# Remarks on the factorization and monotonicity method for inverse acoustic
scatterings

Takashi FURUYA

###### Abstract

We study the factorization and monotonicity method for inverse acoustic
scattering problems. Firstly, we give a new general functional analysis
theorem for the monotonicity method. Comparing with the factorization method,
the general theorem of the monotonicity generates reconstruction schemes under
weaker a priori assumptions for unknown targets, and can directly deal with
mixed problems so that the unknown targets have several different boundary
conditions. Using the general theorem, we give the reconstruction scheme for
the mixed crack that the Dirichlet boundary condition is imposed on one side
of the crack and the Neumann boundary condition on the other side, which is a
new extension of monotonicity method.

##  1 Introduction

In this paper, we study the factorization and monotonicity method for inverse
acoustic scattering problems. The factorization method has first been
introduced by Kirsch ([17]) for the inverse acoustic obstacle scattering. It
has been studied so far by many authors (see e.g., [1, 3, 4, 7, 12, 18, 19,
20, 21, 23, 25]), and it is well-known as one of classical qualitative
methods, which includes the linear sampling method of Colton and Kirsch ([5]),
the singular sources method of Potthast ([24]), the probe method of Ikehata
([16]), etc. The monotonicity method, on the other hand, has been recently
introduced by Harrach in [15] for the electrical impedance tomography. Very
recently, it was extended to the Helmholtz equation in the bounded domain [13,
14], the inverse acoustic obstacle [2], Dirichlet crack [9], and medium
scatterings [11, 22]. It was found from these works that the monotonicity
method has advantage over the factorization method that we can give the
reconstruction scheme under weaker a priori assumptions for unknown targets.

The contributions of this paper are the following.

  * [A]

We give a general functional analysis theorem for the monotonicity method
(Theorem 4.2) including previous works [2, 9, 11] for inverse acoustic
scatterings.

  * [B]

Using the general theorem, we give the reconstruction scheme for the inverse
mixed crack scattering, which is a new extension of the monotonicity method
(Theorems 5.13 and 5.14).

A characteristic of the factorization method is to prepare the general
functional analysis theorem which generates reconstruction schemes by
spectrums of the far-field operator (see Theorem 2.15 of [19]). Based on this
idea, the contribution [A] will be discussed as the version of the
monotonicity method.

The general theorem of the factorization method assumes that the real part of
the middle operator of the far-field operator has a decomposition into a
positive coercive operator and a compact operator, while the imaginary part of
the middle operator becomes strictly positive (see the assumptions (b) and (c)
in Theorem 3.1). This two assumptions cause a priori assumptions for the
unknown target, in particular, the positivity of the imaginary part
corresponds to restrictions for the wave number. For example, it is necessary
for the inverse medium scattering that the wave number is not a transmission
eigenvalue with respect to the unknown medium (see e.g., Theorem 4.10 of
[19]), and for the inverse obstacle scattering that the wave number is not an
eigenvalue with respect to the boundary condition of the unknown obstacle (see
e.g., Corollary 2.16 of [19]). However, the general theorem of the
monotonicity method does not assume the positivity of the imaginary part (see
Theorem 4.2), which means that the monotonicity can essentially avoid
restrictions for the wave number. In fact, monotonicity reconstructions for
inverse obstacle (Theorem 5.3 of [2]) and medium (Theorems 5.1–5.3 of [11])
have been successful without restrictions for the wave number.

The advantage of the monotonicity over the factorization is not only weaker a
priori assumptions, but also to directly deal with mixed problems that the
unknown target consists of two separate components with different boundary
conditions. The real part of middle operator of the far-field operator for
mixed problems is not decomposed into a positive coercive operator and a
compact operator (see e.g., Theorem 3.4 of [19], Theorem 3.2 of [20]). In
order to make a decomposition of the coercivity and the compactness, the
factorization method for mixed problems needs masking approaches that one
component is covered by an artificial domain disjoint with the other component
we want to reconstruct (see e.g., Lemma 3.5 of [19], (3.26) of [20]). However,
the general theorem of the monotonicity method (see (3) of Theorem 4.2) does
not assume such a real part decomposition, which means that the monotonicity
essentially do not need masking approaches. In fact, monotonicity
reconstructions for inverse mixed obstacle (Theorem 5.5 of [2]) have been
successful without the masking approach.

This paper studies not only previous works from the viewpoint of the general
theorem, but also a new extension of the monotonicity to the inverse acoustic
mixed crack scattering, which corresponds to the contribution [B]. The mixed
crack consists of only one component, but imposes the Dirichlet boundary
condition on one side of the unknown crack and the Neumann boundary condition
on the other side (see the beginning of Section 5.5). The factorization method
for the mixed crack has been studied in [25], but an extensive closed curve of
the unknown crack should be known, which is a very restrictive assumption (see
Theorem 3.3 of [25]). Using the general theorem for the monotonicity method,
we give a reconstruction scheme without assuming such an extensive curve (see
Theorems 5.13 and 5.14).

This paper is organized as follows. In Section 2, we define the inverse
acoustic scattering problem. In Section 3, we recall the functional analysis
theorem for the factorization method. In Section 4, we give a new general
functional analysis theorem (Theorem 4.2) for the monotonicity method. In
Section 5, we study several applications of the general theorem. The results
discussed in Sections 5.1–5.4 are the same in previous works [2, 11, 9], while
the one in Section 5.5 is new in the monotonicity for the inverse mixed crack
scattering. Finally in Section 6, we give numerical examples for our
theoretical results.

##  2 Inverse acoustic scattering

First of all, we define the inverse acoustic scattering problem. Let $k>0$ be
the wave number, and let $\theta\in\mathbb{S}^{d-1}$ ($d=2,3$) be the incident
direction. We denote the incident field $u^{inc}(x,\theta)$ with incident
direction $\theta$ by the plane wave of the form

$u^{inc}(x,\theta):=\mathrm{e}^{ikx\cdot\theta},\ x\in\mathbb{R}^{d}.$ (2.1)

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded open set with the smooth
boundary $\partial\Omega$ such that the exterior
$\mathbb{R}^{d}\setminus\overline{\Omega}$ is connected. Here, we denote by
$\overline{\Omega}=\Omega\cup\partial\Omega$. In particular, we discuss the
following two cases. The first case is that the scatterer $\Omega$ is a
penetrable medium, and determine the total field $u=u^{sca}+u^{inc}$ such that

$\Delta u+k^{2}(1+q)u=0\ \mathrm{in}\ \mathbb{R}^{d},$ (2.2)
$\lim_{r:=|x|\to\infty}r^{\frac{d-1}{2}}\biggl{(}\frac{\partial
u^{sca}}{\partial r}-iku^{sca}\biggr{)}=0,$ (2.3)

where $q\in L^{\infty}(\mathbb{R}^{d})$ has a compact support such that
$\Omega=\mathrm{supp}\ q$, and $\Delta$ is the Laplace operator. The
Sommerfeld radiation condition (2.3) holds uniformly in all directions
$\hat{x}:=\frac{x}{|x|}$. The second case is that $\Omega$ is an impenetrable
obstacle, and determine the total field $u=u^{sca}+u^{inc}$ such that

$\Delta u+k^{2}u=0\ \mathrm{in}\ \mathbb{R}^{d}\setminus\overline{\Omega},$
(2.4) $\mathcal{B}u=0\ \mathrm{on}\ \partial\Omega,$ (2.5)

and $u^{sca}$ satisfies the Sommerfeld radiation condition (2.3) where (2.5)
means the boundary conditions, for example, the Dirichlet boundary condition
$\mathcal{B}u=u$, the Neumann boundary condition $\mathcal{B}u=\frac{\partial
u}{\partial\nu}$, etc. In both cases, it is well-known that there exists a
unique solution $u^{sca}$ that has the following asymptotic behaviour (see
e.g., [6])

$u^{sca}(x)=\frac{\mathrm{e}^{ikr}}{r^{\frac{d-1}{2}}}\Bigl{\\{}u^{\infty}(\hat{x},\theta)+O\bigl{(}1/r\bigr{)}\Bigr{\\}},\
r\to\infty.$ (2.6)

The function $u^{\infty}$ is called the far-field pattern of the scattered
field $u^{sca}$. With the far-field pattern $u^{\infty}$, we define the far-
field operator $F:L^{2}(\mathbb{S}^{d-1})\to L^{2}(\mathbb{S}^{d-1})$ by

$Fg(\hat{x}):=\int_{\mathbb{S}^{d-1}}u^{\infty}(\hat{x},\theta)g(\theta)ds(\theta),\
\hat{x}\in\mathbb{S}^{d-1}.$ (2.7)

In the inverse acoustic scattering problem, we reconstruct $\Omega$ from the
far-field pattern $u^{\infty}(\hat{x},\theta)$ for all
$\hat{x},\theta\in\mathbb{S}^{d-1}$, and fixed $k>0$. In other words, given
the far-field operator $F$, we reconstruct $\Omega$.

##  3 The factorization method

Here, we recall the general functional analysis theorem for the factorization
method. The following functional analytic theorem is proved by Theorem 2.15 of
[19] and Theorem 3.1 of [8].

######  Theorem 3.1 (Theorem 2.15 of [19] and Theorem 3.1 of [8]).

Let $X\subset U\subset X^{*}$ be a Gelfand triple with a Hilbert space $U$ and
a reflexive Banach space $X$ such that the embedding is dense. Furthermore,
let Y be a Hilbert space and let $F:Y\to Y$, $G:X\to Y$, $T:X^{*}\to X$ be
linear bounded operators such that

$F=GTG^{*}.$ (3.1)

We make the following assumptions:

(a)

$G$ is compact with dense range in $Y$.

(b)

$\mathrm{Re}T$ has the form $\mathrm{Re}T=C+K$ where $K:X^{*}\to X$ is some
self-adjoint compact operator and $C:X^{*}\to X$ is some positive coercive
operator, i.e., there exists a constant $c>0$ such that

$\langle\varphi,C\varphi\rangle\geq c\left\|\varphi\right\|_{X^{*}}^{2}\ for\
all\ \varphi\in X^{*},$ (3.2)

where $\langle\cdot,\cdot\rangle$ denotes the duality pairing between $X^{*}$
and $X$.

(c)

$\mathrm{Im}\langle\varphi,T\varphi\rangle>0$ for all
$\varphi\in\overline{\mathrm{Ran}(G^{*})}$ with $\varphi\neq 0$.

Then, the operator $F_{\\#}:=\bigl{|}\mathrm{Re}F\bigr{|}+\mathrm{Im}F$ is
non-negative, and the ranges of $G:X\to Y$ and $F_{\\#}^{1/2}:Y\to Y$ coincide
with each other, that is, we have the following range identity;

$\mathrm{Ran}(G)=\mathrm{Ran}(F_{\\#}^{1/2}).$ (3.3)

Here, the real part and the imaginary part of an operator $A$ are self-adjoint
operators given by

$\mathrm{Re}A=\displaystyle\frac{A+A^{*}}{2}\ \ \ \mathrm{and}\ \ \
\mathrm{Im}A=\displaystyle\frac{A-A^{*}}{2i}.$ (3.4)

######  Remark 3.2.

It has been well-known that in Theorem 2.1 of [23] the assumption (c) can be
replaced by the injectivity of $T$, and it has been mainly used especially for
the relaxation of the assumption that the wave number $k>0$ is not a
transmission eigenvalue in inverse medium scatterings (see e.g., [7, 19, 20,
23]). However, it was found that this replacement is not correct (see Remark
3.2 of [8]), thus the factorization method for inverse medium scatterings
essentially needs the assumption of transmission eigenvalues.

##  4 General theorems for the monotonicity method

In this section, we give a new general functional analysis theorem for the
monotonicity method.

######  Definition 4.1.

Let $A,B:H\to H$ be self-adjoint compact linear operators on a Hilbert space
$H$. We write

$A\leq_{\mathrm{fin}}B,$ (4.1)

if $B-A$ has only finitely many negative eigenvalues.

######  Theorem 4.2.

Let $X\subset U\subset X^{*}$, $\tilde{X}\subset\tilde{U}\subset\tilde{X}^{*}$
be Gelfand triples with Hilbert spaces $U$, $\tilde{U}$ and reflexive Banach
spaces $X$, $\tilde{X}$ such that the embeddings are dense. Furthermore, let Y
be a Hilbert space and let $F:Y\to Y$, $\tilde{F}:Y\to Y$, $G:X\to Y$,
$\tilde{G}:\tilde{X}\to Y$, $T:X^{*}\to X$,
$\tilde{T}:\tilde{X}^{*}\to\tilde{X}$ be linear bounded operators such that

$F=GTG^{*},\ \ \ \tilde{F}=\tilde{G}\tilde{T}\tilde{G}^{*}.$ (4.2)

(1)

Assume that

(1a)

$\mathrm{Re}T$ has the form $\mathrm{Re}T=C+K$ where $C:X^{*}\to X$ is some
positive coercive operator and $K:X^{*}\to X$ is some self-adjoint compact
operator.

