ0 is negligible then the reduced wave equation (1.11) reduces to the Helmholtz equation u(x) + ω2 c(x)2 u(x) = 0 , i.e., u(x) + k2 n(x)2 u(x) = 0 (1.12) where k = ω c0 denotes the wavenumber and n(x) = c0 c(x) the index of refraction and c0 the constant speed of sound of free space. In particular, in free space ρ0 is constant and thus (1.12) holds with n = 1, i.e., u + k2 u = 0 .
- 23. 4 Dirichlet and Neumann boundary conditions We emphasize again, that in general the field u is complex valued and the physically relevant time-dependent field is given by U(x, t) = Re u(x) e−iωt (1.13) where we write U instead of p1. In scattering theory u is the sum of an incident field ui and a scattered field us. The incident field is a solution of the Helmholtz equation in free space while the scattered part compensates for the inhomogeneous medium. We call this situation scattering by an inhomogeneous medium. Obstacle scattering occurs if the fields do not penetrate into an obstacle D. In the sound-soft case the pressure p vanishes on the boundary ∂D of D which leads to the Dirichlet boundary condition u = 0 on ∂D. Similarly, the scattering by a sound-hard obstacle leads to a Neumann boundary condition ∂u/∂ν = 0 on ∂D since here the normal component of the velocity v vanishes on ∂D. The vector ν = ν(x) denotes the unit normal vec- tor at x ∈ ∂D. More general boundary conditions can be formulated as impedance boundary conditions where the normal component of v is proportional to the pressure. This is formulated as ∂u/∂ν + λu = 0 on ∂D with some (possibly space-dependent) impedance λ. 1.2 The direct scattering problem Let D ⊂ R3 be an open and bounded domain with C2-boundary such that the exterior R3 D of D is connected. Here and throughout the monograph we denote by D the closure of the set D of points in R3. A confusion with the complex conjugate z of z ∈ C is not expected. Furthermore, let k 0 be the (real-valued) wavenumber and ui (x, θ) = exp(ikx · θ) , x ∈ R3 , (1.14) be the incident plane wave of direction θ ∈ S2. Here, S2 = {x ∈ R3 : |x| = 1} denotes the unitsphereinR3.TheobstacleD givesrisetoascatteredfieldus ∈ C2(R3D)∩C(R3D) which superposes ui and results in the total field u = ui +us which satisfies the Helmholtz equation u + k2 u = 0 outside D , (1.15) and the Dirichlet boundary condition u = 0 on . (1.16) The scattered field us satisfies the Sommerfeld radiation condition ∂us ∂r − ik us = O r−2 for r = |x| → ∞ (1.17) uniformly with respect to x̂ = x/|x|.
- 24. The direct scattering problem 5 Figure 1.1 Incident, scattered, and total field We illustrate this situation in Figure 1.1 which shows the real parts of the incident plane wave ui (left picture), the scattered wave us (middle picture), and the total field u (right picture), respectively, in two dimensions1. Observing that the incident field ui satisfies the Helmholtz equation (1.15) in all of R3 we note that the scattered field us solves the following exterior boundary value problem for f = −ui: Given f ∈ H1/2() find v ∈ H1 loc(R3 D) such that v + k2 v = 0 outside D, (1.18) v = f on . (1.19) and ∂v ∂r − ik v = O r−2 for r = |x| → ∞ (1.20) uniformly with respect to x̂ = x/|x|. We note that the solution of (1.18) is understood in the variational sense. Indeed, using the Helmholtz equation in Green’s first formula for the region DR = x ∈ R3 D : |x| R DR ϕ v + ∇ϕ · ∇v dx = R ϕ ∂v ∂ν ds (1.21) yields R3D ∇ϕ · ∇v − k2 ϕv dx = 0 (1.22) for any test function ϕ with compact support in R3 D (choose R such that that the ball of radius R contains the support of ϕ). Here and throughout this monograph, ν = ν(x) denotes the unit normal vector at x ∈ ∂D directed into the exterior of D. It follows from 1 The setting of the two-dimensional scattering problem, i.e., where x, θ ∈ R2 and D ⊂ R2, differs from the one in three dimensions only in the radiations condition which has now the form ∂us/∂r − ik us = O r−3/2 . We refer to [43] and Section 1.7.
- 25. 6 Dirichlet and Neumann boundary conditions interior regularity results for elliptic differential equations (cf. [70]) that any solution of (1.22) is a classical solution of the Helmholtz equation in R3D.We call v ∈ H1 loc(R3D) a variational solution of (1.18), (1.19), and (1.20) if v solves (1.22) for all ϕ ∈ H1 0 (R3D) which vanish outside of some ball, and v = f on in the sense of the trace theorem and v satisfies Sommerfeld’s radiation condition (1.20). Thefundamentalresultsonuniquenessandexistencearesummarizedinthefollowing theorem. Theorem 1.1 For any f ∈ H1/2() there exists a unique (variational) solution v ∈ H1 loc(R3 D) of (1.18), (1.19), and (1.20). Furthermore, if the boundary data f is continuous on then v ∈ C2(R3 D) ∩ C(R3 D). If f is even continuously differentiable on then the normal derivative ∂v/∂ν exists and is continuous on . For a proof we refer to [150] and [43] (see also Chapter 2, Section 2.1 where we show existence and uniqueness for the Robin boundary conditions). Solutions of the Helmholtz equation (1.18) in some exterior domain which satisfy also Sommerfeld’s radiation condition (1.20) will be referred to as radiating solutions of (1.18). The uniqueness part in the proof of the previous theorem makes essential use of the following result which is due to Rellich [166] (cf. [43]). Lemma 1.2 Let v be a solution of the Helmholtz equation (1.18) in some region of the form {x ∈ R3 : |x| R} satisfying lim r→∞ |x|=r |v(x)|2 ds = 0 . Then v vanishes for |x| R. For a proof we refer to [43]. Green’s formula is the essential tool also in the proof of the following representation formula for radiating solutions of the Helmholtz equation which sometimes is called “Green’s representation formula”. Theorem 1.3 Let v ∈ C2(R3 D) ∩ C(R3 D) be a radiating solution of (1.18) such that v posseses a normal derivative on the boundary in the sense that the limit ∂v(x) ∂ν = lim h→+0 ν(x) · ∇v x + hν(x) , x ∈ , exists uniformly with respect to x ∈ . Then Green’s formula holds in the form v(x) = v(y) ∂(x, y) ∂ν(y) − ∂v(y) ∂ν (x, y) ds(y) , x / ∈ D . (1.23)
- 26. The far field patterns and the inverse problem 7 Again, ν(y) denotes the exterior unit normal vector at y ∈ and the fundamental solution of the Helmholtz equation in R3 given by (x, y) = exp(ik|x − y|) 4π|x − y| , x, y ∈ R3 , x = y . (1.24) For a proof we refer to [43, Theorem 2.4]. As a direct consequence of this theorem one has the following result. Its proof can again be found in [43], Theorems 2.5 and 2.6. Theorem 1.4 Let v ∈ C2(R3 D) ∩ C(R3 D) be a radiating solution of (1.18) such that v posses a normal derivative on the boundary in the sense of Theorem 1.3. Then v is analytic in R3 D and has the asymptotic behavior v(x) = exp(ik|x|) 4π|x| v∞ (x̂) + O(|x|−2 ) , |x| → ∞ , (1.25) uniformly with respect to x̂ = x/|x| ∈ S2. The function v∞ : S2 → C is analytic and is called the far field pattern of v. It has the form v∞ (x̂) = v(y) ∂ ∂ν(y) e−ikx̂·y − ∂v(y) ∂ν e−ikx̂·y ds(y) , x̂ ∈ S2 . (1.26) As a consequence of the analyticity of v and Rellich’s Lemma 1.2 we have Corollary 1.5 (a) If v vanishes on some open subset of R3 D then v vanishes everywhere in R3 D. (Note that we always assume that the exterior of D is connected.) (b) If v∞ vanishes on an open part of S2 (open relative to S2) then v vanishes in the exterior of D. Application of these results to the scattering problem (1.15), (1.16), and (1.17) assures existence of a unique solution u for any incident field ui. Its dependence on the incident direction is indicated by writing u = u(·, θ). Analogously, the far field pattern u∞ of us dependsonthetwoangularvaluesx̂ andθ.Weindicatethisbywritingu∞ = u∞(x̂, θ)and note that u∞ depends analytically on both variables.This follows, e.g., fromTheorem 1.6 below. 1.3 The far field patterns and the inverse problem First, we will prove a reciprocity principle for u∞. It states the (physically obvious) fact that it is the same if we illuminate an object from the direction θ and observe it in the direction −x̂ or the other way around: illumination from x̂ and observation in −θ. Theorem 1.6 (First reciprocity principle) Let u∞ x̂, θ be the far field pattern corre- sponding to the direction x̂ of observation and the direction θ of the incident plane wave. Then u∞ −x̂, θ = u∞ −θ, x̂ for all x̂, θ ∈ S2 . (1.27)
- 27. 8 Dirichlet and Neumann boundary conditions Proof: Application of Green’s second theorem to ui and us in the interior and exterior of D, respectively, yields 0 = ui (y, θ) ∂ ∂ν ui (y, x̂) − ui (y, x̂) ∂ ∂ν ui (y, θ) ds(y) , 0 = us (y, θ) ∂ ∂ν us (y, x̂) − us (y, x̂) ∂ ∂ν us (y, θ) ds(y) . (More precisely, to prove the second equation, one applies Green’s second theorem to us in the region {x ∈ R3 D : |x| R} with R large enough and lets R tend to infinity.) Now we use the representation (1.26) for the far field patterns u∞ −x̂, θ and u∞ −θ, x̂ : u∞ (−x̂, θ) = us (y, θ) ∂ ∂ν ui (y, x̂) − ui (y, x̂) ∂ ∂ν us (y, θ) ds(y) , −u∞ (−θ, x̂) = ui (y, θ) ∂ ∂ν us (y, x̂) − us (y, x̂) ∂ ∂ν ui (y, θ) ds(y) . Adding these four equations yields u∞ (−x̂, θ) − u∞ (−θ, x̂) = u(y, θ) ∂ ∂ν u(y, x̂) − u(y, x̂) ∂ ∂ν u(y, θ) ds(y) . (1.28) So far, we have not used the boundary condition on . With u(y, x̂) = 0 and u(y, θ) = 0 on the assertion follows. There exists an interesting second reciprocity principle which relates the scattered field us = us(x, θ) corresponding to the plane wave of direction θ as incident field to the far field pattern v∞ = v∞(x̂, z) which corresponds to the point source (·, z) as inci- dent field. Indeed, by the same arguments as in Theorem 1.6 one can show (cf. [160], Section 2.1): Theorem 1.7 (Second or mixed reciprocity principle) Let us(z, −θ) be the scattered field at z ∈ R3 D which corresponds to the incident field ui(x, −θ) = exp(−ikx · θ), x ∈ R3. Furthermore, let v∞(θ, z) be the far field pattern of the scattered field vs at θ which corresponds to the incident field vi(x, z) = (x, z), x ∈ R3. Then us (z, −θ) = v∞ (θ, z) for all θ ∈ S2 and z / ∈ D . (1.29) The far field patterns u∞(x̂, θ), x̂, θ ∈ S2, define the integral operator F : L2(S2) → L2(S2) by (Fg)(x̂) = S2 u∞ (x̂, θ) g(θ) ds(θ) for x̂ ∈ S2 , (1.30)
- 28. The far field patterns and the inverse problem 9 which we will call the far field operator. It is certainly compact in L2(S2) since its kernel is analytic in both variables and is related to the scattering operator S : L2(S2) → L2(S2) by S = I + ik 8π2 F (1.31) where I denotes the identity. In the next theorem we collect some properties of these operators. Theorem 1.8 (a) The far field operator F satisfies F − F∗ = ik 8π2 F∗ F (1.32) where F∗ denotes the L2 – adjoint of F. (b) The scattering operator S = I + ik 8π2 F is unitary, i.e., S∗S = S S∗ = I. (c) The far field operator F is normal, i.e., F∗F = F F∗. (d) Assume that there exists no non-trivial Herglotz wave function vg, i.e., a function of the form vg(x) = S2 eikx·θ g(θ) ds(θ), x ∈ R3 , (1.33) with density g ∈ L2(S2) which vanishes on . In particular, such a function does not exists if k2 is not a Dirichlet eigenvalue of − in D.2 Then F is one-to-one and its range R(F) is dense in L2(S2). Proof: (a) For g, h ∈ L2(S2), define the Herglotz wave functions vi and wi by vi (x) = S2 eikx·θ g(θ) ds(θ), x ∈ R3 , wi (x) = S2 eikx·θ h(θ) ds(θ), x ∈ R3 , respectively. Let v and w be the solutions of the scattering problem (1.15), (1.16), and (1.17) corresponding to incident fields vi and wi, respectively, with corresponding scat- tered fields vs = v − vi, ws = w − wi and far field patterns v∞, w∞, respectively. Then, by linearity, v∞ = Fg and w∞ = Fh. Green’s formula in DR = {x ∈ R3 D : |x| R} and the boundary conditions yield 0 = DR v w − w v dx = |x|=R v ∂w ∂ν − w ∂v ∂ν ds . (1.34) 2 k2 is called a Dirichlet eigenvalue of − in D if there exists a non-trivial solution u ∈ C2(D) ∩ C(D) of the Helmholtz equation in D such that u vanishes on .