(1b)

There exists a compact operator $R:\tilde{X}\to X$ such that $\tilde{G}=GR$.

Then,

$\mathrm{Re}\tilde{F}\leq_{\mathrm{fin}}\mathrm{Re}F.$ (4.3)

(2)

Assume that

(2a)

$\mathrm{Re}\tilde{T}$ has the form $\mathrm{Re}\tilde{T}=\tilde{C}+\tilde{K}$
where $\tilde{C}:\tilde{X}^{*}\to\tilde{X}$ is some positive coercive operator
and $\tilde{K}:\tilde{X}^{*}\to\tilde{X}$ is some self-adjoint compact
operator.

(2b)

There exists an infinite dimensional subspace $W$ in $\mathrm{Ran}(\tilde{G})$
such that $W\cap\mathrm{Ran}(G)=\\{0\\}$.

Then,

$\mathrm{Re}\tilde{F}\not\leq_{\mathrm{fin}}\mathrm{Re}F.$ (4.4)

(3)

Let $X_{j}\subset U_{j}\subset X^{*}_{j}$ ($j=1,2$) be a Gelfand triple with a
Hilbert space $U_{j}$ and a reflexive Banach space $X_{j}$ such that the
embedding is dense. Let $F^{Mix}:Y\to Y$, $G^{Mix}:X_{1}\times X_{2}^{*}\to
Y$, $T^{Mix}:X_{1}^{*}\times X_{2}\to X_{1}\times X_{2}^{*}$ be linear bounded
operators such that

$F^{Mix}=G^{Mix}T^{Mix}G^{Mix\ *}.$ (4.5)

Assume that

(3a)

$\mathrm{Re}T^{Mix}$ has the form
$\mathrm{Re}T^{Mix}=\left(\begin{array}[]{cc}C_{11}&C_{12}\\\
C_{21}&C_{22}\end{array}\right)+K^{Mix}$ where $C_{11}:X_{1}^{*}\to X_{1}$ is
some positive coercive operator, $C_{12}:X_{2}\to X_{1}$ and
$C_{21}:X_{1}^{*}\to X_{2}^{*}$ are some linear bounded operators,
$C_{22}:X_{2}\to X_{2}^{*}$ is some negative coercive operator (that is,
$-C_{22}$ is positive coercive), and $K^{Mix}:X_{1}^{*}\times X_{2}\to
X_{1}\times X_{2}^{*}$ is some self-adjoint compact operator.

(3b)

There exists an infinite dimensional subspace $W_{1}$ in
$\mathrm{Ran}(G^{Mix}R_{1}^{*})$ such that
$W_{1}\cap\mathrm{Ran}\left(\left[\tilde{G},G^{Mix}R_{2}^{*}\right]\right)=\\{0\\}$
where $R_{1}:X_{1}^{*}\times X_{2}\to X_{1}^{*}$ is defined by
$R_{1}\left(\begin{array}[]{ccc}f\\\ g\\\ \end{array}\right):=f$, and its
adjoint operator $R^{*}_{1}:X_{1}\to X_{1}\times X_{2}^{*}$ is given by
$R_{1}^{*}\varphi:=\left(\begin{array}[]{ccc}\varphi\\\ 0\\\
\end{array}\right)$.

Then,

$\mathrm{Re}F^{Mix}\not\leq_{\mathrm{fin}}\mathrm{Re}\tilde{F}.$ (4.6)

######  Remark 4.3.

If the assumption (3b) is replaced by

(3b)’

There exists a finite dimensional subspace $W_{2}$ in
$\mathrm{Ran}(G^{Mix}R_{2}^{*})$ such that
$W_{2}\cap\mathrm{Ran}\left(\left[\tilde{G},G^{Mix}R_{1}^{*}\right]\right)=\\{0\\}$
where $R_{2}:X_{1}^{*}\times X_{2}\to X_{2}$ is defined by
$R_{2}\left(\begin{array}[]{ccc}f\\\ g\\\ \end{array}\right):=g$, and its
adjoint operator $R^{*}_{2}:X^{*}_{2}\to X_{1}\times X_{2}^{*}$ is given by
$R_{2}^{*}\psi:=\left(\begin{array}[]{ccc}0\\\ \psi\\\ \end{array}\right)$,

then, we can show that
$-\mathrm{Re}F^{Mix}\not\leq_{\mathrm{fin}}\mathrm{Re}\tilde{F}$ by the same
argument.

We recall the following technical lemmas which will be useful to prove Theorem
4.2.

######  Lemma 4.4 (Corollary 3.3 of [14]).

Let $A,B:H\to H$ be self-adjoint compact linear operators on a Hilbert space
$H$ with an inner product $(\cdot,\cdot)_{H}$. Then, the following statements
are equivalent:

(1)

$A\leq_{\mathrm{fin}}B$

(2)

There exists a finite dimensional subspace $V$ in $H$ such that

$\left((B-A)v,v\right)_{H}\geq 0,$ (4.7)

for all $v\in V^{\bot}$.

######  Lemma 4.5 (Lemma 4.6 in [14]).

Let $X$, $Y$, and $Z$ be Hilbert spaces, and let $A:X\to Y$ and $B:X\to Z$ be
bounded linear operators. Then,

$\exists C>0:\ \left\|Ax\right\|_{Y}^{2}\leq C\left\|Bx\right\|_{Z}^{2}\ for\
all\ x\in X\ \ \ \ \Longleftrightarrow\ \ \ \
\mathrm{Ran}(A^{*})\subset\mathrm{Ran}(B^{*}).$ (4.8)

######  Lemma 4.6 (Lemma 4.7 in [14]).

Let $X$, $Y$, $V\subset Z$ be subspaces of a vector space $Z$. If

$X\cap Y=\\{0\\},\ \ \ \ and\ \ \ \ X\subset Y+V,$ (4.9)

then, $\mathrm{dim}(X)\leq\mathrm{dim}(V)$.

###### Proof of Theorem 4.2.

(1) Since the restriction $C\bigl{|}_{U}:U\to U$ is positive, there exists a
positive square root $\hat{W}:U\to U$, i.e., $C\bigl{|}_{U}=\hat{W}^{2}$.
Since we have,

$\|\hat{W}\varphi\|^{2}_{U}=(\varphi,\hat{W}^{2}\varphi)_{U}=(\varphi,C\bigl{|}_{U}\varphi)_{U}=\left<\varphi,C\varphi\right>\leq\|C\|\|\varphi\|_{X^{*}}^{2},\
\ for\ all\ \varphi\in U,$ (4.10)

and the embedding $U\subset X^{*}$ is dense, $\hat{W}$ has a bounded extension
$W:X^{*}\to U$ of $\hat{W}$. By the positive coercivity of $C$, there exists a
constant $c>0$ such that for all $\varphi\in U$,

$c\|\varphi\|^{2}_{X^{*}}\leq\langle\varphi,C\varphi\rangle=(\varphi,C\bigl{|}_{U}\varphi)_{U}=\|\hat{W}\varphi\|_{U}^{2}.$
(4.11)

Hence, by the dense embedding $U\subset X^{*}$, we have
$c\|\varphi\|^{2}_{X^{*}}\leq\|W\varphi\|_{U}^{2}$ for all $\varphi\in X^{*}$,
which implies that the extension $W:X^{*}\to U$ of $\hat{W}$ is bounded
invertible. It is easy to check that $C=W^{*}W$. By this, the factorization
(4.2) of operators $F$ and $\tilde{F}$, assumptions (1a) and (1b), we have

$\displaystyle\mathrm{Re}F-\mathrm{Re}\tilde{F}=G\left[C+\mathrm{Re}K-R(\mathrm{Re}\tilde{T})R^{*}\right]G^{*}$
(4.12) $\displaystyle=$
$\displaystyle\left[GW^{*}\right]\left[W^{*^{-1}}CW^{-1}+W^{*^{-1}}\left\\{\mathrm{Re}K-R(\mathrm{Re}\tilde{T})R^{*}\right\\}W^{-1}\right]\left[GW^{*}\right]^{*}$
$\displaystyle=:$ $\displaystyle\hat{G}[I_{U}+\hat{K}]\hat{G}^{*}.$

Let $\left\\{\mu_{j},\phi_{j}\right\\}$ be an eigensystem of the self-adjoint
compact operator $\hat{K}:U\to U$, and let

$V:=\mathrm{span}\left\\{\phi_{j}:\mu_{j}\leq-\displaystyle\frac{1}{2}\right\\}.$
(4.13)

Then, $V$ is a finite dimensional subspace of $U$, and for all
$\varphi\in\left[\mathrm{Ran}\left(\hat{G}\bigl{|}_{V}\right)\right]^{\bot}$,
which is equivalent to $\hat{G}^{*}\varphi\in
V^{\bot}=\mathrm{span}\left\\{\phi_{j}:\mu_{j}>-\displaystyle\frac{1}{2}\right\\}$,

$((\mathrm{Re}F-\mathrm{Re}\tilde{F})\varphi,\varphi)_{Y}=((I_{U}+\hat{K})\hat{G}^{*}\varphi,\hat{G}^{*}\varphi)_{U}\geq\displaystyle\frac{1}{2}\|\hat{G}^{*}\varphi\|_{U}^{2}\geq
0.$ (4.14)

From
$\mathrm{dim}\left[\mathrm{Ran}\left(\hat{G}\bigl{|}_{V}\right)\right]<\infty$
and Lemma 4.4, we conclude (4.3).

(2) Assume on the contrary that
$\mathrm{Re}\tilde{F}\leq_{\mathrm{fin}}\mathrm{Re}F.$ Then by Lemma 4.4,
there exists a finite dimensional subspace $V_{1}$ in $Y$ such that

$(\mathrm{Re}\tilde{F}\varphi,\varphi)_{Y}\leq\left(\mathrm{Re}F\varphi,\varphi\right)_{Y},$
(4.15)

for all $\varphi\in V^{\bot}_{1}$. By the same argument in the beginning of
(1), there exists a bounded invertible operator
$\tilde{W}:\tilde{X}^{*}\to\tilde{U}$ and a self-adjoint compact operator $K$
such that

$\mathrm{Re}\tilde{F}=[\tilde{G}\tilde{W}^{*}][I_{\tilde{U}}+K][\tilde{G}\tilde{W}^{*}]^{*},$
(4.16)

which implies that by the same argument in (4.13)–(4.14) there exists a finite
dimensional subspace $V_{2}$ in $Y$ and a constant $c>0$ such that

$(\mathrm{Re}\tilde{F}\varphi,\varphi)_{Y}\geq\displaystyle\frac{1}{2}\|\tilde{W}\tilde{G}^{*}\varphi\|_{\tilde{U}}^{2}\geq
c\|\tilde{G}^{*}\varphi\|_{\tilde{X}^{*}}^{2},$ (4.17)

for all $\varphi\in V^{\bot}_{2}$. Setting $V:=V_{1}\cup V_{2}$, $V$ is a
finite dimensional subspace in $Y$. Then by (4.15) and (4.17) we have

$c\|\tilde{G}^{*}\varphi\|_{\tilde{X}^{*}}^{2}\leq(\mathrm{Re}F\varphi,\varphi)_{Y}\leq\|\mathrm{Re}T\|\|G^{*}\varphi\|_{X^{*}}^{2},$
(4.18)

for all $\varphi\in V^{\bot}$.

On the other hand, by the assumption (2b) and Lemma 4.6, we have
$W\not\subset\mathrm{Ran}(G)+V$, which implies that by
$W\subset\mathrm{Ran}(\tilde{G})$

$\mathrm{Ran}(\tilde{G})\not\subset\mathrm{Ran}(G)+V=\mathrm{Ran}\left([G,P_{V}]\right),$
(4.19)

where $P_{V}$ denotes the orthogonal projection on $V$. By this and Lemma 4.5,
there exists a sequence $\displaystyle(\varphi_{n})_{n\in\mathbb{N}}\subset Y$
such that

$\|\tilde{G}^{*}\varphi_{n}\|_{\tilde{X}^{*}}^{2}>n^{2}\left(\|G^{*}\varphi_{n}\|_{X^{*}}^{2}+\|P_{V}\varphi_{n}\|_{Y}^{2}\right),$
(4.20)

for all $n\in\mathbb{N}$. Setting
$\psi_{n}:=\displaystyle\frac{n^{\frac{1}{2}}\varphi_{n}}{\|\tilde{G}^{*}\varphi_{n}\|_{\tilde{X}^{*}}}$,
we obtain

$\|G^{*}\psi_{n}\|_{X^{*}}^{2}+\|P_{V}\psi_{n}\|_{Y}^{2}=n\displaystyle\frac{\|G^{*}\varphi_{n}\|_{X^{*}}^{2}+\|P_{V}\varphi_{n}\|_{Y}^{2}}{\|\tilde{G}^{*}\varphi_{n}\|^{2}_{\tilde{X}^{*}}}\leq\frac{1}{n},\
\ \ \ \|\tilde{G}^{*}\psi_{n}\|_{X^{*}}^{2}=n.$ (4.21)

Setting $\tilde{\psi}_{n}:=(I-P_{V})\psi_{n}\in V^{\bot}$, we finally obtain

$\|\tilde{G}^{*}\tilde{\psi}_{n}\|_{\tilde{X}^{*}}\geq\|\tilde{G}^{*}\psi_{n}\|_{\tilde{X}^{*}}-\|\tilde{G}^{*}\|\|P_{V}\psi_{n}\|_{Y}\to\infty,\
\ \ \mathrm{as}\ n\to\infty,$ (4.22)
$\|G^{*}\tilde{\psi}_{n}\|_{X^{*}}\leq\|G^{*}\psi_{n}\|_{X^{*}}+\|G^{*}\|\|P_{V}\psi_{n}\|_{Y}^{2}\to
0,\ \ \ \mathrm{as}\ n\to\infty,$ (4.23)

which contradicts (4.18). Therefore, we conclude (4.4).

(3) Assume on the contrary that
$\mathrm{Re}F^{Mix}\leq_{\mathrm{fin}}\mathrm{Re}\tilde{F}$. Then by Lemma
4.4, there exists a finite dimensional subspace $V_{1}$ in $Y$ such that

$(\mathrm{Re}F^{Mix}\varphi,\varphi)_{Y}\leq(\mathrm{Re}\tilde{F}\varphi,\varphi)_{Y},$
(4.24)

for all $\varphi\in V^{\bot}_{1}$. Since $C_{11}:X_{1}^{*}\to X_{1}$ and
$-C_{22}^{-1}:X_{2}^{*}\to X_{2}$ are positive coercive, by the same argument
in the beginning of (1) there exists a bounded invertible operator
$W_{jj}:X^{*}_{j}\to U_{j}$ such that

$C_{11}=W_{11}^{*}W_{11},\ \ \ \ \ -C_{22}^{-1}=W_{22}^{*}W_{22},$ (4.25)

We denote by the operator $W:=\left(\begin{array}[]{cc}W_{11}^{*}&0\\\
0&W_{22}^{-1}\end{array}\right):U_{1}\times U_{2}\to X_{1}\times X_{2}^{*}$,
hence, we have

$\displaystyle\mathrm{Re}F^{Mix}$ $\displaystyle=$
$\displaystyle[G^{Mix}W]\left[W^{-1}\left\\{\left(\begin{array}[]{cc}C_{11}&C_{12}\\\
C_{21}&C_{22}\end{array}\right)+K^{Mix}\right\\}W^{*^{-1}}\right][G^{Mix}W]^{*}$
(4.28) $\displaystyle=:$
$\displaystyle\hat{G}^{Mix}\left[\left(\begin{array}[]{cc}I_{U_{1}}&\hat{C}_{12}\\\
\hat{C}_{21}&-I_{U_{2}}\end{array}\right)+\hat{K}^{Mix}\right]\hat{G}^{Mix\
*},$ (4.31)

where $\hat{C}_{12}:=W_{11}^{*^{-1}}C_{12}W_{22}^{*}:U_{2}\to U_{1}$,
$\hat{C}_{21}:=W_{22}C_{21}W_{11}^{*}:U_{1}\to U_{2}$, and
$\hat{K}^{Mix}=W^{-1}K^{Mix}W^{*^{-1}}:U_{1}\times U_{2}\to U_{1}\times
U_{2}$. Since $\hat{K}^{Mix}$ is a self-adjoint compact operator, by the same
argument in (4.13)–(4.14), there exists a finite dimensional subspace $V_{2}$
in Y, and constants $c_{1},c_{2},c_{3}>0$ such that

$\displaystyle(\mathrm{Re}F^{Mix}\varphi,\varphi)_{Y}$ (4.38) $\displaystyle=$
$\displaystyle(\left[\left(\begin{array}[]{cc}I_{U_{1}}&\hat{C}_{12}\\\
\hat{C}_{21}&-I_{U_{2}}\end{array}\right)+\hat{K}^{Mix}\right]\hat{G}^{Mix\
*}\varphi,\hat{G}^{Mix\ *}\varphi)_{U_{1}\times U_{2}}$ $\displaystyle\geq$
$\displaystyle\|W_{11}R_{1}G^{Mix\
*}\varphi\|^{2}_{U_{1}}-\|W_{22}^{*^{-1}}R_{2}G^{Mix\ *}\varphi\|^{2}_{U_{2}}$
$\displaystyle-\frac{1}{2}\left\\{\|W_{11}R_{1}G^{Mix\
*}\varphi\|^{2}_{U_{1}}+\|W_{22}^{*^{-1}}R_{2}G^{Mix\
*}\varphi\|^{2}_{U_{2}}\right\\}$
$\displaystyle+(\left(\begin{array}[]{cc}0&C_{12}\\\
C_{21}&0\end{array}\right)G^{Mix\ *}\varphi,G^{Mix\ *}\varphi)_{U_{1}\times
U_{2}}$ $\displaystyle\geq$ $\displaystyle c_{1}\|R_{1}G^{Mix\
*}\varphi\|^{2}_{X_{1}^{*}}-c_{2}\|R_{2}G^{Mix\ *}\varphi\|^{2}_{X_{2}}$
$\displaystyle-c_{3}\|R_{1}G^{Mix\ *}\varphi\|_{X_{1}^{*}}\|R_{2}G^{Mix\
*}\varphi\|_{X_{2}},$

for all $\varphi\in V_{2}^{\bot}$. Setting $V:=V_{1}\cup V_{2}$, $V$ is finite
dimensional subspace in $Y$. Then by (4.24) and (4.38) we have

$\displaystyle c_{1}\|R_{1}G^{Mix\ *}\varphi\|_{\tilde{X}^{*}_{1}}^{2}$
$\displaystyle\leq$ $\displaystyle c_{2}\|R_{2}G^{Mix\
*}\varphi\|^{2}_{X_{2}}$ $\displaystyle+$ $\displaystyle c_{3}\|R_{1}G^{Mix\
*}\varphi\|_{X_{1}^{*}}\|R_{2}G^{Mix\
*}\varphi\|_{X_{2}}+\|\mathrm{Re}\tilde{T}\|\|\tilde{G}^{*}\varphi\|_{\tilde{X}^{*}}^{2},$

for all $\varphi\in V^{\bot}$.

On the other hand, by the assumption (3b) and Lemma 4.6, we have
$W_{1}\not\subset\mathrm{Ran}(\left[\tilde{G},G^{Mix}R_{2}^{*}\right])+V$,
which implies that by $W_{1}\subset\mathrm{Ran}(G^{Mix}R_{1}^{*})$

$\mathrm{Ran}(G^{Mix}R_{1}^{*})\not\subset\mathrm{Ran}\left([\tilde{G},G^{Mix}R_{2}^{*}]\right)+V=\mathrm{Ran}\left([\tilde{G},G^{Mix}R_{2}^{*},P_{V}]\right).$
(4.40)

By this and Lemma 4.5, there exists sequence
$\displaystyle(\varphi_{n})_{n\in\mathbb{N}}\subset Y$ such that

$\|R_{1}G^{Mix\
*}\varphi_{n}\|_{X^{*}_{1}}^{2}>n^{2}\left(\|\tilde{G}^{*}\varphi_{n}\|_{\tilde{X}^{*}}^{2}+\|R_{2}G^{Mix\
*}\varphi_{n}\|_{X_{2}}^{2}+\|P_{V}\varphi_{n}\|_{Y}^{2}\right),$ (4.41)

for all $n\in\mathbb{N}$. Setting
$\psi_{n}:=\displaystyle\frac{n^{\frac{1}{2}}\varphi_{n}}{\|R_{1}G^{Mix\
*}\varphi_{n}\|_{X^{*}_{1}}}$, we obtain

$\|\tilde{G}^{*}\psi_{n}\|_{\tilde{X}^{*}}^{2}+\|R_{2}G^{Mix\
*}\psi_{n}\|_{X_{2}}^{2}+\|P_{V}\psi_{n}\|_{Y}^{2}\leq\frac{1}{n},\ \ \ \
\|R_{1}G^{Mix\ *}\varphi_{n}\|_{X^{*}_{1}}^{2}=n.$ (4.42)

Setting $\tilde{\psi}_{n}:=(I-P_{V})\psi_{n}\in V^{\bot}$, we finally obtain
as $\ n\to\infty$

$\|R_{1}G^{Mix\ *}\tilde{\psi}_{n}\|_{X^{*}_{1}}\geq\|R_{1}G^{Mix\
*}\psi_{n}\|_{X^{*}_{1}}-\|R_{1}G^{Mix\ *}\|\|P_{V}\psi_{n}\|_{Y},$ (4.43)
$\|\tilde{G}^{*}\tilde{\psi}_{n}\|_{\tilde{X}^{*}}+\|R_{2}G^{Mix\
*}\tilde{\psi}_{n}\|_{X_{2}}\leq\|\tilde{G}^{*}\psi_{n}\|_{\tilde{X}^{*}}+\|\tilde{G}^{*}\|\|P_{V}\psi_{n}\|_{Y}$
$\hskip 56.9055pt+\|R_{2}G^{Mix\
*}\psi_{n}\|_{X_{2}}+\|\tilde{G}^{*}\|\|P_{V}\psi_{n}\|_{Y}\to 0,$ (4.44)
$\displaystyle\|R_{1}G^{Mix\ *}\tilde{\psi}_{n}\|_{X^{*}_{1}}\|R_{2}G^{Mix\
*}\tilde{\psi}_{n}\|_{X_{2}}$ $\displaystyle\leq$
$\displaystyle\left(\|R_{1}G^{Mix\ *}\psi_{n}\|+\|R_{1}G^{Mix\
*}\|\|P_{V}\psi_{n}\|\right)$ $\displaystyle\times\left(\|R_{2}G^{Mix\
*}\psi_{n}\|+\|R_{2}G^{Mix\ *}\|\|P_{V}\psi_{n}\|\right)\to 1+\|R_{2}G^{Mix\
*}\|,$

which contradicts (LABEL:(3)-3). Therefore, we conclude (4.6).

∎

##  5 Applications of the general theorem

In the following, we study many applications of Theorem 4.2 to inverse
acoustic scatterings.

###  5.1 Dirichlet obstacle

Let $F^{Dir}_{\Omega}$ be the far-field operator for a Dirichlet obstacle
$\Omega$, that is, $F^{Dir}_{\Omega}$ is the far-field operator defined by
(2.7) corresponding to the solution of (2.4)–(2.5) where $\mathcal{B}u=u$.
$F^{Dir}_{\Omega}$ has the factorization (see Theorem 1.15 of [19])

$F^{Dir}_{\Omega}=-G^{Dir}_{\Omega}S^{*}_{\Omega}G^{Dir\ *}_{\Omega},$ (5.1)

where $G^{Dir}_{\Omega}:H^{1/2}(\partial\Omega)\to L^{2}(\mathbb{S}^{d-1})$ is
the data-to-pattern operator defined by $G^{Dir}_{\Omega}f:=v^{\infty}$ where
$v^{\infty}$ is the far-field pattern of a radiating solution $v$ (that is,
$v$ satisfies the Sommerfeld radiation condition (2.3)) such that

$\Delta v+k^{2}v=0\ \mathrm{in}\ \mathbb{R}^{d}\setminus\overline{\Omega},\ \
\ \ v=f\ \mathrm{on}\ \partial\Omega,$ (5.2)

and $S_{\Omega}:H^{-1/2}(\partial\Omega)\to H^{1/2}(\partial\Omega)$ is the
single layer boundary operator defined by

$S_{\Omega}\varphi(x):=\int_{\partial\Omega}\varphi(y)\Phi(x,y)ds(y),\
x\in\partial\Omega,$ (5.3)

where $\Phi(x,y)$ denotes the fundamental solution for the Helmholtz equation
in $\mathbb{R}^{d}$, i.e.,

$\Phi(x,y):=\left\\{\begin{array}[]{ll}\displaystyle\frac{i}{4}H^{(1)}_{0}(k|x-y|),&\quad\mbox{$d=2$},\\\
\displaystyle\frac{e^{ik|x-y|}}{4\pi|x-y|},&\quad\mbox{$d=3$},\\\
\end{array}\right.$ (5.4)

where $H^{(1)}_{0}$ is the Hankel function of the first kind of order one. The
single layer boundary operator $S_{\Omega}$ is of the form (see Lemma 1.14 of
[19])

$S_{\Omega}=C_{\Omega}+K_{\Omega},$ (5.5)

where $C_{\Omega}:H^{-1/2}(\partial\Omega)\to H^{1/2}(\partial\Omega)$ is some
positive coercive operator and $K_{\Omega}:H^{-1/2}(\partial\Omega)\to
H^{1/2}(\partial\Omega)$ is some compact operator.