- 29. 10 Dirichlet and Neumann boundary conditions The integral on the right hand side is split into four parts by decomposing v = vi + vs and w = wi + ws. The integral |x|=R vi ∂wi ∂ν − wi ∂vi ∂ν ds vanishes by Green’s second theorem in {x : |x| R} since vi and wi are solutions of the Helmholtz equation (1.18). We note that by our normalization of the far field pattern vs (x) ∂ws(x) ∂r − ws(x) ∂vs(x) ∂r = − 2ik (4πr)2 v∞ x̂ w∞ x̂ + O 1 r3 . From this we conclude that |x|=R vs ∂ws ∂ν − ws ∂vs ∂ν ds −→ − ik 8π2 S2 v∞ w∞ ds = − ik 8π2 (Fg, Fh)L2(S2) as R tends to infinity. Finally, we use the definition of vi and wi and the representation (1.26) to compute |x|=R vi ∂ws ∂ν − ws ∂vi ∂ν ds = S2 g(θ) |x|=R eikx·θ ∂ws(x) ∂ν − ws(x) ∂ ∂ν eikx·θ ds(x) ds(θ) = − S2 g(θ) w∞(θ) d(θ) = −(g, Fh)L2(S2) . Analogously, we have that |x|=R vs ∂wi ∂ν − wi ∂vs ∂ν ds = (Fg, h)L2(S2) . Taking the limit R → ∞ yields 0 = − ik 8π2 (Fg, Fh)L2(S2) − (g, Fh)L2(S2) + (Fg, h)L2(S2) . (1.35) This holds for all g, h ∈ L2(S2). From this the assertion (a) follows. (b) We compute S∗ S = I − ik 8π2 F∗ I + ik 8π2 F = I + ik 8π2 (F − F∗ ) + k2 64π4 F∗ F and thus S∗S = I with part (a). This implies injectivity of S and thus also surjectivity since S is a compact perturbation of the identity. Therefore, S∗ = S−1 and thus also SS∗ = I.
- 30. The far field patterns and the inverse problem 11 (c) This follows now by comparing the forms of S∗S and SS∗. (d) Let g ∈ L2(S2) be such that Fg = 0 on S2. From the definition of the far field operator we note that Fg = v∞ where v∞ is the far field pattern which corresponds to the incident field vi(x) = S2 exp(ikx · θ) g(θ) ds(θ), x ∈ R3. Rellich’s Lemma 1.2 and analytic continuation imply that the scattered field vs vanishes outside of D.3 From the boundary condition we conclude that the incident field vi vanishes on , i.e., vi is a Dirichlet eigenfunction of − in D. From the assumption on the wavenumber vi has to vanish in D and thus everywhere by analytic continuation. Expansion into spherical wave functions by using the Jacobi–Anger expansion (see (1.82) below) yields that g vanishes on S2. Finally, we show that the adjoint F∗ of F is one-to-one as well which proves denseness of the range of F. Indeed, F∗g = 0 yields by using the reciprocity relation (1.27) that 0 = (F∗ g)(x̂) = S2 u∞(θ, x̂) g(θ) ds(θ) = S2 u∞(−x̂, −θ) g(θ) ds(θ) , i.e., Fg̃ = 0 with g̃(θ) = g(−θ). Injectivity of F yields that g̃ = 0 and thus also g = 0. Now we turn to the formulation of the inverse scattering problem for which we will introduce the factorization methods in Section 1.4. Inverse Scattering Problem: Given the wavenumber k 0 and the far field patterns u∞(x̂, θ) for all x̂, θ ∈ S2 determine the shape of the scattering obstacle D! The following uniqueness theorem, taken from [43], assures that in principle the data set u∞(x̂, θ) : x̂, θ ∈ S2 is sufficient to determine D. Theorem 1.9 For fixed wavenumber k 0 the far field patterns u∞(x̂, θ) for all x̂, θ ∈ S2 uniquely determine the shape of the scattering obstacle D, i.e., if there are two obstacles D1 and D2 with corresponding far field patterns u∞ 1 (x̂, θ) and u∞ 2 (x̂, θ), respectively, then u∞ 1 (x̂, θ) = u∞ 2 (x̂, θ) for all x̂, θ ∈ S2 implies that D1 = D2. As in Figure 1.1 we want to illustrate also the inverse scattering problem with two examples, again in two dimensions. In this case, the far field patterns u∞ depend on the two variables x̂ and θ from the unit circle in R2 which we identify with the interval [0, 2π]. Furthermore, we identify x̂ = (cos φ, sin φ) with φ ∈ [0, 2π] and the unit vector θ with the angle θ ∈ [0, 2π]. Figures 1.2 and 1.3 show contour plots of the real and imaginary parts, respectively, of the far field patterns for two examples. The inverse scattering problem is to identify the obstacle D from these plots. In the first example (Figure 1.2) the contour lines are straight lines, i.e., u∞ is constant along lines of the form φ − θ = const. In terms of the unit vectors x̂ and θ this can be written in the form x̂·θ = const. By the following result of Karp [108] (which is also true for the two-dimensional case) we conclude that this first example corresponds to D being a disk. The second example corresponds to a domain D which is certainly not a disk.4 3 Here we make use of the assumption that the exterior of D is connected. 4 This belongs to the domain D of Figure 1.7.
- 31. 12 Dirichlet and Neumann boundary conditions 300 200 100 0 0 100 200 300 0 50 100 150 200 250 300 350 350 300 250 200 150 100 50 0 Figure 1.2 Real (left) and imaginary (right) parts of u∞ = u∞(φ, θ) for φ, θ ∈ [0, 2π] 350 300 250 200 150 100 50 0 0 100 200 300 300 200 100 0 0 50 100 150 200 250 300 350 Figure 1.3 Real (left) and imaginary (right) parts of u∞ = u∞(φ, θ) for φ, θ ∈ [0, 2π] Theorem 1.10 Let k 0 and u∞ = u∞(x̂, θ) for x̂, θ ∈ S2 be the far field patterns corresponding to some domain D ⊂ R3. If u∞ (x̂, θ) = u∞ (Qx̂, Qθ) for all x̂, θ ∈ S2 and all rotations, i.e., all real orthogonal matrices Q ∈ R3×3 with det Q = 1, then D is a ball with center zero. The proof is a simple consequence of the uniqueness result of Theorem 1.9, see Section 5.1 of [43].