For a bounded domain $B\subset\mathbb{R}^{d}$ with the smooth boundary, we
define the Herglotz operator $H_{\partial B}:L^{2}(\mathbb{S}^{d-1})\to
L^{2}(\partial B)$ by

$H_{\partial B}g(x):=\int_{\mathbb{S}^{1}}\mathrm{e}^{ik\theta\cdot
x}g(\theta)ds(\theta),\ x\in\partial B,$ (5.6)

If the range is restricted to the space $H^{1/2}(\partial B)$, we denote its
Herglotz operator by $\hat{H}_{\partial B}:L^{2}(\mathbb{S}^{d-1})\to
H^{1/2}(\partial B)$. From the definition, we have $\hat{H}_{\partial
B}^{*}=G^{Dir}_{B}S_{B}$ (see e.g, Theorem 1.15 of [19]). Let $J_{\partial
B}:H^{1/2}(\partial B)\to H^{-1/2}(\partial B)$ be a self-adjoint compact
embedding. Then, we have

$\displaystyle H^{*}_{\partial B}H_{\partial B}$ $\displaystyle=$
$\displaystyle\hat{H}^{*}_{\partial B}J_{\partial B}\hat{H}_{\partial
B}=G^{Dir}_{B}S_{B}J_{\partial B}S_{B}^{*}G^{Dir\ *}_{B}$ (5.7)
$\displaystyle=$ $\displaystyle G^{Dir}_{B}\left[C_{B}J_{\partial
B}C_{B}+\hat{K}_{B}\right]G^{Dir\ *}_{B},$

where $C_{B}J_{\partial B}C_{B}$ is a positive coercive operator and
$\hat{K}_{B}$ is some self-adjoint compact operator.

Assume that $B\subset\Omega$. Then, we can define $R:H^{1/2}(\partial B)\to
H^{1/2}(\partial\Omega)$ by $Rf:=v\bigl{|}_{\partial\Omega}$ where $v$ is a
radiating solution $v$ of (5.2) replacing $\Omega$ with $B$. Since
$v\bigl{|}_{\partial\Omega}\in C^{\infty}(\partial\Omega)$, $R$ is a compact
operator, and by the definition we have

$G^{Dir}_{B}=G^{Dir}_{\Omega}R,$ (5.8)

which corresponds to the assumption (1a) of Theorem 4.2. Applying (1) of
Theorem 4.2 as

$F=-F^{Dir}_{\Omega}=G^{Dir}_{\Omega}(C_{\Omega}+K^{*}_{\Omega})G^{Dir\
*}_{\Omega},$ $\tilde{F}=H^{*}_{\partial B}H_{\partial
B}=G^{Dir}_{B}\left[C_{B}J_{\partial B}C_{B}+\hat{K}_{B}\right]G^{Dir\
*}_{B},$

we have

$H^{*}_{\partial B}H_{\partial
B}\leq_{\mathrm{fin}}-\mathrm{Re}F^{Dir}_{\Omega}.$ (5.9)

Assume that $B\not\subset\Omega$. Then, there exists a bounded domain
$B_{0}\Subset B$ such that $B_{0}\cap\Omega=\emptyset$. We set
$W:=\mathrm{Ran}(G^{Dir}_{B_{0}})\subset\mathrm{Ran}(G^{Dir}_{B})$, then, $W$
is an infinite dimensional subspace of $L^{2}(\mathbb{S}^{d-1})$ because
$G^{Dir}_{B_{0}}$ is injective (see e.g., Lemma 1.13 of [19]). From
$B_{0}\cap\Omega=\emptyset$, we obtain

$W\cap\mathrm{Ran}(G^{Dir}_{\Omega})=\\{0\\},$ (5.10)

(see e.g., Lemma 4.2 of [2]) which corresponds to the assumption (2b) of
Theorem 4.2. Applying (2) of Theorem 4.2 as

$F=-F^{Dir}_{\Omega}=G^{Dir}_{\Omega}(C_{\Omega}+K^{*}_{\Omega})G^{Dir\
*}_{\Omega},$ $\tilde{F}=H^{*}_{\partial B}H_{\partial
B}=G^{Dir}_{B}\left[C_{B}J_{\partial B}C_{B}+\hat{K}_{B}\right]G^{Dir\
*}_{B},$

we have

$H^{*}_{\partial B}H_{\partial
B}\not\leq_{\mathrm{fin}}-\mathrm{Re}F^{Dir}_{\Omega}.$ (5.11)

From the above discussion, we conclude the following theorem, which is the
same result as Theorem 5.3 of [2].

######  Theorem 5.1 (Theorem 5.3 of [2]).

Let $B\subset\mathbb{R}^{d}$ be a bounded domain with the smooth boundary.
Then,

$B\subset\Omega\ \ \ \ \Longleftrightarrow\ \ \ \ H^{*}_{\partial
B}H_{\partial B}\leq_{\mathrm{fin}}-\mathrm{Re}F^{Dir}_{\Omega},$ (5.12)

By the same argument in Theorem 5.1, one can apply (1) and (2) of Theorem 4.2
as

$F=H^{*}_{\partial B}H_{\partial B}=G^{Dir}_{B}\left[C_{B}J_{\partial
B}C_{B}+\hat{K}_{B}\right]G^{Dir\ *}_{B},$
$\tilde{F}=-F^{Dir}_{\Omega}=G^{Dir}_{\Omega}(C_{\Omega}+K^{*}_{\Omega})G^{Dir\
*}_{\Omega}.$

Then, we also conclude the following theorem.

######  Theorem 5.2.

Let $B\subset\mathbb{R}^{d}$ be a bounded domain with the smooth boundary.
Then,

$\Omega\subset B\ \ \ \ \Longleftrightarrow\ \ \ \
-\mathrm{Re}F^{Dir}_{\Omega}\leq_{\mathrm{fin}}H^{*}_{\partial B}H_{\partial
B},$ (5.13)

######  Remark 5.3.

We remark that the factorization reconstruction for the inverse obstacle
scattering needs to assume that $k^{2}$ is not a Dirichlet eigenvalue of
$-\Delta$ in $\Omega$ (see e.g., Corollary 2.16 of [19]). While, the
monotonicity reconstruction (Theorems 5.1 and 5.2) does not require the
assumption of Dirichlet eigenvalues.

###  5.2 Inhomogeneous medium

Let $F^{Med}_{\Omega}$ be the far-field operator for an inhomogeneous medium
$\Omega$ with the function $q\in L^{\infty}(\Omega)$, that is,
$F^{Med}_{\Omega}$ is the far-field operator defined by (2.7) corresponding to
the solution of (2.2)–(2.3). Throughout this section, we assume that there
exists a constant $q_{0}>0$ such that $q\geq q_{0}$ in $\Omega$.
$F^{Med}_{\Omega}$ has the following factorization by the same argument in
Theorem 4.5 of [19]

$F^{Med}_{\Omega}=H^{*}_{\Omega}T_{\Omega}H_{\Omega},$ (5.14)

where $H_{\Omega}:L^{2}(\mathbb{S}^{d-1})\to L^{2}(\Omega)$ is the Herglotz
operator defined by

$H_{\Omega}g(x):=\int_{\mathbb{S}^{1}}\mathrm{e}^{ik\theta\cdot
x}g(\theta)ds(\theta),\ x\in\Omega,$ (5.15)

and some operator $T_{\Omega}:L^{2}(\Omega)\to L^{2}(\Omega)$ is of the form

$T_{\Omega}=k^{2}qI_{L^{2}(\Omega)}+K_{\Omega},$ (5.16)

where $K_{\Omega}$ is some compact operator.

Let $B\subset\mathbb{R}^{d}$ be a bounded domain with the smooth boundary.
Assume that $B\subset\Omega$. We define the restriction operator
$R:L^{2}(\Omega)\to L^{2}(B)$ by $Rf:=f\bigl{|}_{B}$. Then by the definition
we have $H_{B}=RH_{\Omega}$, and for $\alpha\in(0,k^{2}q_{0})$ we have

$F^{Med}_{\Omega}-\alpha
H_{B}^{*}H_{B}=H^{*}_{\Omega}\left[k^{2}qI_{L^{2}(\Omega)}-\alpha
R^{*}R+K_{\Omega}\right]H_{\Omega},$ (5.17)

where the operator $k^{2}qI_{L^{2}(\Omega)}-\alpha R^{*}R$ is positive
coercive when $\alpha\in(0,k^{2}q_{0})$. Applying (1) of Theorem 4.2 as

$F=F^{Med}_{\Omega}-\alpha
H_{B}^{*}H_{B}=H^{*}_{\Omega}\left[k^{2}qI_{L^{2}(\Omega)}-\alpha
R^{*}R+K_{\Omega}\right]H_{\Omega},$ $\tilde{F}=0,$

we have for $\alpha\in(0,k^{2}q_{0})$

$\alpha H^{*}_{B}H_{B}\leq_{\mathrm{fin}}\mathrm{Re}F^{Med}_{\Omega}.$ (5.18)

Assume that $B\not\subset\Omega$. Then, there exists a bounded domain
$B_{0}\Subset B$ such that $B_{0}\cap\Omega=\emptyset$. We set
$W:=\mathrm{Ran}(H^{*}_{B_{0}})\subset\mathrm{Ran}(H^{*}_{B})$, then, $W$ is
an infinite dimensional subspace of $L^{2}(\mathbb{S}^{1})$ because
$H^{*}_{B_{0}}$ is injective. From $B_{0}\cap\Omega=\emptyset$, we obtain

$W\cap\mathrm{Ran}(H^{*}_{\Omega})=\\{0\\}.$ (5.19)

(see e.g., Lemma 4.3 of [11]). Applying (2) of Theorem 4.2 as

$F=F^{Med}_{\Omega}=H^{*}_{\Omega}T_{\Omega}H_{\Omega},$ $\tilde{F}=\alpha
H^{*}_{B}H_{B},$

we have for $\alpha\in(0,k^{2}q_{0})$,

$\alpha H^{*}_{B}H_{B}\not\leq_{\mathrm{fin}}\mathrm{Re}F^{Med}_{\Omega}.$
(5.20)

From the above discussion, we conclude the following theorem, which is the
same result as Theorem 5.1 of [11].

######  Theorem 5.4 (Theorem 5.1 of [11]).

Let $B\subset\mathbb{R}^{d}$ be a bounded domain with the smooth boundary.
Then, for $\alpha\in(0,k^{2}q_{0})$

$B\subset\Omega\ \ \ \ \Longleftrightarrow\ \ \ \ \alpha
H^{*}_{B}H_{B}\leq_{\mathrm{fin}}\mathrm{Re}F^{Med}_{\Omega}.$ (5.21)

Assume that $\Omega\subset B$. Then, we can define the compact operator
$R:L^{2}(\Omega)\to H^{1/2}(\partial B)$ by $Rg:=w\bigl{|}_{\partial B}$ where
$w$ is a radiating solution $w$ of

$\Delta w+k^{2}(1+q)w=-k^{2}qg\ \mathrm{in}\ \mathbb{R}^{d},$ (5.22)

From the definition we obtain $G^{Med}_{\Omega}=G^{Dir}_{B}R$ where the data-
to-pattern operator $G^{Med}_{\Omega}$ is defined by
$G^{Med}_{\Omega}g:=w^{\infty}$ and $G^{Dir}_{B}$ is defined by (5.2)
replacing $\Omega$ by $B$. Since $G^{Med}_{\Omega}=H_{\Omega}^{*}T_{\Omega}$
and $T_{\Omega}$ is bounded invertible (see e.g., the arguments of Theorem 4.5
of [19]), we have

$H_{\Omega}^{*}=G^{Med}_{\Omega}T^{-1}_{\Omega}=G^{Dir}_{B}RT^{-1}_{\Omega}.$
(5.23)

Applying (1) of Theorem 4.2 as

$F=H^{*}_{\partial B}H_{\partial B}=G^{Dir}_{B}\left[C_{B}J_{\partial
B}C_{B}+\hat{K}_{\Omega}\right]G^{Dir\ *}_{B},$
$\tilde{F}=F^{Med}_{\Omega}=H^{*}_{\Omega}T_{\Omega}H_{\Omega}=H^{*}_{\Omega}\left[k^{2}qI_{L^{2}(\Omega)}+K_{\Omega}\right]H_{\Omega},$

we have

$\mathrm{Re}F^{Med}_{\Omega}\leq_{\mathrm{fin}}H^{*}_{\partial B}H_{\partial
B}.$ (5.24)

Assume that $\Omega\not\subset B$. Then, there exists a bounded domain
$\Omega_{0}\Subset\Omega$ such that $\Omega_{0}\cap B=\emptyset$. We set
$W:=\mathrm{Ran}(G^{Med}_{\Omega_{0}})\subset\mathrm{Ran}(G^{Med}_{\Omega})=\mathrm{Ran}(H^{*}_{\Omega})$,
then, $W$ is an infinite dimensional subspace of $L^{2}(\mathbb{S}^{1})$
because $G^{Med}_{\Omega_{0}}$ is injective. From $\Omega_{0}\cap
B=\emptyset$, we obtain

$W\cap\mathrm{Ran}(G^{Dir}_{B})=\\{0\\},$ (5.25)

(see e.g., Lemma 4.3 of [11]). Applying (2) of Theorem 4.2 as

$F=H^{*}_{\partial B}H_{\partial B}=G^{Dir}_{B}\left[C_{B}J_{\partial
B}C_{B}+\hat{K}_{\Omega}\right]G^{Dir\ *}_{B},$
$\tilde{F}=F^{Med}_{\Omega}=H^{*}_{\Omega}T_{\Omega}H_{\Omega}=H^{*}_{\Omega}\left[k^{2}qI_{L^{2}(\Omega)}+K_{\Omega}\right]H_{\Omega},$

we have

$\mathrm{Re}F^{Med}_{\Omega}\not\leq_{\mathrm{fin}}H^{*}_{\partial
B}H_{\partial B}.$ (5.26)

From the above discussion, we conclude the following theorem.

######  Theorem 5.5.

Let $B\subset\mathbb{R}^{d}$ be a bounded domain with the smooth boundary.
Then,

$\Omega\subset B\ \ \ \ \Longleftrightarrow\ \ \ \
\mathrm{Re}F^{Med}_{\Omega}\leq_{\mathrm{fin}}H^{*}_{\partial B}H_{\partial
B},$ (5.27)

######  Remark 5.6.

We remark that the factorization reconstruction for the inverse medium
scattering needs to assume that $k^{2}$ is not a transmission eigenvalue in
$\Omega$ (see e.g., Theorem 4.10 of [19]). While, the monotonicity
reconstruction (Theorems 5.4 and 5.5) does not require the assumption of
transmission eigenvalues.

###  5.3 Dirichlet crack

Let $F^{Dir}_{\Gamma}$ be the far-field operator for a Dirichlet crack
$\Gamma$ where $\Gamma\subset\mathbb{R}^{d}$ is a smooth non-intersecting open
arc ($d=2$) or surface ($d=3$), and we assume that $\Gamma$ can be extended to
some smooth, connected, closed curve ($d=2$) or surface ($d=3$)
$\partial\Omega$ enclosing a bounded domain $\Omega$ in $\mathbb{R}^{d}$. The
corresponding far-field pattern is defined by solving the scattering problem
(2.4)–(2.5) where $\Omega$ in (2.4) is replaced by $\Gamma$ and the boundary
condition (2.5) is replaced by

$u_{-}=0\ \mathrm{on}\ \Gamma,\ \ \ \ \ \ \ \ u_{+}=0\ \mathrm{on}\ \Gamma,$
(5.28)

where we denote by $u_{\pm}$ the limit of $u$ approaching the boundary from
exterior (+) and interior (-) of an extensive domain $\Omega$ (see Figure 1).
$F^{Dir}_{\Gamma}$ has the factorization (see Lemma 3.4 of [21])

Figure 1: The assumption for $\Gamma$ and the boundary condition.

$F^{Dir}_{\Gamma}=-G^{Dir}_{\Gamma}S^{*}_{\Gamma}G^{Dir\ *}_{\Gamma},$ (5.29)

where $G^{Dir}_{\Gamma}:H^{1/2}(\Gamma)\to L^{2}(\mathbb{S}^{d-1})$ is the
data-to-pattern operator corresponding to the crack $\Gamma$, and
$S_{\Gamma}:\tilde{H}^{-1/2}(\Gamma)\to H^{1/2}(\Gamma)$ is the single layer
boundary operator corresponding to the crack $\Gamma$ where we denote by

$H^{1/2}(\Gamma):=\\{u\bigl{|}_{\Gamma}:u\in H^{1/2}(\partial\Omega)\\},$
(5.30) $\tilde{H}^{1/2}(\Gamma):=\\{u\bigl{|}_{\Gamma}:u\in
H^{1/2}(\partial\Omega),\ supp(u)\subset\overline{\Gamma}\\},$ (5.31)

and $H^{-1/2}(\Gamma)$ and $\tilde{H}^{-1/2}(\Gamma)$ the dual spaces of
$\tilde{H}^{1/2}(\Gamma)$ and $H^{1/2}(\Gamma)$, respectively. We have the
following inclusion relation

$\tilde{H}^{1/2}(\Gamma)\subset H^{1/2}(\Gamma)\subset
L^{2}(\Gamma)\subset\tilde{H}^{-1/2}(\Gamma)\subset H^{-1/2}(\Gamma).$ (5.32)

The single layer boundary operator $S_{\Gamma}$ is of the form (see Lemma 3.2
of [21])

$S_{\Gamma}=C_{\Gamma}+K_{\Gamma},$ (5.33)

where $C_{\Gamma}:\tilde{H}^{-1/2}(\Gamma)\to H^{1/2}(\Gamma)$ is some
positive coercive operator and $K_{\Gamma}:\tilde{H}^{-1/2}(\Gamma)\to
H^{1/2}(\Gamma)$ is some compact operator.

Let $\sigma\subset\mathbb{R}^{d}$ be a smooth arc ($d=2$) or surface ($d=3$).
Assume that $\sigma\subset\Gamma$. Then, we define $R:L^{2}(\Gamma)\to
L^{2}(\sigma)$ by $Rf:=f\bigl{|}_{\sigma}$. Let $J:H^{1/2}(\Gamma)\to
L^{2}(\Gamma)$ be the compact embedding. Since
$\hat{H}_{\Gamma}^{*}=G^{Dir}_{\Gamma}S_{\Gamma}$ (see e.g., Lemma 3.4 of
[21]), we have

$H_{\sigma}=RH_{\Gamma}=RJ\hat{H}_{\Gamma}=RJS_{\Gamma}^{*}G^{Dir\
*}_{\Gamma},$ (5.34)

where $H_{\Gamma}:L^{2}(\mathbb{S}^{d-1})\to L^{2}(\Gamma)$ is the Herglotz
operator corresponding to $\Gamma$ and
$\hat{H}_{\Gamma}:L^{2}(\mathbb{S}^{d-1})\to H^{1/2}(\Gamma)$ is the Herglotz
operator that its range is restricted to the space $H^{1/2}(\Gamma)$. Applying
(1) of Theorem 4.2 as

$F=-F^{Dir}_{\Gamma}=G^{Dir}_{\Gamma}(C_{\Gamma}+K^{*}_{\Gamma})G^{Dir\
*}_{\Gamma},$ $\tilde{F}=H^{*}_{\sigma}H_{\sigma},$

we have

$H^{*}_{\sigma}H_{\sigma}\leq_{\mathrm{fin}}-\mathrm{Re}F^{Dir}_{\Gamma}.$
(5.35)

Assume that $\sigma\not\subset\Gamma$. Then, there exists
$\sigma_{0}\Subset\sigma$ such that $\sigma_{0}\cap\Gamma=\emptyset$. We set
$W:=\mathrm{Ran}(H^{*}_{\sigma_{0}})\subset\mathrm{Ran}(H^{*}_{\sigma})$,
then, $W$ is an infinite dimensional subspace of $L^{2}(\mathbb{S}^{1})$
because $H^{*}_{\sigma_{0}}$ is injective. From
$\sigma_{0}\cap\Gamma=\emptyset$, we obtain

$W\cap\mathrm{Ran}(G^{Dir}_{\Gamma})=\\{0\\}.$ (5.36)

(see e.g., Lemma 4.1 of [9]). Applying (2) of Theorem 4.2 as

$F=-F^{Dir}_{\Gamma}=G^{Dir}_{\Gamma}(C_{\Gamma}+K^{*}_{\Gamma})G^{Dir\
*}_{\Gamma},$ $\tilde{F}=H^{*}_{\sigma}H_{\sigma},$

we have

$H^{*}_{\sigma}H_{\sigma}\not\leq_{\mathrm{fin}}-\mathrm{Re}F^{Dir}_{\Gamma}.$
(5.37)

From the above discussion, we conclude the following theorem, which is the
same result as Theorem 1.1 of [9].

######  Theorem 5.7 (Theorem 1.1 of [9]).

Let $\sigma\subset\mathbb{R}^{d}$ be a smooth arc ($d=2$) or surface ($d=3$).
Then,

$\sigma\subset\Gamma\ \ \ \ \Longleftrightarrow\ \ \ \
H^{*}_{\sigma}H_{\sigma}\leq_{\mathrm{fin}}-\mathrm{Re}F^{Dir}_{\Gamma}.$
(5.38)

By the same argument in Theorem 5.7, one can apply (1) and (2) of Theorem 4.2
as

$F=H^{*}_{\partial B}H_{\partial B}=G^{Dir}_{B}\left[C_{B}J_{\partial
B}C_{B}+\hat{K}_{B}\right]G^{Dir\ *}_{B},$
$\tilde{F}=-F^{Dir}_{\Gamma}=G^{Dir}_{\Gamma}(C_{\Gamma}+K^{*}_{\Gamma})G^{Dir\
*}_{\Gamma}.$

Then, we also conclude the following theorem, which is the same result as
Theorem 1.2 of [9].

######  Theorem 5.8 (Theorem 1.2 of [9]).

Let $B\subset\mathbb{R}^{d}$ be a bounded domain with the smooth boundary.
Then,

$\Gamma\subset B\ \ \ \ \Longleftrightarrow\ \ \ \
-\mathrm{Re}F^{Dir}_{\Gamma}\leq_{\mathrm{fin}}H^{*}_{\partial B}H_{\partial
B}.$ (5.39)

######  Remark 5.9.

We remark that the factorization reconstruction for the inverse crack
scattering also does not restrict the wave number. (see Theorem 3.9 of [21]).
One of the advantage of the monotonicity over the factorization is to have not
only inside tests (Theorem 5.7), but also outside tests (Theorem 5.8).

###  5.4 Mixed obstacle

Let $F^{Mix}_{\Omega_{1},\Omega_{2}}$ be the far-field operator for the mixed
obstacle $\Omega=\Omega_{1}\cup\Omega_{2}$ with the Dirichlet part
$\Omega_{1}$ and the Neumann part $\Omega_{2}$ where $\Omega_{1},\Omega_{2}$
are bounded domains with the smooth boundary such that
$\Omega_{1}\cup\Omega_{2}=\emptyset$. The corresponding far-field pattern is
defined by solving the scattering problem (2.4)–(2.5) where the boundary
condition (2.5) is replaced by

$u=0\ \mathrm{on}\ \partial\Omega_{1},\ \ \ \ \ \ \ \ \frac{\partial
u}{\partial\nu}=0\ \mathrm{on}\ \partial\Omega_{2}.$ (5.40)

$F^{Mix}_{\Omega_{1},\Omega_{2}}$ has the factorization (see Theorem 3.4 of
[19])

$F^{Mix}_{\Omega_{1},\Omega_{2}}=-G^{Mix}_{\Omega_{1},\Omega_{2}}T^{Mix}_{\Omega_{1},\Omega_{2}}G^{Mix\
*}_{\Omega_{1},\Omega_{2}},$ (5.41)

where $G^{Mix}_{\Omega_{1},\Omega_{2}}:H^{1/2}(\partial\Omega_{1})\times
H^{-1/2}(\partial\Omega_{2})\to L^{2}(\mathbb{S}^{d-1})$ is the data-to-
pattern operator for the mixed obstacle $\Omega$, i.e., defined by
$G^{Mix}_{\Omega_{1},\Omega_{2}}\left(\begin{array}[]{cc}f_{1}\\\
f_{2}\end{array}\right):=v^{\infty}$ where $v^{\infty}$ is the far-field
pattern of a radiating solution $v$ such that

$\Delta v+k^{2}v=0\ \mathrm{in}\ \mathbb{R}^{d}\setminus\overline{\Omega},\ \
v=f_{1}\ \mathrm{on}\ \partial\Omega_{1},\ \ \frac{\partial
v}{\partial\nu}=f_{2}\ \mathrm{on}\ \partial\Omega_{2},$ (5.42)

and some operator
$T^{Mix}_{\Omega_{1},\Omega_{2}}:H^{-1/2}(\partial\Omega_{1})\times
H^{1/2}(\partial\Omega_{2})\to H^{1/2}(\partial\Omega_{1})\times
H^{-1/2}(\partial\Omega_{2})$ has the form (Theorem 3.4 of [19])

$T^{Mix}_{\Omega_{1},\Omega_{2}}=\left(\begin{array}[]{cc}C^{+}_{\Omega_{1}}&0\\\
0&C^{-}_{\Omega_{2}}\end{array}\right)+K^{Mix}_{\Omega_{1},\Omega_{2}},$
(5.43)

where $C^{+}_{\Omega_{1}}:H^{-1/2}(\partial\Omega_{1})\to
H^{1/2}(\partial\Omega_{1})$ is some positive coercive operator,
$C^{-}_{\Omega_{2}}:H^{1/2}(\partial\Omega_{2})\to
H^{-1/2}(\partial\Omega_{2})$ is some negative coercive operator, and
$K^{Mix}:H^{-1/2}(\partial\Omega_{1})\times H^{1/2}(\partial\Omega_{2})\to
H^{1/2}(\partial\Omega_{1})\times H^{-1/2}(\partial\Omega_{2})$ is some
compact operator.