- 32. Factorization methods 13 Remarks: (a) We note that by the analyticity of u∞ with respect to both variables it is sufficient to require that u∞ 1 (x̂, θ) and u∞ 2 (x̂, θ) coincide for all x̂ and θ from open subsets of S2 or even from an infinite number of pairs (x̂, θm). We refer to [130] for more details. (b) The historically first uniqueness result for this inverse scattering problem is due to Schiffer (see remark in [138], Section V.5). The proof depends crucially on the com- pact imbedding property of H1 0 (G) in L2(G) for any bounded open set G to ensure that the spectrum of − in G with respect to Dirichlet boundary conditions is dis- crete. The analogous argument for other boundary conditions such as the Neumann boundary condition requires the compact imbedding property of H1(G) in L2(G) which holds only under smoothness assumptions on the boundary of G. Since the argument is applied to G being the (set-)difference of two obstacles D1 and D2 smoothness of this difference cannot be assured. Therefore, Schiffer’s proof cannot be transfered to other boundary conditions. Based on results of Isakov [105] for penetrable obstacles Kirsch and Kress [124] obtained uniqueness results for several kinds of boundary conditions. (c) To the authors knowledge it is an open problem whether the far field patterns u∞(x̂, θ) for all x̂ ∈ S2 but only one incident wave ui(x) = exp(ikx · θ) uniquely determines D if no a priori information on D is available. Partial results are known if a priori information on the obstacle D is available. It has been shown by Colton and Sleeman in [57] (cf. [43], Theorem 5.2) that the scatterer is uniquely determined by the far field patterns of a finite number of incident plane waves provided a priori information on the size of the obstacle D is available. In recent papers by Elschner, Yamamoto, Liu, and others (see [61, 140]) uniqueness of the inverse scattering problem for one incident plane wave and polyhedral scatterers is shown. (d) A similar theorem shows that the scatterer is uniquely determined by the far field patterns for an infinite number of incident plane waves with distinct wavenumbers from a bounded interval in R0. We refer again to [130] for more details. 1.4 Factorization methods We recall from (1.30) that the far field patterns u∞ define the far field operator F : L2(S2) → L2(S2) by (Fg)(x̂) = S2 u∞ x̂, θ g(θ) ds(θ) , x̂ ∈ S2 . (1.36) With respect to the inverse problem, this operator contains the known data. It is the aim to give explicit characterizations of the unknown domain D by this “data operator” F. This section is organized as follows. In Subsection 1.4.1 we will derive a factorization of the operator F in the form F = G T G∗ (1.37)
- 33. 14 Dirichlet and Neumann boundary conditions with some compact operator G and some isomorphism T between suitable spaces which depend on D, of course. This factorization is the basis of all versions of the Factorization Methods and is responsible for its name. From this factorization we observe already that the range of the operator F is contained in the range of G. There is a simple – and very explicit – relationship between the range of the operator G and the shape of D. Let us first define the operator G. Definition 1.11 Let the data-to-pattern operator G : H1/2() → L2(S2) be defined by Gf = v∞ where v∞ ∈ L2(S2) is the far field pattern of the solution v of the exterior Dirichlet problem (1.18), (1.19), and (1.20) with boundary data f ∈ H1/2(), i.e., v ∈ H1 loc(R3 D) solves v + k2 v = 0 outside D , (1.38) v = f on , (1.39) and ∂v ∂r − ik v = O r−2 for r = |x| → ∞ (1.40) uniformly with respect to x̂ = x/|x|. We note that we do not indicate the type of boundary condition by writing GDir for the Dirichlet boundary condition. Later in this chapter (in Section 1.6) we will introduce the analogous operator for the Neumann boundary condition and denote it also by G. In Chapter 2, however, we will indicate the type of boundary condition by writing GDir and GNeu, respectively. Then we have: Theorem 1.12 Let G : H1/2() → L2(S2) be defined by Definition 1.11. For any z ∈ R3 define the function φz ∈ L2(S2) by φz(x̂) := e−ikx̂·z , x̂ ∈ S2 . (1.41) Then φz belongs to the range R(G) of G if, and only if, z ∈ D. Proof: Let first z ∈ D and define v(x) := (x, z) = exp(ik|x − z|) 4π|x − z| , x / ∈ D , and f := v|. Then f ∈ H1/2() and the far field pattern of v is given by v∞ (x̂) = e−ikx̂·z , x̂ ∈ S2 , which coincides with φz, i.e., Gf = v∞ = φz, i.e., φz ∈ R(G). Let now z / ∈ D and assume on the contrary that there exists f ∈ H1/2() with Gf = φz. Let v be the solution of the exterior Dirichlet problem with boundary data f and v∞ = Gf be its far field pattern. Since φz is the far field pattern of (·, z) we
- 34. Factorization methods 15 conclude by Rellich’s Lemma 1.2 that v(x) = (x, z) for all x outside of any sphere containing D and z. Finally, by analytic continuation5 we conclude that v and (·, z) coincide on R3 (D ∪ {z}). If z / ∈ D this contradicts the fact that v is analytic in R3 D and (x, z) is singular at x = z. If z ∈ we have that (x, z) = f (x) for x ∈ , x = z, i.e., the function x → (x, z) is in H1/2(). This contradicts the fact that this function is certainly not in H1(D) or H1 loc(R3 D) since ∇(x, z) = O(1/|x − z|2) as x → z. The main work of the Factorization Methods is to relate the range of G to the known data operator F (or some operator which can be derived from F). In this chapter we will do this in two possible ways which will be presented in Subsections 1.4.2 and 1.4.3, respectively. Each of these subsections will begin with an abstract result from functional analysis, formulated in general Hilbert or Banach spaces. Application of this abstract result to the operators G and F will lead to a precise characterization of the range R(G) of G by the data operator F. Combination of this result with Theorem 1.12 will give fairly explicit formulas for the characteristic function χD of D which will solely be expressed by quantities computable from F. 1.4.1 Factorization of the far field operator We recall Definition 1.11 of the crucial data-to-pattern operator G : H1/2() → L2(S2). It is defined by Gf = v∞ where v∞ ∈ L2(S2) is the far field pattern of the solution v of the exterior Dirichlet problem (1.38), (1.39), and (1.40) with boundary data f ∈ H1/2(). Properties of G are collected in the following lemma. Lemma 1.13 The data-to-pattern operator G : H1/2() → L2(S2) is compact, one-to- one with dense range in L2(S2). Proof: First, injectivity is a direct consequence of Rellich’s Lemma and analytic continuation, see Corollary 1.5. To prove compactness we choose a ball B = B(0, R) of radius R centered at 0 which contains D in its interior. Using the representation (1.26) we can decompose G as G = G2G1 where G1 : H1/2() → C(∂B)×C(∂B) and G2 : C(∂B)×C(∂B) → L2(S2) are defined by G1f = v|∂B, ∂v/∂ν|∂B and G2(g, h)(x̂) = ∂B g(y) ∂ ∂ν(y) e−ikx̂·y − h(y) e−ikx̂·y ds(y) , x̂ ∈ S2 , respectively, and where again v denotes the solution of the exterior Dirichlet boundary value problem with boundary values f . Then G1 is bounded by interior regularity results and G2 compact which proves compactness of G. 5 Note that it is here where we make use of the assumption that the exterior of D is connected.
- 35. 16 Dirichlet and Neumann boundary conditions To prove denseness of the range of G we recall the definition of the spherical harmonics Ym n defined by Ym n (x̂) = Ym n (φ, ϕ) = 2n + 1 4π (n − |m|)! (n + |m|)! P|m| n (cos φ) eimϕ , (1.42) for −n ≤ m ≤ n and n ∈ N. Here (φ, ϕ) are the spherical polar coordinates of x̂ ∈ S2 and Pm n the associated Legendre functions. The spherical harmonics Ym n are normalized such that they form a complete orthonormal system in L2(S2). Furthermore, we denote by hn the spherical Hankel functions of the first kind and order n ∈ N. We refer to [43], Sections 2.3 and 2.4, for a brief introduction to these functions. Now we make use of the fact that every element f in L2(S2) can be approximated by a finite linear combination of the spherical harmonics, i.e., for every ε 0 there exists N ∈ N and constants cm n ∈ C with f − N n=0 |m|≤n cm n Ym n L2(S2) ≤ ε . We choose the origin inside of D and define the function v by v(x) = k 4π N n=0 |m|≤n cm n ei(n+1)π/2 hn(k|x|)Ym n (x̂) , x = 0 . Thefarfieldpatternofhn(k|x|)Ym n (x̂)isdeterminedbytheasymptoticbehaviorofhn(t)as t tends to infinity and is given by (4π/k) exp −i(n+1)π/2 Ym n (x̂) (cf. [43]). Therefore, v∞ = N n=0 |m|≤n cm n Ym n and thus f − v∞L2(S2) ≤ ε. Observing that v∞ = Gv| yields the assertion. The operator T in (1.37) will be the adjoint of the single layer boundary operator S : H−1/2() → H1/2(), defined by (Sϕ)(x) := (x, y) ϕ(y) ds(y) , x ∈ , (1.43) where denotes again the fundamental solution of the Helmholtz equation in three dimensions as defined in (1.24). Note that we write this as an ordinary integral although, strictly speaking, this has to be understood as the bounded extension of the classical operator S defined on L2(). In the following lemma we summarize some of the well-known properties of S which will imply, in particular, that T in (1.37) is an isomorphism. By H−1/2(), H1/2() we denote the dual form which, in our setting, is the extension of the inner product of L2(). In particular, this dual form is sesqui-linear, i.e., the mappings ϕ → ϕ, ψ and ψ → ϕ, ψ are linear where the bar denotes the complex conjugation.
- 36. Factorization methods 17 Lemma 1.14 Assume that k2 is not a Dirichlet eigenvalue of − in D. Then the following holds. (a) S is an isomorphism from the Sobolev space H−1/2() onto H1/2(). (b) Imϕ, Sϕ 0 for all ϕ ∈ H−1/2() with ϕ = 0. Again, ·, · denotes the duality pairing in H−1/2(), H1/2(). (c) Let Si be the single layer boundary operator (1.43) corresponding to the wavenumber k = i. The operator Si is self-adjoint and coercive as an operator from H−1/2() onto H1/2(), i.e., there exists c0 0 with ϕ, Siϕ ≥ c0ϕ2 H−1/2() for all ϕ ∈ H−1/2 () . (1.44) (d) The difference S − Si is compact from H−1/2() into H1/2(). Proof: (a) The mapping properties of S in Sobolev spaces are intensively studied in [144]. (b) Define, for any ϕ ∈ H−1/2(), the single layer potential v by v(x) = ϕ(y) (x, y) ds(y) , x ∈ R3 . (1.45) Then v is a solution of the Helmholtz equation in R3 . From the theory of potentials with H−1/2()−densities it is known (see [111, 144]) that v ∈ H1(D) ∩ H1 loc(R3 D), that the traces v± and ∂v±/∂ν exist in the variational sense with v± = Sϕ and ϕ = ∂v−/∂ν − ∂v+/∂ν. Here, v± denotes the limit from the exterior (+) and interior (−), respectively. Therefore, using Green’s formula in D and in DR := x ∈ R3 D : |x| R we conclude that ϕ, Sϕ = ∂v− ∂ν − ∂v+ ∂ν , v (1.46) = D∪DR |∇v|2 − k2 |v|2 dx − |x|=R v ∂v ∂r ds (1.47) = D∪DR |∇v|2 − k2 |v|2 dx − ik |x|=R |v|2 ds + O 1 R (1.48) as R tends to infinity. Taking the imaginary part yields Imϕ, Sϕ = −k lim R→∞ |x|=R |v|2 ds = − k (4π)2 S2 |v∞ |2 ds ≤ 0 . (1.49) Let now Imϕ, Sϕ = 0 for some ϕ ∈ H−1/2(). Then v∞ = 0. From (1.49), Rellich’s Lemma 1.2, and unique continuation we conclude that v vanishes outside of D.Therefore, Sϕ = 0 on by the trace theorem. Since S is an isomorphism ϕ has to vanish.