Assume that $\Omega_{1}\subset B$. Then, there exists bounded domain
$\tilde{B}\Subset B$ such that $\tilde{B}\cap\Omega_{2}=\emptyset$ and
$\Omega_{1}\subset\tilde{B}$. Define $R:H^{1/2}(\partial\Omega_{1})\times
H^{-1/2}(\partial\Omega_{2})\to H^{1/2}(\partial\tilde{B})\times
H^{-1/2}(\partial\Omega_{2})$ by $R\left(\begin{array}[]{cc}f_{1}\\\
f_{2}\end{array}\right):=\left(\begin{array}[]{cc}v\bigl{|}_{\partial\tilde{B}}\\\
f_{2}\end{array}\right)$. Then, $R$ has the form
$R=\left(\begin{array}[]{cc}0&0\\\ 0&I\end{array}\right)+\mathrm{compact}$,
and

$G^{Mix}_{\Omega_{1},\Omega_{2}}=G^{Mix}_{\tilde{B},\Omega_{2}}R,$ (5.44)

where $G^{Mix}_{\tilde{B},\Omega_{2}}:H^{1/2}(\partial\tilde{B})\times
H^{-1/2}(\partial\Omega_{2})\to L^{2}(\mathbb{S}^{d-1})$ is the data-to-
pattern operator corresponding to $\tilde{B}\cup\Omega_{2}$ with the Dirichlet
part $\tilde{B}$ and the Neumann part $\Omega_{2}$. We also define
$\tilde{R}:H^{1/2}(\partial\tilde{B})\to H^{1/2}(\partial\tilde{B})\times
H^{-1/2}(\partial\Omega_{2})$ by $\tilde{R}g:=\left(\begin{array}[]{cc}g\\\
\frac{\partial w}{\partial\nu}\bigl{|}_{\partial\Omega_{2}}\end{array}\right)$
where $w$ is a radiating solution $w$ of

$\Delta w+k^{2}w=0\ \mathrm{in}\ \mathbb{R}^{d}\setminus\overline{\tilde{B}},\
\ w=g\ \mathrm{on}\ \partial\tilde{B},$ (5.45)

Then, $\tilde{R}$ has the form $\tilde{R}=\left(\begin{array}[]{cc}I\\\
0\end{array}\right)+\mathrm{compact}$, and

$G^{Dir}_{\tilde{B}}=G^{Mix}_{\tilde{B},\Omega_{2}}\tilde{R}.$ (5.46)

By these and (5.7), we have

$\displaystyle F^{Mix}+H^{*}_{\partial\tilde{B}}H_{\partial\tilde{B}}$ (5.49)
$\displaystyle=$ $\displaystyle
G^{Mix}_{\tilde{B},\Omega_{2}}\left[R(-T^{Mix}_{\Omega_{1},\Omega_{2}})R^{*}+\tilde{R}S_{\tilde{B}}J_{\partial\tilde{B}}S^{*}_{\tilde{B}}\tilde{R}^{*}\right]G^{Mix\
*}_{\tilde{B},\Omega_{2}}$ $\displaystyle=$ $\displaystyle
G^{Mix}_{\tilde{B},\Omega_{2}}\left[\left(\begin{array}[]{cc}C_{\tilde{B}}J_{\partial\tilde{B}}C_{\tilde{B}}&0\\\
0&-C_{\Omega_{2}^{-}}\end{array}\right)+\tilde{K}^{Mix}\right]G^{Mix\
*}_{\tilde{B},\Omega_{2}},$

where
$\left(\begin{array}[]{cc}C_{\tilde{B}}J_{\partial\tilde{B}}C_{\tilde{B}}&0\\\
0&-C_{\Omega_{2}^{-}}\end{array}\right)$ is a positive coercive operator and
$\tilde{K}^{Mix}$ is some compact operator. Applying (1) of Theorem 4.2 as

$F=F^{Mix}_{\Omega_{1},\Omega_{2}}+H^{*}_{\partial\tilde{B}}H_{\partial\tilde{B}}=G^{Mix}_{\tilde{B},\Omega_{2}}\left[\left(\begin{array}[]{cc}C_{\tilde{B}}J_{\partial\tilde{B}}C_{\tilde{B}}&0\\\
0&-C_{\Omega_{2}^{-}}\end{array}\right)+\tilde{K}^{Mix}\right]G^{Mix\
*}_{\tilde{B},\Omega_{2}},$ $\tilde{F}=0,$

we have

$-\mathrm{Re}F^{Mix}_{\Omega_{1},\Omega_{2}}\leq_{\mathrm{fin}}H^{*}_{\partial\tilde{B}}H_{\partial\tilde{B}}.$
(5.50)

Since we have $\tilde{B}\subset B$, one can show by the same argument in (5.8)
that there exists a compact operator $R_{B}:H^{1/2}(\partial\tilde{B})\to
H^{1/2}(\partial B)$ such that

$G^{Dir}_{\tilde{B}}=G^{Dir}_{B}R_{B}.$ (5.51)

Then, applying (1) of Theorem 4.2 as $F=H^{*}_{\partial B}H_{\partial B}$,
$\tilde{F}=H^{*}_{\partial\tilde{B}}H_{\partial\tilde{B}}$ (remark (5.7)), it
follows that

$H^{*}_{\partial\tilde{B}}H_{\partial\tilde{B}}\leq_{\mathrm{fin}}H^{*}_{\partial
B}H_{\partial B},$ (5.52)

which implies that with (5.50) we conclude that

$-\mathrm{Re}F^{Mix}_{\Omega_{1},\Omega_{2}}\leq_{\mathrm{fin}}H^{*}_{\partial
B}H_{\partial B}.$ (5.53)

Assume that $\Omega_{1}\not\subset B$ and
$\mathbb{R}^{d}\setminus(\overline{B\cup\Omega})$ is connected. Then, there
exists $\Gamma\Subset\partial\Omega_{1}$ such that $\Gamma\cap B=\emptyset$.
Define $E_{\Gamma}:H^{1/2}(\Gamma)\to H^{1/2}(\partial\Omega_{1})$ by
$E_{\Gamma}f=f$ on $\Gamma$, otherwise zero. Denote by $X_{\Gamma}\subset
H^{1/2}(\Gamma)$ the subspace of piecewise linear continuous functions on
$\Gamma$ that vanish on $\partial\Gamma$. Set
$W:=\mathrm{Ran}\left(G^{Mix}R_{1}^{*}E_{\Gamma}\big{|}_{X_{\Gamma}}\right)\subset\mathrm{Ran}(G^{Mix}R_{1}^{*})$,
it is infinite dimensional because $X_{\Gamma}$ is infinite dimensional and
the operator $G^{Mix}R_{1}^{*}E_{\Gamma}$ is injective. By Lemma 4.6 of [2],
we have

$W\cap\mathrm{Ran}(G^{Dir}_{B},G^{Mix}R_{2}^{*})=\\{0\\}.$ (5.54)

Applying (3) of Theorem 4.2 as

$F^{Mix}=-F^{Mix}_{\Omega_{1},\Omega_{2}}=G^{Mix}_{\Omega_{1},\Omega_{2}}\left(\left(\begin{array}[]{cc}C^{+}_{\Omega_{1}}&0\\\
0&C^{-}_{\Omega_{2}}\end{array}\right)+K^{Mix}_{\Omega_{1},\Omega_{2}}\right)G^{Mix\
*}_{\Omega_{1},\Omega_{2}},$ $\tilde{F}=H^{*}_{\partial B}H_{\partial
B}=G^{Dir}_{B}\left[C_{B}J_{\partial B}C_{B}+\hat{K}_{B}\right]G^{Dir\
*}_{B},$

we have

$-\mathrm{Re}F^{Mix}_{\Omega_{1},\Omega_{2}}\not\leq_{\mathrm{fin}}H^{*}_{\partial
B}H_{\partial B}.$ (5.55)

From the above discussion, we conclude the following theorem, which is the
same result as Theorem 5.5 of [2]. We remark that the monotonicity
reconstruction discussed here was succeeded without assuming that the
wavenumber $k^{2}$ is neither a Dirichlet eigenvalue of $-\Delta$ in
$\Omega_{1}$ and $B$, nor a Neumann eigenvalue in $\Omega_{2}$, although
Theorem 5.5 of [2] assumed it.

######  Theorem 5.10 (Theorem 5.5 of [2]).

Let $B\subset\mathbb{R}^{d}$ be a bounded domain with the smooth boundary such
that $\mathbb{R}^{d}\setminus(\overline{B\cup\Omega})$ is connected. Then,

$\Omega_{1}\subset B\ \ \ \ \Longleftrightarrow\ \ \ \
-\mathrm{Re}F^{Mix}_{\Omega_{1},\Omega_{2}}\leq_{\mathrm{fin}}H^{*}_{\partial
B}H_{\partial B}.$ (5.56)

By the same argument in Theorem 5.10, one can apply (1) and (3) replacing (3b)
by (3b)’ of Theorem 4.2 as

$F^{Mix}=F^{Mix}_{\Omega_{1},\Omega_{2}}=-G^{Mix}_{\Omega_{1},\Omega_{2}}\left(\left(\begin{array}[]{cc}C^{+}_{\Omega_{1}}&0\\\
0&C^{-}_{\Omega_{2}}\end{array}\right)+K^{Mix}_{\Omega_{1},\Omega_{2}}\right)G^{Dir\
*}_{\Omega_{1},\Omega_{2}},$ $\tilde{F}=H^{*}_{\partial B}H_{\partial
B}=G^{Dir}_{B}\left[C_{B}J_{\partial B}C_{B}+\hat{K}_{B}\right]G^{Dir\
*}_{B},$

for the reconstruction of the Neumann part $\Omega_{2}$. Then, we also
conclude the following theorem, which is the same result as Theorem 5.5 of [2]
without the restriction of the wave number $k>0$ as well as Theorem 5.10.

######  Theorem 5.11 (Theorem 5.5 of [2]).

Let $B\subset\mathbb{R}^{d}$ be a bounded domain with the smooth boundary such
that $\mathbb{R}^{d}\setminus(\overline{B\cup\Omega})$ is connected. Then,

$\Omega_{2}\subset B\ \ \ \ \Longleftrightarrow\ \ \ \
\mathrm{Re}F^{Mix}_{\Omega_{1},\Omega_{2}}\leq_{\mathrm{fin}}H^{*}_{\partial
B}H_{\partial B}.$ (5.57)

######  Remark 5.12.

We remark that the factorization reconstruction for the mixed obstacle has to
assume that one component is covered by some artificial domain which is
disjoint with the other one we want to reconstruct, and furthermore assume
that $k^{2}$ is neither a Dirichlet eigenvalue of $-\Delta$ in $\Omega_{1}$
and artificial covering domain, nor a Neumann eigenvalue in $\Omega_{2}$ (see
Lemma 3.5 of [19]). However, the monotonicity reconstruction (Theorems 5.10
and 5.11) does not require both of them.