- 37. 18 Dirichlet and Neumann boundary conditions (c) For k = i the same arguments as above yield ϕ, Siϕ = D∪DR |∇v|2 + |v|2 dx + |x|=R |v|2 ds + O 1 R , R → ∞ , and thus as R → ∞ (note that v decays exponentially): ϕ, Siϕ = R3 |∇v|2 + |v|2 dx = v2 H1(R3) . The trace theorem and the boundedness of S−1 i yields the existence of c 0 and c0 0 with ϕ, Siϕ ≥ c v2 H1/2() = c Siϕ2 H1/2() ≥ c0 ϕ2 H−1/2() . (d) This follows from the fact that the kernel of S − Si is of the form exp(ik|x − y|) − exp(−|x − y|) 4π |x − y| = |x − y| A(|x − y|2 ) + B(|x − y|2 ) with analytic functions A and B. Now we are able to derive the fundamental factorization of F. Theorem 1.15 The following relation holds between F, G and S: F = −G S∗ G∗ (1.50) where G∗ : L2(S2) → H−1/2() and S∗ : H−1/2() → H1/2() are the adjoints of G and S, respectively, with respect to L2(S2) and the dual pairing6 H−1/2(), H1/2(). Proof: As an auxiliary operator we define H : L2(S2) → H1/2() by, (Hg)(x) := S2 g(θ) eikx·θ ds(θ) , x ∈ . (1.51) Hg is the trace on of the Herglotz wave function (1.33) with density g. Its adjoint H∗ : H−1/2() → L2(S2) is given by (H∗ ϕ)(x̂) = ϕ(y) e−ikx̂·y ds(y) , x̂ ∈ S2 . (1.52) We note that by the asymptotic behavior of the fundamental solution H∗ϕ is just the far field pattern of the single layer potential (1.45). The single layer potential (1.45) with continuous density ϕ is continuous in R3 D and thus H∗ϕ = GSϕ, i.e., by a density argument H∗ = GS and therefore H = S∗ G∗ . (1.53) 6 We recall again that in our setting the dual form is sesqui-linear rather than bi-linear, see the remark preceding Lemma 1.14.
- 38. Factorization methods 19 L2(S2) L2(S2) H1/2() H1/2() G* G S* F Figure 1.4 The factorization F = −G S∗ G∗ Now we observe that Fg is the far field pattern of the solution of the exterior Dirichlet problem with boundary data − S2 g(θ) eikx·θ ds(θ) = −(Hg)(x) , x ∈ . This shows that Fg = −GHg . (1.54) Substituting H from (1.53) into (1.54) yields the assertion. We sketched the factorization in Figure 1.4. Remark: If k2 is not a Dirichlet eigenvalue of − in D we can solve (1.53) for G and arrive at the factorization F = −H∗ S−1 H (1.55) with the explicitly given operator H from (1.51). 1.4.2 The inf-criterion Motivated by Theorem 1.12 we will now give a first expression of the range of G by the criterion which depends solely on F. Although not very helpful from the compu- tational point of view it is nevertheless quite general and will lead to more elegant characterizations in the forthcoming subsections. The method is based on the following result from functional analysis. Theorem 1.16 Let X , Y be (complex) reflexive Banach spaces with duals X ∗, Y∗, respec- tively, and dual forms ·, · in X ∗, X and Y∗, Y. Furthermore, let F : Y∗ → Y and B : X → Y linear operators with F = B A B∗ for some linear and bounded operator A : X ∗ → X (1.56) which satisfies a coercivity condition of the form: There exists c 0 with ϕ, Aϕ ≥ cϕ2 X ∗ for all ϕ ∈ R(B∗ ) ⊂ X ∗ . (1.57)
- 39. 20 Dirichlet and Neumann boundary conditions Then, for any φ ∈ Y, φ = 0, φ ∈ R(B) if and only if inf ψ, Fψ : ψ ∈ Y∗ , ψ, φ = 1 0 . (1.58) Here again, R(B) denotes the range of the operator B : X → Y. Furthermore, if φ = Bϕ0 ∈ R(B) for some ϕ0 ∈ X then inf ψ, Fψ : ψ ∈ Y∗ , ψ, φ = 1 ≥ c ϕ02 X . (1.59) Proof: First, we observe that ψ, Fψ = B∗ ψ, AB∗ ψ ≥ cB∗ ψ2 X ∗ for all ψ ∈ Y∗ . (1.60) Let now φ = Bϕ0 for some ϕ0 ∈ X . For ψ ∈ Y∗ with ψ, φ = 1 we have ψ, Fψ ≥ cB∗ ψ2 X ∗ = c ϕ02 X B∗ ψ2 X ∗ ϕ02 X ≥ c ϕ02 X B∗ ψ, ϕ0 2 = c ϕ02 X ψ, Bϕ0 =φ 2 = c ϕ02 X . This provides the lower bound of (1.59). Second, assume that φ / ∈ R(B). Define the closed subspace V := ψ ∈ Y∗ : ψ, φ = 0 . We show that B∗(V) is dense in R(B∗) ⊂ X ∗. This is equivalent to the statement that the annihilators B∗(V) ⊥ and R(B∗) ⊥ = N(B) coincide. Therefore, let ϕ ∈ B∗(V) ⊥ , i.e., B∗ψ, ϕ = 0 for all ψ ∈ V, i.e., ψ, Bϕ = 0 for all ψ ∈ V, i.e., Bϕ ∈ V⊥ = span {φ}. Since φ / ∈ R(B) this implies Bϕ = 0, i.e., ϕ ∈ N(B). By a consequence of the Hahn–Banach Theorem one can find φ̂ ∈ Y∗ with φ̂, φ = 1. Choose a sequence {ψ̂n} in V such that B∗ ψ̂n −→ −B∗ φ̂ as n → ∞ . We set ψn = ψ̂n +φ̂. Then ψn, φ = 1 and B∗ψn → 0. From the first equation of (1.60) we conclude that ψn, Fψn ≤ A B∗ ψn2 X ∗ and thus ψn, Fψn −→ 0, n → ∞, which proves that inf ψ, Fψ : ψ ∈ Y∗ , ψ, φ = 1 = 0 . In our applications we will often prove the coercivity condition (1.57) with the help of the following lemma. Lemma 1.17 Let X be a reflexive Banach space and A, A0 : X ∗ → X be linear and bounded operators such that (i) ϕ, Aϕ ∈ C (−∞, 0] for all ϕ ∈ closure R(B∗) with ϕ = 0, (ii) ϕ, A0ϕ is real-valued, and there exists c0 0 with ϕ, A0ϕ ≥ c0ϕ2 X ∗ for all ϕ ∈ R(B∗ ), (1.61) (iii) A − A0 is compact.
- 40. Factorization methods 21 Then (1.57) holds, i.e., there exists c 0 with ϕ, Aϕ ≥ c ϕ2 X ∗ for all ϕ ∈ R(B∗ ) . (1.62) Proof: If there exists no constant c with (1.62) then there exists a sequence {ϕn} in R(B∗) with ϕnX ∗ = 1 and ϕn, Aϕn −→ 0 as n tends to infinity. Since the unit ball in X ∗ is weakly compact there exists a subsequence which converges weakly to some ϕ ∈ closure R(B∗). We denote this subsequence again by {ϕn}. The compactness of A − A0 yields that (A − A0)ϕn → (A − A0)ϕ in the norm of X . We conclude that ϕn, (A − A0)(ϕ − ϕn) −→ 0. By linearity, ϕ − ϕn, A0(ϕ − ϕn) = ϕ, A0(ϕ − ϕn) − ϕn, (A0 − A)(ϕ − ϕn) + ϕn, Aϕn − ϕn, Aϕ . The first three terms on the right hand side converge to zero, the forth term to ϕ, Aϕ. Assumption (i) implies that ϕ vanishes. Therefore, c0ϕn2 X ∗ ≤ ϕn, A0ϕn ≤ ϕn, (A0 − A)ϕn + ϕn, Aϕn which tends to zero as n → ∞.Therefore, also ϕn → 0 which contradicts the assumption ϕnX ∗ = 1. We wish to apply Theorem 1.16 to the factorization (1.50) of the far field operator F. We choose Y = L2(S2) and X = H1/2(), identify Y∗ with L2(S2), and set B = G. We have to show the coercivity condition (1.62) for A = −S∗. This property follows immediatelyfromthepreviouslemmaincombinationwithLemma1.14whereweproved the properties of the operator S. We formulate the result as a corollary. Corollary 1.18 Assume that k2 is not a Dirichlet eigenvalue of − in D. Then there exists c 0 with |ϕ, Sϕ| ≥ c ϕ2 H−1/2() for all ϕ ∈ H−1/2 () . (1.63) Here again, ·, · denotes the duality pairing in H−1/2(), H1/2(). Application of Theorem 1.16 yields the following result. Theorem 1.19 Assume that k2 is not a Dirichlet eigenvalue of − in D. Then for any φ ∈ L2(S2) with φ = 0 the following holds: φ ∈ R(G) if, and only if, inf (ψ, Fψ)L2(S2) : ψ ∈ L2 (S2 ), (ψ, φ)L2(S2) = 1 0 , (1.64)
- 41. 22 Dirichlet and Neumann boundary conditions where again G : H1/2() → L2(S2) is defined by Gf = v∞ and v solves the boundary value problem (1.38), (1.39), and (1.40). Furthermore, if φ = Gf for some f ∈ H1/2() then inf (ψ, Fψ)L2(S2) : ψ ∈ L2 (S2 ), (ψ, φ)L2(S2) = 1 ≥ c f 2 H1/2() (1.65) for some c 0 independent of φ. Combining this with the characterization of Theorem 1.12 of D by the range of G yields immediately the main result of this subsection. Theorem 1.20 Assume that k2 is not a Dirichlet eigenvalue of − in D. For any z ∈ R3 define again φz ∈ L2(S2) by (1.41), i.e., φz(x̂) := e−ikx̂·z , x̂ ∈ S2 . Then z ∈ D if, and only if, inf (ψ, Fψ)L2(S2) : ψ ∈ L2 (S2 ), (ψ, φz)L2(S2) = 1 0 . (1.66) Therefore, the characteristic function of D is given by χD(z) = sign inf (ψ, Fψ)L2(S2) : ψ ∈ L2 (S2 ), (ψ, φz)L2(S2) = 1 , z ∈ R3 . Furthermore, for z ∈ D we have the estimate: inf (ψ, Fψ)L2(S2) : ψ ∈ L2 (S2 ), (ψ, φz)L2(S2) = 1 ≥ c (·, z)2 H1/2() (1.67) for some constant c 0 which is independent of z. Proof: It remains to prove the estimate (1.67). It follows directly from (1.65) and the observation that φz = G(·, z)| for z ∈ D (see proof of Theorem 1.12). We note again that the evaluation of the form of χD(z) uses only known information on the far field operator F. Although satisfactory from the theoretical point of view there is a major drawback with respect to the computationally point of view since it is very time consuming to solve a minimization problem for every sampling point z. We conclude this subsection with the remark that Theorem 1.20 provides an alter- native – and very explicit – proof of uniqueness of the inverse scattering problem under the assumption that k2 is not a Dirichlet eigenvalue of − in D. 1.4.3 The (F ∗F )1/4-method As an important observation from Theorem 1.16 we note that the inf-criterion in the char- acterization (1.58) depends only on F and not on the factorization itself. This observation leads directly to the first part of the following result (see [71]). Theorem 1.21 Let H be a Hilbert space and let F : H → H have two factorizations of the form F = B1 A1 B∗ 1 = B2 A2 B∗ 2 (1.68)
- 42. Factorization methods 23 with linear operators Bj : Xj → H, j = 1, 2, from reflexive Banach spaces Xj into H and linear operators Aj : X ∗ j → Xj, j = 1, 2, which both satisfy the coercivity condition (1.57), i.e., ϕ, Ajϕ ≥ cϕ2 X ∗ j for all ϕ ∈ R(B∗ j ) and j = 1, 2 . (1.69) Then the ranges of B1 and B2 coincide. If in addition B1 and B2 are one-to-one then B−1 2 B1 and B−1 1 B2 are (topological) isomorphisms from X1 onto X2 and from X2 onto X1, respectively. Proof: It remains to prove the second part. From B−1 2 B1 B−1 1 B2 = IX2 and B−1 1 B2 B−1 2 B1 = IX1 we observe that B−1 2 B1 and B−1 1 B2 are algebraical isomor- phisms. It remains to show that they are bounded. Let ϕ1 ∈ X1 and set ϕ2 = B−1 2 B1ϕ1. Then B2ϕ2 = B1ϕ1 and thus B2ϕ2 ∈ R(B1). In particular, from Theorem 1.16 we conclude that c ϕ12 X1 ≤ inf (ψ, Fψ)H : ψ ∈ H, (ψ, B2ϕ2)H = 1 = inf B∗ 2ψ, A2B∗ 2ψ : ψ ∈ H, B∗ 2ψ, ϕ2 = 1 = inf φ, A2φ : φ ∈ X ∗ 2 , φ, ϕ2 = 1 since the range of B∗ 2 is dense in X ∗ 2 . By a well-known result from functional analysis (application of the theorem of Hahn–Banach) there exists φ2 ∈ X ∗ 2 with φ2X ∗ 2 = 1 and φ2, ϕ2 = ϕ2X2 . Choosing φ = φ2/ϕ2X2 in the last estimate yields (since φ, ϕ2 = 1) c ϕ12 X1 ≤ 1 ϕ22 X2 φ2, A2φ2 ≤ A2 ϕ22 X2 φ22 X ∗ 2 = A2 ϕ22 X2 . This yields ϕ2X2 ≤ A2 c ϕ1X1 which proves continuity of B−1 2 B1. The proof for the continuity of B−1 1 B2 follows by interchanging the roles of B1 and B2. As a first application of this theorem we formulate a result for self-adjoint and non- negative operators F which will be applied for problems in impedance tomography (see Chapter 6). Corollary 1.22 Let F : H → H be a compact and self-adjoint operator from the Hilbert space H into itself which has a factorization of the form F = B A B∗ with some operator B : X → H (where again X is a reflexive Banach space) and some self-adjoint operator A : X ∗ → X which is coercive on R(B∗), i.e., there exists c 0 with ϕ, Aϕ ≥ cϕ2 X ∗ for all ϕ ∈ R(B∗). Then the ranges of B and F1/2 coincide.
- 43. 24 Dirichlet and Neumann boundary conditions Remark: The operator F1/2 can be defined by using an eigensystem of F. Indeed, if λj ≥ 0 are the eigenvalues of the non-negative (!) operator F with corresponding normalized eigenfunctions ψj ∈ H then F has the form Fψ = j λj (ψ, ψj)H ψj , ψ ∈ H , (1.70) and thus F1/2 ψ = j λj (ψ, ψj)H ψj , ψ ∈ H . Proof of the corollary: The operator F admits a second factorization in the form F = F1/2F1/2. The assertion follows directly from Theorem 1.21 because the operators A and the identity satisfy both the coercivity condition (1.69). The far field operator F : L2(S2) → L2(S2) for the Dirichlet boundary condition fails to be self-adjoint and this corollary is not applicable. However, it is normal and – even more – the operator I + ik 8π2 F is unitary. For this situation there exists a corresponding result which we first formulate and prove in the general setting before we apply it to the factorization of the far field operator. Theorem 1.23 Let H be a Hilbert space, X a reflexive Banach space and let the compact operator F : H → H have a factorization of the form F = B A B∗ with operators B : X → H and A : X ∗ → X such that Imϕ, Aϕ = 0 for all ϕ ∈ closure R(B∗) with ϕ = 0. Let furthermore A be of the form A = A0 + C for some compact operator C and some self-adjoint operator A0 which is coercive on R(B∗) in the sense of (1.61). Finally, assume that F is one-to-one and I + irF is unitary for some r 0. Then the ranges of B and (F∗F)1/4 coincide. Furthermore, the operators (F∗F)−1/4B and B−1(F∗F)1/4 are isomorphisms from X onto H and from H onto X , respectively. Proof: First we note that by Lemma 1.17 the operator A satisfies the coercivity condition (1.69), i.e., there exists c 0 with ϕ, Aϕ ≥ cϕ2 X ∗ for all ϕ ∈ R(B∗ ) . (1.71) The unitarity of I + irF implies that F is normal. Therefore, there exists a complete set of orthonormal eigenfunctions ψj ∈ H with corresponding eigenvalues λj ∈ C, j = 1, 2, 3, . . . (see, e.g., [168]). Furthermore, since the operator I + irF is unitary the eigenvalues λj of F lie on the circle of radius 1/r and center i/r. The spectral theorem for normal operators yields that F has the form (1.70), i.e., Fψ = ∞ j=1 λj(ψ, ψj)H ψj , ψ ∈ H . (1.72)
- 44. Factorization methods 25 From this we conclude that F has a second factorization in the form F = (F∗ F)1/4 A2 (F∗ F)1/4 , (1.73) where the operator (F∗F)1/4 : H → H is given by (F∗ F)1/4 ψ = ∞ j=1 |λj| (ψ, ψj)H ψj , ψ ∈ H , (1.74) and the signum A2 : H → H of F is given by A2ψ = ∞ j=1 λj |λj| (ψ, ψj)H ψj , ψ ∈ H . (1.75) In order to apply Theorem 1.21 with X2 = H and B2 = (F∗F)1/4 we have to show that also A2 satisfies the coercivity condition (1.69) on H. We set sj = λj/|λj| for abbreviation. From the facts that λj − i r = 1 r and that λj tends to zero as j tends to infinity we conclude that the only accumulation points of the sequence {sj} can be +1 or −1. This situation is illustrated in Figure 1.5. The main part of the proof consists of showing that the only accumulation point is +1. Before we show this we define the functions ϕj ∈ X ∗ by ϕj = 1 λj B∗ ψj , j ∈ N , where the branch of the square root is chosen such that Im λj 0. The following argument proves a kind of orthogonality relation of ϕj: ϕj, Aϕ = 1 λj √ λ B∗ ψj , A B∗ ψ = 1 λj √ λ (ψj , B A B∗ ψ)H = λ λj √ λ (ψj, ψ)H , λ1 λ2 λ3 λj–1 λj+1 λj δ 0 –1 +1 0 s1 s2 s3 sj–1 sj sj+1 Figure 1.5 Eigenvalues {λj} of F (left) and sj = λj/|λj| (right)
- 45. 26 Dirichlet and Neumann boundary conditions i.e., ϕj, Aϕ = sj δj for j, ∈ N . (1.76) From this condition and (1.71) we conclude that the sequence {ϕj} is bounded: c ϕj2 X ∗ ≤ ϕj, Aϕj = |sj| = 1 for all j . Now we assume that −1 is an accumulation point of {sj}. Then there exists a subsequence of {sj} which converges to −1. We denote this by writing sj → −1 again. Since the sequence {ϕj} is bounded there exists a further subsequence such that ϕj converges weakly in X ∗ to some ϕ ∈ closure R(B∗). From (1.76) we have that ϕj, Aϕj = ϕj, A0ϕj + ϕj, Cϕj −→ −1 , j → ∞ . (1.77) Since C is compact from X into X ∗ we conclude that Cϕj converges to Cϕ and thus ϕj, Cϕj = ϕj, Cϕ + ϕj, C(ϕj − ϕ) . The second term on the right hand side converges to zero by the Cauchy-Schwarz inequality and the first term to ϕ, Cϕ by the weak convergence of ϕj to ϕ, i.e., ϕj, Cϕj −→ ϕ, Cϕ , j → ∞ . Comparing the imaginary parts of this and of (1.77) implies that Imϕ, Aϕ vanishes. From our assumption we conclude that ϕ has to vanish. Then (1.77) yields that ϕj , A0ϕj −→ −1 which is impossible since the left-hand side is bounded below by zero. This proves that the sequence {sj} converges to +1. Now we proceed with the proof of the estimate (1.69) for A2. Let ψ = ∞ j=1 cj ψj with ψ2 H = ∞ j=1 |cj|2 = 1. We compute (A2ψ, ψ)H = ∞ j=1 sj cj ψj , ∞ j=1 cj ψj = ∞ j=1 sj|cj|2 . The complex number ∞ j=1 sj|cj|2 belongs to the convex hull M = conv{sj : j ∈ N} ⊂ C of all numbers sj. We conclude that (A2ψ, ψ)H ≥ inf{|z| : z ∈ M } . The set M is contained in the part of the upper half-disk which is above the line = ts1 + (1 − t)1 : t ∈ R passing through s1 and 1. The distance of the origin to this convex hull M is given by inf{|z| : z ∈ M } = inf{|z| : z ∈ } = sin δ 2 , (1.78) where π − δ ∈ (0, π) is the argument of s1, i.e., s1 = − cos δ + i sin δ (see Figure 1.6). Therefore, we arrive at the estimate (A2ψ, ψ)H ≥ sin δ 2 ψ2 H .