###  5.5 Mixed crack

Let $F^{Mix}_{\Gamma}$ be the far-field operator for the mixed crack $\Gamma$
(The assumption for $\Gamma$ is the same as Section 5.3). The corresponding
far-field pattern is defined by solving the scattering problem (2.4)–(2.5)
where $\Omega$ in (2.4) is replaced by $\Gamma$ and the boundary condition
(2.5) is replaced by

$u_{-}=0\ \mathrm{on}\ \Gamma,\ \ \ \ \ \ \ \ \frac{\partial
u_{+}}{\partial\nu}=0\ \mathrm{on}\ \Gamma.$ (5.58)

$F^{Mix}_{\Gamma}$ has the factorization (see (3.6) and (3.13) of [25])

$F^{Mix}_{\Gamma}=-G^{Mix}_{\Gamma}M^{Mix}_{\Gamma}G^{Mix\ *}_{\Gamma},$
(5.59)

where $G^{Mix}_{\Gamma}:H^{1/2}(\Gamma)\times H^{-1/2}(\Gamma)\to
L^{2}(\mathbb{S}^{d-1})$ is the data-to-pattern operator for the mixed crack
$\Gamma$, i.e., defined by $G^{Mix}_{\Gamma}\left(\begin{array}[]{cc}f_{1}\\\
f_{2}\end{array}\right):=v^{\infty}$ where $v^{\infty}$ is the far-field
pattern of a radiating solution $v$ such that

$\Delta v+k^{2}v=0\ \mathrm{in}\ \mathbb{R}^{d}\setminus\Gamma,\ \
v_{-}=f_{1}\ \mathrm{on}\ \Gamma,\ \ \frac{\partial v_{+}}{\partial\nu}=f_{2}\
\mathrm{on}\ \Gamma,$ (5.60)

and $M^{Mix}_{\Gamma}:\tilde{H}^{-1/2}(\Gamma)\times\tilde{H}^{1/2}(\Gamma)\to
H^{1/2}(\Gamma)\times H^{-1/2}(\Gamma)$ has the form (see (3.12) of [25])

$M^{Mix}_{\Gamma}=\left(\begin{array}[]{cc}C^{+}_{\Gamma}&-I\\\
-I&C^{-}_{\Gamma}\end{array}\right)+K^{Mix}_{\Gamma},$ (5.61)

where $C^{+}_{\Gamma}:\tilde{H}^{-1/2}(\Gamma)\to H^{1/2}(\Gamma)$ is some
positive coercive operator, $C^{-}_{\Gamma}:\tilde{H}^{1/2}(\Gamma)\to
H^{-1/2}(\Gamma)$ is some negative coercive operator, and
$K^{Mix}_{\Gamma}:\tilde{H}^{-1/2}(\Gamma)\times\tilde{H}^{1/2}(\Gamma)\to
H^{1/2}(\Gamma)\times H^{-1/2}(\Gamma)$ is some compact operator.

Assume that $\Gamma\subset B$. Define $R:H^{1/2}(\Gamma)\times
H^{-1/2}(\Gamma)\to H^{1/2}(\partial B)$ by
$R\left(\begin{array}[]{cc}f_{1}\\\
f_{2}\end{array}\right):=v\bigl{|}_{\partial B}$, then, $R$ is a compact
operator and

$G^{Mix}_{\Gamma}=G^{Dir}_{B}R.$ (5.62)

Applying (1) of Theorem 4.2 as

$F=H^{*}_{\partial B}H_{\partial B}=G^{Dir}_{B}\left[C_{B}J_{\partial
B}C_{B}+\hat{K}_{B}\right]G^{Dir\ *}_{B},$
$\tilde{F}=-F^{Mix}_{\Gamma}=G^{Mix}_{\Gamma}M^{Mix}_{\Gamma}G^{Mix\
*}_{\Gamma},$

we have

$-\mathrm{Re}F^{Mix}_{\Gamma}\leq_{\mathrm{fin}}H^{*}_{\partial B}H_{\partial
B}.$ (5.63)

Assume that $\Gamma\not\subset B$. Then, there exists
$\tilde{\Gamma}\Subset\Gamma$ such that $\tilde{\Gamma}\cap B=\emptyset$.
Setting
$W:=\mathrm{Ran}\left(G^{Mix}_{\Gamma}R_{1}^{*}E_{\tilde{\Gamma}}\big{|}_{X_{\tilde{\Gamma}}}\right)\subset\mathrm{Ran}(G^{Mix}_{\Gamma}R_{1}^{*})$,
it is infinite dimensional because $X_{\tilde{\Gamma}}$ is infinite
dimensional and the operator $G^{Mix}_{\Gamma}R_{1}^{*}E_{\tilde{\Gamma}}$ is
injective. By the same argument in Lemma 4.6 of [2], we have

$W\cap\mathrm{Ran}(G^{Dir}_{B},G^{Mix}_{\Gamma}R_{2}^{*})=\\{0\\}.$ (5.64)

Applying (3) of Theorem 4.2 as

$F^{Mix}=-F^{Mix}_{\Gamma}=G^{Mix}_{\Gamma}\left(\left(\begin{array}[]{cc}C^{+}_{\Gamma}&-I\\\
-I&C^{-}_{\Gamma}\end{array}\right)+K^{Mix}_{\Gamma}\right)G^{Mix\
*}_{\Gamma},$ $\tilde{F}=H^{*}_{\partial B}H_{\partial
B}=G^{Dir}_{B}\left[C_{B}J_{\partial B}C_{B}+\hat{K}_{B}\right]G^{Dir\
*}_{B},$

we have

$-\mathrm{Re}F^{Mix}_{\Gamma}\not\leq_{\mathrm{fin}}H^{*}_{\partial
B}H_{\partial B}.$ (5.65)

From the above discussion, we conclude the following theorem.

######  Theorem 5.13.

Let $B\subset\mathbb{R}^{d}$ be a bounded domain. Then,

$\Gamma\subset B\ \ \ \ \Longleftrightarrow\ \ \ \
-\mathrm{Re}F^{Mix}_{\Gamma}\leq_{\mathrm{fin}}H^{*}_{\partial B}H_{\partial
B}.$ (5.66)

By the same argument in Theorem 5.13, one can apply (1) and (3) replacing (3b)
by (3b)’ of Theorem 4.2 as

$F^{Mix}=F^{Mix}_{\Gamma}=-G^{Mix}_{\Gamma}\left(\left(\begin{array}[]{cc}C^{+}_{\Gamma}&-I\\\
-I&C^{-}_{\Gamma}\end{array}\right)+K^{Mix}_{\Gamma}\right)G^{Mix\
*}_{\Gamma},$ $\tilde{F}=H^{*}_{\partial B}H_{\partial
B}=G^{Dir}_{B}\left[C_{B}J_{\partial B}C_{B}+\hat{K}_{B}\right]G^{Dir\
*}_{B},$

Then, we also conclude the following theorem.

######  Theorem 5.14.

Let $B\subset\mathbb{R}^{d}$ be a bounded domain. Then,

$\Gamma\subset B\ \ \ \ \Longleftrightarrow\ \ \ \
\mathrm{Re}F^{Mix}_{\Gamma}\leq_{\mathrm{fin}}H^{*}_{\partial B}H_{\partial
B}.$ (5.67)

######  Remark 5.15.

The factorization reconstruction for the mixed crack has been studied in [25],
but an extensive closed curve of the unknown crack should be known, which is a
very restrictive assumption (see Theorem 3.3 of [25]). However, the
monotonicity reconstruction (Theorems 5.13 and 5.14) does not require it. We
also remark that our works in this section would be a new extension of the
monotonicity to the inverse acoustic mixed crack scattering.

##  6 Numerical examples

In this section, we study numerical examples of our theoretical results in 2
dimensions. The far-field operator $F$ is approximated by the matrix

$F\approx\frac{2\pi}{N}\bigl{(}u^{\infty}(\hat{x}_{l},\theta_{m})\bigr{)}_{1\leq
l,m\leq N}\in\mathbb{C}^{N\times N},$ (6.1)

where $\hat{x}_{l}=\bigl{(}\mathrm{cos}(\frac{2\pi
l}{N}),\mathrm{sin}(\frac{2\pi l}{N})\bigr{)}$ and
$\theta_{m}=\bigl{(}\mathrm{cos}(\frac{2\pi m}{N}),\mathrm{sin}(\frac{2\pi
m}{N})\bigr{)}$. For the far-field pattern $u^{\infty}(\hat{x},\theta)$ of the
Dirichlet obstacle and the Dirichlet crack, we numerically compute the
integral

$u^{\infty}(\hat{x},\theta)=\frac{e^{i\frac{\pi}{4}}}{\sqrt{8\pi
k}}\int_{\Gamma}e^{-ik\hat{x}y}\varphi_{\theta}(y)ds(y),$ (6.2)

where $\varphi_{\theta}$ is given by solving

$-e^{ikx\cdot\theta}=\int_{\Gamma}\Phi(x,y)\varphi_{\theta}(y)ds(y),\
x\in\Gamma.$ (6.3)

For the inhomogeneous medium, we numerically compute the integral

$u^{\infty}(\hat{x},\theta)=\frac{e^{i\frac{\pi}{4}}}{\sqrt{8\pi
k}}\int_{\Omega}e^{-ik\hat{x}y}q(y)(u_{\theta}(y)+e^{iky\cdot\theta})dy,$
(6.4)

where $u_{\theta}$ is given by solving

$u(x)=k^{2}\int_{\Omega}\Phi(x,y)q(y)(u_{\theta}(y)+e^{iky\cdot\theta})dy,\
x\in\Omega.$ (6.5)

For a bounded domain (or a smooth curve) $B$, the operator $H^{*}_{B}H_{B}$ is
approximated by

$H^{*}_{B}H_{B}\approx\frac{2\pi}{N}\biggl{(}\int_{B}e^{iky\cdot(\theta_{m}-\hat{x}_{l})}dy\biggr{)}_{1\leq
l,m\leq N}\in\mathbb{C}^{N\times N}.$ (6.6)

We denote by the sampling square region $[-R,R]^{2}$, which includes the
unknown target. We also denote by a grid point
$z_{i,j}:=(\frac{Ri}{M},\frac{Rj}{M})$ ($i,j=-M,-M+1,...,M$) in the sampling
square region $[-R,R]^{2}$ (see Figure 2). Throughout our examples, we fix
parameters $R=1.5$, $M=100$, and $N=20$.

Figure 2: The grid points in the sampling square region.

###  6.1 Dirichlet obstacle and inhomogeneous medium

We consider the Dirichlet obstacle and the inhomogeneous medium detections
discussed in Sections 5.1 and 5.2. The following shapes $\partial\Omega_{j}$
($j=1,2$) are considered (see Figure 6):

$\partial\Omega_{1}=\left\\{\left(0.7\mathrm{cos}(\pi s),0.7\mathrm{sin}(\pi
s)\right)|-1\leq s\leq 1\right\\}.$ (6.7)
$\partial\Omega_{2}=\left\\{\left(0.3\mathrm{cos}(\pi s)-0.7,\
0.3\mathrm{sin}(\pi s)\right)|-1\leq s\leq 1\right\\}$
$\cup\left\\{\left(0.3\mathrm{cos}(\pi s)+0.7,\ 0.3\mathrm{sin}(\pi
s)\right)|-1\leq s\leq 1\right\\}.$ (6.8)

Based on Theorems 5.1 and 5.4, the indicator functions for the Dirichlet
obstacle and inhomogeneous medium are given by

$I^{MM}_{dir}(B):=\\#\left\\{\mathrm{negative}\ \mathrm{eigenvalues}\
\mathrm{of}\ -\mathrm{Re}F^{Dir}_{\Omega}-H^{*}_{\partial B}H_{\partial
B}\right\\},$ (6.9) $I^{MM}_{med}(B):=\\#\left\\{\mathrm{negative}\
\mathrm{eigenvalues}\ \mathrm{of}\ \mathrm{Re}F^{Med}_{\Omega}-\alpha
H^{*}_{\partial B}H_{\partial B}\right\\},$ (6.10)

for a bounded domain $B$, respectively. For the medium, we always consider
$q=1$ in $\Omega_{j}$, and $\alpha=1$. Here, $B$ is chosen as a square, i.e.,
$B=B_{i,j}(r):=z_{i,j}+[-\frac{r}{2},\frac{r}{2}]\times[-\frac{r}{2},\frac{r}{2}]$
where a grid point $z_{i,j}$ is the center of a square (see Figure 3).

Figure 3: The square test.

Then, we can compute the integral

$\int_{B_{i,j}(r)}e^{iky\cdot(\theta_{m}-\hat{x}_{l})}dy=r^{2}e^{ik(\theta_{m}-\hat{x}_{l})\cdot
z_{i,j}}\mathrm{sinc}\left(\frac{kr}{2}(\theta_{m}-\hat{x}_{l})_{1}\right)\mathrm{sinc}\left(\frac{kr}{2}(\theta_{m}-\hat{x}_{l})_{2}\right).$
(6.11)

Figures 7 and 8 are given by plotting the values of the indicator functions

$I^{MM}_{dir,r}(z_{i,j}):=I^{MM}_{dir}(B_{i,j}(r)),\ \ \mathrm{for\ each}\
i,j=-100,-99,...,100,$ (6.12)
$I^{MM}_{med,r}(z_{i,j}):=I^{MM}_{med}(B_{i,j}(r)),\ \ \mathrm{for\ each}\
i,j=-100,-99,...,100,$ (6.13)

respectively, for different lengths $r=0.1,0.5$, wavenumbers $k=1,5$, and
shapes $\Omega_{1},\Omega_{2}$.

We also plot the values of the indicator function for the factorization method

$I^{FM}(z_{i,j}):=\left(\sum_{n=1}^{\infty}\frac{|\langle\phi_{z_{i,j}},\varphi_{n}\rangle_{L^{2}(\mathbb{S}^{1})}|^{2}}{\mu_{n}}\right)^{-1},\
\ \mathrm{for\ each}\ i,j=-100,-99,...,100,$ (6.14)

with $\left\\{\mu_{n},\phi_{n}\right\\}$ an eigensystem of the self-adjoint
compact operator $|\mathrm{Re}F|+|\mathrm{Im}F|$ where $F$ is some far-field
operator, and $\phi_{z_{i,j}}$ is defined by

$\phi_{z_{i,j}}(\hat{x}):=e^{-ik\hat{x}\cdot
z_{i,j}},\hat{x}\in\mathbb{S}^{1},$ (6.15)

Figures 9 and 10 are corresponding to $F=F^{Dir}_{\Omega}$ for the Dirichlet
obstacle and $F=F^{Med}_{\Omega}$ for the inhomogeneous medium, respectively.
For details of introductions of the indicator function $I^{FM}$, we refer to
Corollary 2.16 and Theorem 4.9 of [19].