- 46. Factorization methods 27 +1 s1 s2 s3 sj–1 sj sj+1 –1 0 δ Figure 1.6 The distance from conv{sj} to the origin is positive Therefore, all the assumptions of Theorem 1.21 are satisfied. It’s application yields the assertion. We apply this abstract result to the factorization (1.50) with H = L2(S2), X = H1/2(), B = G, and A = −S∗. The assumptions on A = −S∗ are satisfied by Lemma 1.14. Furthermore, we note that B = G and (F∗F)1/4 are one-to-one by Lemma 1.13 and Theorem 1.8, respectively, if k2 is not a Dirichlet eigenvalue. Therefore we have: Theorem 1.24 Assume that k2 is not a Dirichlet eigenvalue of − in D. Then the ranges of G and (F∗F)1/4 coincide. Furthermore, the operators (F∗F)−1/4G and G−1(F∗F)1/4 are isomorphisms from H1/2() onto L2(S2) and from L2(S2) onto H1/2(), respectively. We note that the range of G is expressed by a characterization which depends solely on the data operator F – just as in the case of Theorem 1.19. The combination of this result with Theorem 1.12 yields the main result of this subsection. Theorem 1.25 Assume that k2 is not a Dirichlet eigenvalue of − in D. For any z ∈ R3 define again φz ∈ L2(S2) by (1.41), i.e., φz(x̂) := e−ikx̂·z , x̂ ∈ S2 . Then z ∈ D ⇐⇒ φz ∈ R (F∗ F)1/4 (1.79) ⇐⇒ W(z) := j (φz, ψj)L2(S2) 2 |λj| −1 0 . (1.80) Here, λj ∈ C are the eigenvalues of the normal operator F with corresponding normalized eigenfunctions ψj ∈ L2(S2). Therefore, χD(z) = sign W(z) is the characteristic function of D. Proof: It remains to prove the characterization (1.80). We note from the characterization (1.79) that a point z ∈ R3 belongs to D if, and only if, the equation (F∗ F)1/4 g = φz (1.81)
- 47. 28 Dirichlet and Neumann boundary conditions is solvable in L2(S2). We write φz in spectral form as φz = j (φz, ψj)L2(S2) ψj . From (1.74) we observe that (1.81) is solvable if, and only if, the series j (φz, ψj)L2(S2) 2 |λj| converges,7 and in this case g = j (φz, ψj)L2(S2) |λj| ψj is the solution of (1.81). Therefore, a point z ∈ R3 belongs to D if, and only if, the series j (φz, ψj)L2(S2) 2 |λj| converges which proves the characterization (1.80). The essential assumption under which the characterization (1.80) had been derived was the normality of the far field operator F and the unitarity of the scattering operator S. In many cases the operator F fails to be normal. Examples of theses cases are “absorbing” media D or limited angle data u∞(x̂, θ). If the far field operator fails to be normal not very much is known about eigenvalues. We refer to [43] for some results. In particular, a complete set of eigenfunctions usually does not exist. Therefore, the technique of this subsection does not work. As we will see later, the minimization approach is still appli- cable but, as we mentioned already, it is very time consuming from the computational point of view. Before we modify the approach of this subsection in the next chapter appropriately we observe that the convergence of the series in (1.80) depends only on the rates of decay of the eigenvalues |λj| and the expansion coefficients. From the obvious estimate 1 √ 2 | Re λj| + | Im λj| ≤ |λj| ≤ | Re λj| + | Im λj| and the observation that Im λj 0 we note that we can replace |λj| in (1.80) by | Re λj|+ Im λj. Furthermore, we observe that | Re λj|+Im λj are the eigenvalues of the self-adjoint and positive operator F# = | Re F| + Im F where the self-adjoint parts Re F and Im F are defined by Re F = 1 2 F + F∗ and Im F = 1 2i F − F∗ . 7 This is just Picard’s criterion, see [113].
- 48. An explicit example 29 Therefore, we can replace the operator (F∗F)1/4 in (1.81) by F 1/2 # . We will see in the next chapters that the characterization (1.81) by F 1/2 # has a much wider applicability. We continue with the example of D being the unit ball in R3. 1.5 An explicit example Let D be the unit ball in R3 centered at the origin. We will compute the quantities which appear in the series (1.80). First, we expand the incident and scattered fields into spherical wave functions. Let jn and hn be the, respectively, spherical Bessel functions and spherical Hankel functions of the first kind and order n ∈ N and let Ym n (x̂) be the spherical harmonics of order n normalized such that they form a complete orthonormal system in L2(S2) (see (1.42)). The Jacobi–Anger expansion (cf. [43]) has the form ui (x) = eikx·θ = 4π ∞ n=0 |m|≤n in jn(k|x|) Ym n (x̂) Ym n (θ) , x ∈ R3 . (1.82) Again, the unit vector θ ∈ S2 denotes the direction of incidence and x̂ = x/|x|. It is immediately seen (at least formally) that the scattered field is given by us (x) = −4π ∞ n=0 |m|≤n in jn(k) hn(k) hn(k|x|) Ym n (x̂) Ym n (θ) , |x| ≥ 1 . (1.83) The far field pattern of hn(k|x|) Ym n (x̂) is again given by (4π/k) exp −i(n+1)π/2 Ym n (x̂) (compare proof of Lemma 1.13). As shown rigorously in [43] the far field pattern of us is derived by the term-by-term asymptotics of hn(k|x|) Ym n (x̂), i.e., u∞ (x̂, θ) = − (4π)2 k ∞ n=0 |m|≤n in jn(k) hn(k) e−i(n+1)π/2 Ym n (x̂) Ym n (θ) = (4π)2 i k ∞ n=0 |m|≤n jn(k) hn(k) Ym n (x̂) Ym n (θ) , x̂, θ ∈ S2 . (1.84) From this we observe that the far field operator F : L2(S2) → L2(S2) from (1.36) is given by (Fg)(x̂) = S2 u∞ x̂, θ g(θ) ds(θ) = (4π)2 i k ∞ n=0 |m|≤n jn(k) hn(k) gm n Ym n (x̂) , x̂ ∈ S2 , (1.85) where gm n = S2 g(θ) Ym n (θ) ds(θ) , |m| ≤ n , n ∈ N ,
- 49. 30 Dirichlet and Neumann boundary conditions are the expansion coefficients of g ∈ L2(S2). From (1.85) we observe that λn = (4π)2 i k jn(k) hn(k) , n ∈ N , (1.86) are the eigenvalues of F of multiplicity 2n + 1. The asymptotic behavior jn(k) = kn 1 · 3 · · · (2n + 1) 1 + O 1 n , (1.87) hn(k) = 1 · 3 · · · (2n − 1) i kn+1 1 + O 1 n yields that λn = − (4π)2 k2n (2n − 1)!! (2n + 1)!! 1 + O 1 n , (1.88) where we used the convenient notation p!! = 1 · 3 · 5 · · · p for any odd number p. Next, we compute the expansion coefficients of the functions (1.41) by the Jacobi–Anger expansion (1.82), i.e., φz(x̂) = e−ikz·x̂ = 4π ∞ n=0 |m|≤n (−i)n jn(k|z|) Ym n (x̂) Ym n (ẑ) , x̂ ∈ S2 , where ẑ = z/|z|. From this we conclude that (φz, Ym n )L2(S2) = 4π (−i)n jn(k|z|) Ym n (ẑ) . Using the formula |m|≤n Ym n (ẑ) 2 = 2n + 1 4π which is a special form of the addition theorem (see [43]) and the asymptotic form (1.87) of jn(k|z|) we conclude that |m|≤n (φz, Ym n )L2(S2) 2 = 4π(2n + 1) (k|z|)2n [(2n + 1)!!]2 1 + O(1/n) . Combining this with (1.88) yields |m|≤n (φz, Ym n )L2(S2) 2 |λn| = 2n + 1 4π k2n (2n − 1)!! (2n + 1)!! (k|z|)2n 1 + O 1 n = |z|2n 4π 1 + O 1 n .
- 50. The Neumann boundary condition 31 Here we observe directly that the series ∞ n=0 |m|≤n (φz, Ym n )L2(S2) 2 |λn| converges if, and only if, |z| 1, i.e., z is inside D. We finally remark that if |z| 1 then the series behaves as ∞ n=0 |z|2n 4π = 1 4π (1−|z|2) , i.e., W(z) behaves as 4π (1 − |z|2) as z approaches the boundary of D. 1.6 The Neumann boundary condition In this section we discuss briefly the obstacle scattering with respect to Neumann boundary conditions. In the direct scattering problem the incident plane wave ui(x) = exp(ikx · θ) and the obstacle D ⊂ R3 are again given and the total wave u ∈ C2(R3 D) ∩ C1(R3 D) has to be determined with u + k2 u = 0 outside D (1.89) and the Neumann boundary condition ∂u ∂ν = 0 on . (1.90) Furthermore, the scattered field us = u−ui satisfies the Sommerfeld radiation condition ∂us ∂r − ik us = O r−2 for r = |x| → ∞ (1.91) uniformly with respect to x̂ = x/|x|. Again, the scattered field us satisfies the following exterior boundary value problem for f = −∂ui/∂ν: Given f ∈ H−1/2() find v ∈ H1 loc(R3 D) such that v + k2 v = 0 outside D , (1.92) ∂v ∂ν = f on , (1.93) and ∂v ∂r − ik v = O r−2 for r = |x| → ∞ (1.94) uniformly with respect to x̂ = x/|x|. The data-to-pattern operator G : H−1/2() → L2(S2) is now defined to map f ∈ H−1/2() into the far field pattern v∞ = Gf of the exterior Neumann boundary value problem (1.92), (1.93), and (1.94).