###  6.2 Dirichlet crack

We consider the Dirichlet crack detection discussed in Section 5.3. The
following shapes $\Gamma_{j}$ ($j=1,2$) are considered (see Figure 11):

$\Gamma_{1}=\left\\{\left(\mathrm{cos}(2s),\mathrm{sin}(2s)\right)|-1\leq
s\leq 1\right\\}.$ (6.16) $\Gamma_{2}=\left\\{\left(-0.4\mathrm{cos}(2s)-0.7,\
0.4\mathrm{sin}(2s)\right)|-1\leq s\leq 1\right\\}$
$\cup\left\\{\left(0.4\mathrm{cos}(2s)+0.7,\ 0.4\mathrm{sin}(2s)\right)|-1\leq
s\leq 1\right\\}.$ (6.17)

Based on Theorem 5.7, the indicator function is given by

$I^{MM}_{dir}(\sigma):=\\#\left\\{\mathrm{negative}\ \mathrm{eigenvalues}\
\mathrm{of}-\mathrm{Re}F^{Dir}_{\Gamma}-H^{*}_{\sigma}H_{\sigma}\right\\}.$
(6.18)

for a smooth arc $\sigma$. Here, $\sigma$ is chosen as a line segment, i.e.,
$\sigma=\sigma_{i,j}(\eta,r):=z_{i,j}+L(\eta,r)$ where a grid point $z_{i,j}$
is the center of line segments, and $L(\eta,r)$ is defined by

$L(\eta,r):=\left\\{(s,s\mathrm{tan}(\eta))\bigl{|}-\frac{r}{2}\mathrm{cos}(\eta)\leq
s\leq\frac{r}{2}\mathrm{cos}(\eta)\right\\},$ (6.19)

where $\eta\in[0,\pi]$ is the angle and $r>0$ is the length of the line
segment (see Figure 4).

Figure 4: The line segment test.

Then, we can compute the integral

$\int_{\sigma_{i,j}(\eta,r)}e^{iky\cdot(\theta_{m}-\hat{x}_{l})}ds(y)=re^{ik(\theta_{m}-\hat{x}_{l})\cdot
z_{i,j}}\mathrm{sinc}\biggl{(}\frac{rk}{2}\Bigl{(}\mathrm{cos}(\eta)(\theta_{m}-\hat{x}_{l})_{1}+\mathrm{sin}(\eta)(\theta_{m}-\hat{x}_{l})_{2}\Bigr{)}\biggr{)}.$
(6.20)

Figures 12 and 13 are given by plotting the values of the indicator function
for $\Gamma_{1}$ and $\Gamma_{2}$

$I^{MM}_{dir,\eta,r}(z_{i,j}):=I^{MM}_{dir}(\sigma_{i,j}(\eta,r)),\ \
\mathrm{for\ each}\ i,j=-100,-99,...,100,$ (6.21)

respectively, for different angles $\eta=0,\pi/2$, lengths $r=0.01,0.1$, and
wavenumbers $k=1,5$.

In Figure 14 we also plot the values of the indicator function for the
factorization method

$I^{FM}_{dir}(z_{i,j}):=\left(\sum_{n=1}^{\infty}\frac{|\langle\phi_{z_{i,j}},\varphi_{n}\rangle_{L^{2}(\mathbb{S}^{1})}|^{2}}{\mu_{n}}\right)^{-1},\
\ \mathrm{for\ each}\ i,j=-100,-99,...,100,$ (6.22)

with $\left\\{\mu_{n},\phi_{n}\right\\}$ an eigensystem of the self-adjoint
compact operator $|\mathrm{Re}F^{Dir}_{\Gamma}|+|\mathrm{Im}F^{Dir}_{\Gamma}|$
and $\phi_{z_{i,j}}$ is defined in (6.15). For details of introductions of the
indicator function $I^{FM}_{dir}$, we refer to Theorem 3.9 of [21].

###  6.3 Mixed crack

We consider the mixed crack detection discussed in Section 5.5. The following
shape $\Gamma_{3}$ is considered (see Figure 15):

$\Gamma_{3}=\left\\{\left(0.5\mathrm{cos}(2s),0.5\mathrm{sin}(2s)\right)|-1\leq
s\leq 1\right\\}.$ (6.23)

Based on Theorem 5.13, the indicator function is given by

$I^{MM}_{mix}(B):=\\#\left\\{\mathrm{negative}\ \mathrm{eigenvalues}\
\mathrm{of}\ \mathrm{Re}F^{Mix}_{\Gamma}+H^{*}_{\partial B}H_{\partial
B}\right\\}.$ (6.24)

for a bounded domain $B$. Here, $B$ is chosen as a circle, i.e., $B=B_{r}(z)$
is an open circle with center $z\in\mathbb{R}^{2}$ and radius $r>0$. Then, we
can compute the integral

$\int_{\partial B_{r}(z)}e^{iky\cdot(\theta_{m}-\hat{x}_{l})}ds(y)=2\pi
re^{ik(\theta_{m}-\hat{x}_{l})\cdot z}J_{0}(kr|\theta_{m}-\hat{x}_{l}|),$
(6.25)

where $J_{0}$ is the Bessel function of the first kind for the zero order.
Figures 16 and 17 are given by plotting the values of two types (see Figure 5)
of indicator functions

$I^{MM}_{mix,r}(z_{i,j}):=I^{MM}_{mix}(B_{r}(z_{i,j})),\ \ \mathrm{for\ each}\
i,j=-100,-99,...,100,$ (6.26)
$I^{MM}_{mix,p}(z_{i,j}):=I^{MM}_{mix}(B_{|z_{i,j}-p|}(p)),\ \ \mathrm{for\
each}\ i,j=-100,-99,...,100,$ (6.27)

respectively, for different radiuses $r=0.25,1$, points $p=(0,0)$, $(1,0)$,
wavenumbers $k=1,5$.

---
Figure 5: The shifting circle test(left) and the shrinking circle test(right).

In Figures 18 we also plot the values of the indicator function for the
factorization method

$I^{FM}_{mix}(z_{i,j}):=\left(\sum_{n=1}^{\infty}\frac{|\langle\phi_{z_{i,j}},\varphi_{n}\rangle_{L^{2}(\mathbb{S}^{1})}|^{2}}{\mu_{n}}\right)^{-1},\
\ \mathrm{for\ each}\ i,j=-100,-99,...,100,$ (6.28)

with $\left\\{\mu_{n},\phi_{n}\right\\}$ an eigensystem of the self-adjoint
compact operator $|\mathrm{Re}\tilde{F}|+|\mathrm{Im}\tilde{F}|$. The operator
$\tilde{F}$ is defined by adding the Herglotz operator correspoding to the
auxiliary $\partial\Omega_{3}$ to the original far-field operator

$\tilde{F}:=F^{Mix}_{\Gamma}-pH^{*}_{\partial\Omega_{3}}H_{\partial\Omega_{3}},$
(6.29)

for $p\in\mathbb{C}\setminus\\{0\\}$. We choose $p=0.01+0.01i$ or $0.1+0.1i$
in our numerical examples. Here, the auxiliary $\partial\Omega_{3}$ is defined
by

$\partial\Omega_{3}=\left\\{\left(0.5\mathrm{cos}(\pi s),0.5\mathrm{sin}(\pi
s)\right)|-1\leq s\leq 1\right\\},$ (6.30)

which is an extension of $\Gamma_{3}$, that is,
$\Gamma_{3}\subset\partial\Omega_{3}$. For details of introductions of the
indicator function $I^{FM}_{mix}$, we refer to Theorem 3.4 of [25].

## Conclusions

In this paper, we studied the factorization and monotonicity method for
inverse acoustic scattering problems. The main contribution was to give a new
general functional analysis theorem (Theorem 4.2) for the monotonicity method,
which can provide reconstruction schemes under weaker a priori assumptions
rather than the factorization method (see the assumptions in Theorems 3.1 and
4.2). Furthermore, we observed that the factorization method needs the real
and imaginary parts of the far-field operator (see (3.3)), while the
monotonicity needs only the real part (see (4.3), (4.4), and (4.6)), which is
also the advantage over the factorization in terms of less data. After proving
the general theorem, we also showed how the general theorem is applied to
three typical inverse scattering problems (obstacle in Sections 5.1 and 5.4,
medium in Section 5.2, and crack in Sections 5.3 and 5.5). However, it can be
applied to other inverse problems as well (especially for inverse problems
that the factorization method already studied, e.g., inverse scatterings by a
layered medium [3], a mixed-type scatterer of a obstacle and a medium [20],
and an obstacle in an homogeneous half-space [10]).

We also provided several numerical examples to compare the factorization
method with the monotonicity method. The factorization method is a point test
which checks whether a point $z$ is included in the unknown target or not,
while the monotonicity is a domain test which checks whether a domain $B$ is
included in the unknown target or not. For the domain test, we have to find
the appropriate choice of $B$. By testing the monotonicity for many different
shapes and sizes of $B$, we obtained as accurate reconstructions as the
factorization (see Figures 7, 8, 9, 10, 12, 13, and 14).

In our numerical examples for the mixture crack, we observed that the
factorization method is more accurate than the monotonicity (see Figures 16,
17, 18). This comes from that the factorization method used the Herglotz
operator corresponding to the auxiliary closed curve that is a extension of
the unknown crack (see (6.29)), which is a very restrictive assumption. On the
other hand, the monotonicity can give the reconstruction scheme without using
the auxiliary closed curve. Its scheme is the outside domain test that checks
whether a domain $B$ contains $\Gamma$ or not. However in our numerical
examples, the outside domain test can not obtain the exact shape of the
unknown crack $\Gamma$ although the location and size can be done (see Figures
16, 17). As we see numerical examples for the mixed obstacle discussed in [2],
it also seems to be difficult to obtain the shape of unknown obstacle from
outside domain tests. The numerical tests for the mixed problem in the
monotonicity has to be developed in the future.

## Acknowledgments

This work was supported by JSPS KAKENHI Grant Number JP19J10238.

## References

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Figure 6: The original domains $\Omega_{1}$ (left) and $\Omega_{2}$ (right).
---
Figure 7: Reconstruction for the Dirichlet obstacle by the monotonicity method
for different lengths $r=0.1,0.5$, wavenumbers $k=1,5$, and shapes
$\Omega_{1},\Omega_{2}$.
---
Figure 8: Reconstruction for the inhomogeneous medium by the monotonicity
method for different lengths $r=0.1,0.5$, wavenumbers $k=1,5$, and shapes
$\Omega_{1},\Omega_{2}$
---
Figure 9: Reconstruction for the Dirichlet obstacle by the factorization
method for different wavenumbers $k=1,5$ and shapes $\Omega_{1},\Omega_{2}$.
---
Figure 10: Reconstruction for the inhomogeneous medium by the factorization
method for different wavenumbers $k=1,5$ and shapes $\Omega_{1},\Omega_{2}$.
---
Figure 11: The original open arcs $\Gamma_{1}$ (left) and $\Gamma_{2}$
(right).
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Figure 12: Reconstruction for the Dirichlet crack $\Gamma_{1}$ by the
monotonicity method for different angles $\eta=0,\pi/2$, lengths $r=0.01,0.1$,
and wavenumbers $k=1,5$.
---
Figure 13: Reconstruction for the Dirichlet crack $\Gamma_{2}$ by the
monotonicity method for different angles $\eta=0,\pi/2$, lengths $r=0.01,0.1$,
and wavenumbers $k=1,5$.
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Figure 14: Reconstruction for the Dirichlet crack by the factorization method
for different wavenumbers $k=1,5$ and shapes $\Gamma_{1}$, $\Gamma_{2}$.
Figure 15: The original open arc $\Gamma_{3}$.
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Figure 16: Reconstruction for the mixed crack $\Gamma_{3}$ by the shifting
circle test of monotonicity method for different radiuses $r=0.25,1$ and
wavenumbers $k=1,5$.
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Figure 17: Reconstruction for the mixed crack $\Gamma_{3}$ by the shrinking
circle test of monotonicity method for different points $p=(0,0),(1,0)$ and
wavenumbers $k=1,5$.
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Figure 18: Reconstruction for the mixed crack $\Gamma_{3}$ by the
factorization method for different wavenumbers $k=1,5$ and complex numbers
$p=0.01+0.01i,0.1+0.1i$.