- 51. 32 Dirichlet and Neumann boundary conditions The solution is again understood in the variational sense, i.e., v ∈ H1 loc(R3 D) is a variational solution of (1.92) and (1.93) if it satisfies R3D ∇u · ∇φ − k2 u φ dx = f , φ for all φ ∈ H1(R3 D) with compact support. Here, ·, · denotes again the dual form in H−1/2(), H1/2(). Existence and uniqueness of this exterior Neumann boundary value problem is well established, see [183], Lecture 4, [150], Section 2.6, or Chapter 1, Theorem 2.2. The far field patterns u∞ = u∞(x̂, θ) of the scattered fields us define again the far field operator F : L2(S2) → L2(S2) by (compare (1.36)) (Fg) x̂ = S2 u∞ x̂, θ g θ ds(θ) for x̂ ∈ S2 . (1.95) By literally the same proofs as inTheorems 1.6 and 1.8 one shows reciprocity (1.27) of the far field patterns, normality of F and unitarity of the scattering operator S = I + ik 8π2 F. Furthermore, F is one-to-one if k2 is not a Neumann eigenvalue of − in D.8 Also, the uniqueness result of Theorem 1.9 holds. The analogous results of Lemmas 1.13 and 1.14 and Theorem 1.15 are formulated in the following theorem. Theorem 1.26 (a) The far field operator F : L2(S2) → L2(S2) from (1.95) has a factorization in the form F = −G N∗ G∗ (1.96) where G : H−1/2() → L2(S2) maps f ∈ H−1/2() into the far field pattern v∞ = Gf of the exterior Neumann boundary value problem (1.92), (1.93), and (1.94), and N : H1/2() → H−1/2() is the normal derivative of the double layer potential, defined by (Nϕ)(x) = ∂ ∂ν ϕ(y) ∂ ∂ν(y) (x, y) ds(y) , x ∈ , (1.97) for ϕ ∈ H1/2(). (b) G is compact, one-to-one with dense range in L2(S2). (c) N is an isomorphism from H1/2() onto H−1/2() if k2 is not a Neumann eigenvalue of − in D. (d) ImNϕ, ϕ 0 for all ϕ ∈ H1/2() with ϕ = 0 if k2 is not a Neumann eigenvalue of − in D. Again, ·, · denotes the duality pairing in H−1/2(), H1/2(). 8 k2 is called a Neumann eigenvalue of − in D if there exists a non-trivial solutionu ∈ C2(D) ∩ C1(D) of the Helmholtz equation in D such that ∂u/∂ν vanishes on .
- 52. The Neumann boundary condition 33 (e) Let Ni be the boundary operator (1.97) corresponding to the wavenumber k = i. The operator −Ni is self-adjoint and coercive as an operator from H1/2() onto H−1/2(), i.e., −Niϕ, ϕ ≥ c0ϕ2 H1/2() for all ϕ ∈ H1/2 () . (1.98) (f) The difference N − Ni is compact from H1/2() into H−1/2(). Remark: We note that the classical definition of Nϕ by (1.97) makes only sense for sufficiently smooth densities (i.e., for Hölder continuously differentiable functions on , see [43]). It can be shown that N has a bounded extension to an operator from H1/2() into H−1/2() which we also denote by N (see [144]). Proof: (a) Analogously to H from (1.51) we define the operator ∂H : L2(S2) → H−1/2() by (∂H)g(x) := ∂ ∂ν S2 g(θ) eikx·θ ds(θ) = ik S2 g(θ) (ν(x) · θ) eikx·θ ds(θ) , x ∈ . (1.99) Its adjoint (∂H)∗ : H1/2() → L2(S2) is now given by (∂H)∗ ϕ(x̂) = −ik (ν(y) · x̂) ϕ(y) e−ikx̂·y ds(y) = ϕ(y) ∂ ∂ν(y) e−ikx̂·y ds(y) (1.100) for x̂ ∈ S2. We note that (∂H)∗ϕ is now the far field pattern of the double layer potential v(x) = ϕ(y) ∂ ∂ν(y) (x, y) ds(y) , x ∈ R3 D , (1.101) and ∂v/∂ν = Nϕ provided ϕ ∈ C1,α() (see [42]). This yields (∂H)∗ϕ = GNϕ and thus ∂H = N∗ G∗ by a density argument. Furthermore, we note that F = −G(∂H) and therefore F = −G N∗G∗. (b) This follows from similar the arguments as in the proof of Lemma 1.13. (c) For this property we refer again to [144]. (d) Here we proceed exactly as in the proof of Lemma 1.14. Define, for any ϕ ∈ H1/2() the double layer potential v by v(x) = ϕ(y) ∂(x, y) ∂ν(y) ds(y) , x ∈ R3 . (1.102) Then v ∈ H1(D)∩H1 loc(R3 D) is again a solution of the Helmholtz equation in R3 . The traces v± and ∂v±/∂ν exist in the variational sense with ϕ = v+ − v− and ∂v− ∂ν = ∂v+ ∂ν = Nϕ. Therefore, using Green’s formula in D and in DR := x ∈ R3 D : |x| R
- 53. 34 Dirichlet and Neumann boundary conditions we conclude that Nϕ, ϕ = ∂v ∂ν , v+ − v− (1.103) = − D∪DR |∇v(x)|2 − k2 |v(x)|2 dx + |x|=R v(x) ∂v(x) ∂r ds (1.104) = − D∪DR |∇v|2 − k2 |v|2 dx + ik |x|=R |v|2 ds + O 1 R (1.105) as R tends to infinity. Taking the imaginary part yields ImNϕ, ϕ = k lim R→∞ |x|=R |v|2 ds = k (4π)2 S2 |v∞ |2 ds ≥ 0 . (1.106) Let now ImNϕ, ϕ = 0 for some ϕ ∈ H1/2(). Again, from (1.106), Rellich’s Lemma (see Lemma 1.2) and unique continuation we conclude that v vanishes outside of D. Therefore, Nϕ = 0 on by the trace theorem. Since N is an isomorphism ϕ has to vanish. (e) For k = i the same arguments as above yield Niϕ, ϕ = − D∪DR |∇v|2 + |v|2 dx − |x|=R |v|2 ds + O 1 R , R → ∞ , and thus as R → ∞ (note that v decays exponentially): Niϕ, ϕ = − R3 |∇v(x)|2 + |v(x)|2 dx = −v2 H1(R3) . The trace theorem and the boundedness of N−1 i yields the existence of c 0 and c0 0 with −Niϕ, ϕ ≥ c ∂v/∂ν2 H−1/2() = c Niϕ2 H−1/2() ≥ c0 ϕ2 H1/2() . (f) This follows again from the fact that the kernel of N − Ni is smooth. Now we proceed exactly as in the case of the Dirichlet problem. Indeed, the functional analytic Theorem 1.23 is applicable where now X = H−1/2() and X ∗ = H1/2() and A = −N∗. Moreover, Theorem 1.12 holds also for the Neumann boundary condition. Therefore, we have the following analogy of Theorem 1.25. Theorem 1.27 Assume k2 is not a Neumann eigenvalue of − in D. For any z ∈ R3 define again φz ∈ L2(S2) by φz(x̂) := e−ikx̂·z , x̂ ∈ S2 .
- 54. Additional remarks and numerical examples 35 Then z ∈ D ⇐⇒ φz ∈ R (F∗ F)1/4 . (1.107) Let λj ∈ C be the eigenvalues of the normal operator F with corresponding eigenfunctions ψj ∈ L2(S2). Then the following characterization holds. A point z ∈ R3 belongs to D if, and only if, the series ∞ j=1 (φz, ψj)L2(S2) 2 |λj| converges, i.e., if, and only if, W(z) := ∞ j=1 (φz, ψj)L2(S2) 2 |λj| −1 0 . (1.108) Therefore, χD(z) = sign W(z) is the characteristic function of D. Remark: Of course, the function W is different from the one defined in Theorem 1.25. We do not indicate this by a different symbol. The characterizations of Theorems 1.25 and 1.27 imply again uniqueness of the inverse scattering problem if k2 is not an eigenvalue of the underlying boundary value problem in D. It is remarkable that the characterization of the characteristic function depends only on F and makes no use of the boundary condition. We will exploit this fact further down. 1.7 Additional remarks and numerical examples Before we turn to the presentation of some numerical simulations we want to add some remarks. In this chapter we considered the set u∞(·, θ) : θ ∈ S2 of far field patterns as data for the inverse problem or, equivalently, the far field operator F : L2(S2) → L2(S2). In many applications the incident fields are point sources vi = vi(·, y) given by vi(x, y) = (x, y) for y from a surface which contains D in its interior. The corresponding scattered fields vs = vs(·, y) are measured on the same surface .9 Therefore, the set vs(·, y) : y ∈ are the data for the inverse problem or, equivalently, the near field operator F : L2() → L2() given by (Fϕ)(x) = ϕ(y) vs (x, y) ds(y) , x ∈ . 9 Or even a different one.
- 55. Exploring the Variety of Random Documents with Different Content
- 59. The Project Gutenberg eBook of An Annapolis First Classman
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: An Annapolis First Classman Author: Edward L. Beach Illustrator: Frank T. Merrill Release date: July 29, 2012 [eBook #40368] Most recently updated: October 23, 2024 Language: English Credits: Produced by David Edwards, Katie Hernandez and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive) *** START OF THE PROJECT GUTENBERG EBOOK AN ANNAPOLIS FIRST CLASSMAN ***
- 61. cover HE GLANCED AT THE WRITTEN ORDER
- 63. LT. COM. EDWARD L. BEACH U.S. NAVY Author of AN ANNAPOLIS PLEBE AN ANNAPOLIS YOUNGSTER AN ANNAPOLIS SECOND CLASSMAN Illustrated by Frank T. Merrill THE PENN PUBLISHING COMPANY PHILADELPHIA MCMX
- 64. Introduction This is the fourth and last book of the Annapolis Series. It has been the purpose of the author faithfully to portray the conditions in which our midshipmen live at the Naval Academy. The training given at Annapolis is regulated by the needs of the Fleet, and the Naval Academy in all of its departments is entirely directed and controlled by seagoing naval officers. After the Fleet's world-encircling cruise, many of the officers attached to it were sent to the Naval Academy to instruct midshipmen in navigation and electricity and gunnery and seamanship. In the navy it is believed that the officer who is fresh from drilling a twelve-inch turret or a battery of broadside guns at record and battle target practice, should be well qualified to initiate midshipmen in the beginnings of naval gunnery. It is for this reason that the training at Annapolis reflects the needs of the Fleet, and every officer on duty there has either seen recent sea service or is looking forward to an early sea assignment. Stonewell and Robert Drake by name never existed, but the same thoughts and ambitions that animate them have animated many hundreds of midshipmen; and incidents similar to those described have happened countless times. From this point of view these stories are true stories. The names of their chief characters may be found in no navy list, but the truth of the Annapolis books does not depend upon that. Stonewell and Robert Drake have actually lived many times, and to-day are living at Annapolis. The author hopes he has presented in this book and its three predecessors, An Annapolis Plebe, An Annapolis Youngster, and An Annapolis Second Classman, a fair picture of the life of American midshipmen; and not only of the naval atmosphere which surrounds them, but of that inner life which for the time dominates
- 65. their relations to each other and to the institution made famous as the alma mater of many names illustrious in naval history. Edward L. Beach, Lieutenant-Commander, U.S. Navy.
- 66. Contents I.Glassfell, Drake and Stonewell 9 II.The Commandant of Midshipmen 22 III.A Happy Surprise 36 IV.Academy Life Begins 46 V.A Mysterious Cry 61 VI.The Gates Forward Pass 77 VII.The West Point Game 88 VIII.The Man Wore a Slouch Hat 101 IX.Robert Gets Bad News 111 X.Robert Gets Good News 124 XI.Three Groans for the Superintendent133 XII.Robert Makes a Discovery 142 XIII.Harry Blunt is Rebuffed 155 XIV.A Mystery Solved 166 XV.Stonewell Receives a Letter 181 XVI.Bligh Makes a Friend 194 XVII.An Ill-Favored, Red-Bearded Rogue 205 XVIII.An Old Colored Man is in Trouble 217 XIX.The Kidnappers 227 XX.Six-Pounder Target Practice 237 XXI.A Good Shot with the Six-Pounder 255 XXII.Grice Appears Again 265 XXIII.Robert Resigns 275 XXIV.It Was Stonewell 287 XXV.John 15:13 298 XXVI.Commander Dalton Becomes Angry 305 XXVII.Robert Finally Answers 320 XXVIII.Bligh, Bligh, Bligh! 334
- 67. XXIX.The End of a Long Day 343 XXX.Graduation 350
- 68. Illustrations PAGE He Glanced at the Written Order Frontispiece Around the End 68 The Stranger Threw off His Hat 152 He Half Arose From His Seat 200 He Saw Two Dark Figures 273 That Will Do, Gentlemen 296 It Must Have Been a Very Pretty Speech 354
- 69. An Annapolis First Classman
- 70. CHAPTER I GLASSFELL, DRAKE AND STONEWELL Hello, Stone! Hello, Bob! By George, but I'm glad to see you! Hello, Glass, you old sinner, I can just imagine you've led those dear old aunts of yours a lively life the last two weeks. You'll win, Stone, but you ought to get them to tell you about it; ha, ha, ha! the dear old ladies never dropped once. Explosively enthusiastic greetings were exchanged between three stalwart young men in the Union Station, Chicago, on the twentieth of September, of the year nineteen hundred and something. Passers- by noticed them and smiled, and in approving accents said, College boys! All three were tall, broad-shouldered, bronzed in face, and possessed a lithesomeness of movement that betokened health and strength. Glassfell, Drake and Stonewell were midshipmen on leave from the United States Naval Academy. It was evident that they had met in the Union Station by appointment. Glassfell had just arrived from Wisconsin, and Drake and Stonewell were to leave in two hours for Annapolis. You two chaps are martyrs! exclaimed Glassfell; here you are giving up ten days of glorious leave just to go and train for the football team. Now here I am, cheer leader, head yeller, or whatever you call me, far more important than either of you, you'll admit, and I'm not due at Annapolis until October first. 'Daily News,' last edition, droned a newsboy near by. Don't bother me, boy; Chicago news doesn't interest me. Some new sandbagging on Wabash Avenue, I suppose, and nothing else. Get out.
- 71. A fine cruise, wasn't it, Glass? remarked Robert Drake. By George! I'd had some troubles on my previous cruises, but this went like clockwork; not a single thing happened to worry me, and I certainly had troubles enough on my plebe and youngster cruises. You did indeed, Bob, remarked Stonewell, but you'll have to admit you were fortunate in the wind up. Now Glass, here—— 'Daily News,' last edition, was shouted close to their ears. Stuff that boy. Put a corn-cob down his throat, said Glassfell with an amused glance at the persistent newsboy. Say, fellows, wasn't that a good one I worked on old 'I mean to say'? Ha, ha, ha! Which one, Glass? asked Robert Drake. Oh, the best one, the time I hoisted up two red balls to the masthead when he was on watch in charge of the deck, during drill period. And didn't the captain give him the mischief? An outburst of wild hilarious laughter greeted this reminiscence, as evidently a very humorous episode was recalled. In seagoing language two red balls means that the ship carrying them is not under control; and at the time referred to by Glassfell the red balls had no business to be hoisted, and their presence brought down upon Lieutenant-Commander Gettem, nicknamed I mean to say by the midshipmen, a wrathful reprimand from his captain. That was pretty good, Glass, laughed Stonewell, but you had to own up, and got sanded well for it. 'Daily News,' last edition! screamed a voice interrupting the midshipmen. Look here, boy, how many papers have you to sell? inquired Glassfell. Twenty-five, boss; here's yours, and only one cent. All right. I'll buy your twenty-five papers and give you twenty-five cents besides if you'll make a hundred yard dash for the outside. Give me your papers; here's fifty cents.
- 72. I'm your man, boss, cried the newsboy, handing over his papers, grabbing the fifty-cent piece and making a tremendous bolt toward the exit. He's afraid of a recall, laughed Robert. Say, Glass, are you going to start a wholesale newspaper business? Let's see what the news of the day is, replied Glassfell, unfolding one of the papers and laying the others down on a seat. Here's an alderman up for graft; a bank cashier has gone wrong; hello! My heavens, here's a naval war-ship goes to the bottom with all on board. What ship? what ship? simultaneously cried out Stonewell and Robert, in affrighted tones. The submarine boat 'Holland'! Ha, ha, ha, I got you both that time, didn't I? You chaps will nab any bait that comes along. All three laughed heartily. You're an incorrigible wretch, smiled Robert; I shudder at the idea of spending another year with you at the Academy. But the friendly hug that accompanied these words left no doubt of the affection Robert bore to the jovial Glassfell. By George, fellows, here is an interesting item, 'New cadet officers at the Naval——' You don't sell me again to-day, Glass, grinned Robert. You'll be giving yourself five stripes and Stone a second class buzzard. Pick up a paper and read for yourself, cried out Glassfell excitedly. Farnum gets five stripes! Glassfell read no further, but with an expression of intense disgust threw the paper down and stamped on it. Stonewell and Robert were now eagerly reading the paper. Cadet Commander, commanding the Brigade of Midshipmen, Farnum, read Robert. Cadet Lieutenant-Commanders, commanding first and second battalions, respectively, Stonewell and Sewall; Cadet
- 73. Lieutenant and Brigade Adjutant, Ryerson. Cadet Lieutenant, commanding first company, Blair—— A look of blank astonishment mingled with disdain was to be seen on Robert's face. Well, Stone, he said, the officers have done it again, and I guess they can be relied upon to make chumps of themselves as regularly as they assign the brigade officers. You should be our cadet commander, Stone, our five striper; you know it, every midshipman in the brigade knows it, the officers ought to know it! You are number one man in the class, the leader in Academy athletics, head and shoulders above us all. And here they've picked out a regular 'snide,' a sneak, and have given him the place that belongs to you. Robert spoke passionately; he was intensely disappointed. You are entirely wrong about Farnum, Bob, remarked Stonewell quietly; he's a far better man than you give him credit for. You don't understand Farnum; he'll do credit to his five stripes. I'm entirely satisfied with my four stripes; to be cadet lieutenant-commander is as much as I have any right to expect. You know why you don't get five stripes, don't you? asked Robert vehemently; it's because you took French leave a year ago, and reported yourself for it! And didn't Farnum jump ship at the same time? Only he didn't get spotted for it. You reported yourself for the purpose of explaining my deliberate neglect of duty last year. You were reduced to ranks as a result and Farnum was then given your position as acting senior cadet officer of the summer detail. If he'd had any sense of fitness he would have reported himself rather than have accepted it; that was only a temporary affair, however, and didn't amount to much; but because of that same report it's outrageous that you should be shoved out of the five stripes you've earned by a man who was equally guilty, but didn't have the manhood to report himself when you did. It's rotten, remarked Glassfell. Well, Stone, old chap, he continued, I'm sorry; everybody will be; we all thought you had a cinch on five stripes. But I wouldn't be in Farnum's shoes; everybody
- 74. will know he is a fake. But as long as they didn't make Stonewell cadet commander I'm rather surprised they didn't give the job to me. Look here, Bob, said Stonewell, I have been hoping you would get three stripes—but I'm sorry not to see you down for anything. That's too bad; isn't Bob down for anything? inquired Glassfell. Not even for a second class buzzard, the lowest thing in cadet rank at the Naval Academy, replied Stonewell. I'm sorry to hear that, remarked Glassfell, much concerned. Bob ought to have three stripes, anyway. Don't you worry, fellows, said Robert, cheerily, I haven't expected a thing and am not a bit disappointed. A midshipman cannot live down a 'deliberate neglect of duty' report in one year. Yes, Bob, I know, but I had hoped that your conduct at the fire a year ago and that remarkable trip of yours last June would—— Now, Stone, please don't; you know that is not to be talked about. Of course, but at the same time in spite of that report you ought to get three stripes. That's right, commented Glassfell. The officers only see one side of a midshipman's character; here I am, another martyr to their ignorance; I'm one of the best men in the class, the band master thinks so, and he's the grandest thing I've ever seen at Annapolis; and I'm a private in ranks for another year. But perhaps this report isn't authentic; let's see, the paper says that it is likely that these recommendations will be made to the superintendent by the commandant; the former is away, will not arrive at Annapolis for two days yet—hurrah, I may still get five stripes. Stone, I still hope you may command the brigade of midshipmen our last year, said Robert thoughtfully. This newspaper account does not pretend to be official; it says 'it has leaked out' that the commandant of midshipmen's recommendation of the assignment of
- 75. cadet officers of the brigade will be so and so. Now the superintendent evidently has not seen these recommendations, so they are not as yet finally decided upon. Probably this newspaper list is correct in the main, but it is not final; the superintendent is away on leave and has not yet acted; he has not even seen the commandant's recommendations. If either the superintendent or the commandant were to know that Farnum had been guilty of the same offense which is now to deprive you of the five stripes you otherwise, by every count, had earned, you would never be set aside in favor of a man equally guilty but not so square. It's shameful, that's what it is. Robert boiled over with angry thoughts. Strong feelings dominated his expressive features, and it was with difficulty that he controlled himself. His classmate Stonewell was at once his joy and pride, and he loved him with brotherly affection. Stonewell in his studies towered above all of his classmates; he was the leader in athletics, captain of the football team, and captain of the Academy crew. He was class president and his own class and all midshipmen confidently expected he would be cadet commander in his last year at the Naval Academy. But Robert Drake more than wished for it. Until this moment he had not realized how he longed for it. In the preceding three years at Annapolis Robert had had perhaps more than his own share of troubles, and in them all Stonewell had been to him a mountain of strength and a deep well of affectionate wisdom. Farnum for our five striper! Faugh! The thought of it makes me sick! I'll not stand for it, cried Robert. How can you help it, Bob? queried Glassfell, himself much disappointed, though not nearly so vehement as Robert. I'll tell you what I'm going to do, almost shouted the latter; Stone and I will be in Annapolis the day after to-morrow, and I'm going straight to the commandant and convince him that he's made a big bust. That's what I'm going to do!
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