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Identification of small inclusions from multistatic data using the reciprocity gap concept
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IOP PUBLISHING INVERSE PROBLEMS
Inverse Problems 28 (2012) 045011 (19pp) doi:10.1088/0266-5611/28/4/045011
Identification of small inclusions from multistatic data
using the reciprocity gap concept
Houssem Haddar1
and Ridha Mdimagh2
1 INRIA-Scalay Ile de France/Ecole Polytechnique, CMAP, route de Saclay, 91128, Palaiseau
Cedex, France
2 ENIT-LAMSIN, BP 37, 1002 Tunis, Tunisia
E-mail: houssem.haddar@inria.fr and ridha.mdimagh@lamsin.rnu.tn
Received 23 December 2011, in final form 27 February 2012
Published 27 March 2012
Online at stacks.iop.org/IP/28/045011
Abstract
We consider the problem of identifying small inclusions (or point sources)
from multistatic Cauchy data at given surface measurements associated
with harmonic waves at a fixed frequency. We employ the reciprocity gap
sampling method to recover the location of the inclusions and identify their
equivalent dielectric properties. As opposed to the case of extended obstacles,
no approximation argument is needed in the theoretical justification of the
method. These aspects are numerically validated through multiple numerical
experiments associated with small inclusions.
1. Introduction
We are interested in the identification of dielectric inclusions embedded into a homogeneous
medium from measurements of the total electromagnetic Cauchy data (i.e. tangential electric
and magnetic fields) on the boundary of this medium at a fixed frequency and for several
incident waves. We shall consider the cases where the sizes of the inclusions are small
compared to the wavelength. This configuration is typically the case, for instance, for imaging
experiments based on microwaves used to detect malignancies or tumors [3].
We propose here the use of an algorithm introduced in [9] for the reconstruction of
extended targets, which combines ideas from samplings methods (for instance, linear sampling
method [10], factorization method [17], MUSIC [7, 14, 16], . . .) with the reciprocity gap
concept (first used in numerical reconstruction procedures by [1]). The main advantage of our
algorithm is that it does not require the evaluation of the full background Green tensor (as
required by sampling methods for inverse scattering [4, 14, 2]) or the Dirichlet to Neumann
operator of the host medium (as required by the factorization methods applied to impedance
tomography [6, 13]). The background contribution is implicitly taken into account in our
algorithm, since this contribution cancels when evaluating the reciprocity gap functional (see
section 3). For an application of a similar procedure in layered media, but relying on the
0266-5611/12/045011+19$33.00 © 2012 IOP Publishing Ltd Printed in the UK & the USA 1

Inverse Problems 28 (2012) 045011 H Haddar and R Mdimagh
Ω
x0
Σ
Sj + δDj , j = 1, 2, 3, . . .
Figure 1. Geometry and notation.
MUSIC algorithm, we refer to [12]. For an overview of the vast literature on identification of
small inclusions or sources we refer to [3, 5, 11].
The main focuses of our contribution are twofold. First, we indicate how one can remove
the approximation argument used in the case of extended targets [9, 8] when the small
inclusion asymptotic regime is valid: one obtains an exact characterization of the location of
small objects from the range of constructed sampling operator. Second, we show how one can
identify physical properties of the dielectric inclusions (those contained in the amplitude of
the second asymptotic term) from the computed indicator function. The evaluation of these
quantities requires however the evaluation of the background Green tensor associated with
a source point (or a dipole, depending on the nature of the identified point) located at the
identified location. Let us quote that our method would also apply, with minor adaptations, to
the case of small crack-type inclusions.
We shall restrict ourselves to a 2D setting of the problem and consider the cases of
small inclusions that have monopolar or dipolar source asymptotic behaviors. We formulate a
theorem for the convergence of the indicator function when the size of the inclusions tends to
0. The identification of the equivalent intensity of small inclusions for the monopolar behavior
or the polarization tensor for the dipolar behavior will then provide qualitative information on
its size or index. Numerical tests are given to prove the efficiency of the algorithm and evaluate
its resolution and robustness with respect to the measurement noise.
The outline of this paper is as follows. In section 2, we state the inverse problem and
recall the asymptotic behavior of the scattered field when the size of the inclusions is very
small. In section 3, an explicit identification procedure is proposed for source points and small
inclusion identification. In section 4, we make explicit the identification procedure and give
some numerical validating results.
2. Settings for the inverse problem
We consider a bounded domain of R2
with a sufficiently smooth boundary ∂ , the place
where we will carry out the measurements, holding a homogeneous environment of relative
(constant) electrical permittivity εb and relative (constant) magnetic permeability μb which
can be complex for an absorbent environment. We denote by ν the normal unit vector to
∂ directed to the exterior of . This domain contains m small inclusions Dδ
j := Sj + δDj,
where Sj represents the center, δ is the size of the inclusion (δ λ), where λ is a given
wavelength and Dj is a reference bounded smooth domain containing the origin (see figure 1).
2

We respectively denote by εj and μj the relative electrical permittivity and relative magnetic
permeability of the inclusion Dδ
j, respectively. These quantities are assumed to be constant
and independent from δ. In order to simplify the exposure, we shall assume that the domain
outside is the vacuum. (Our analysis should also carry over the case where the index is
piecewise constant and equal to the vacuum index outside a bounded domain.) We denote by
ε and μ the electrical permittivity and magnetic permeability of the background medium and
introduce ui
b(·, x0) ∈ H1
loc(R2
{x0}), the field generated in the background medium by a point
source at x0 ∈ , where is a curve outside . This field is the solution to
∇ ·
1
μ
∇ui
b(·, x0) + k2
εui
b(·, x0) = −δx0
in R2
, (1)
and satisfies the Sömmerfeld radiation condition
lim
r→∞ |x|=r
∂ui
b
∂r
(x, x0) − ikui
b(x, x0)
2
ds(x) = 0. (2)
where k denotes the wave number in the vacuum. Let uδ(·, x0) ∈ H1
loc(R2
{x0}) be the
field generated with the same sources inside the medium with the small inclusions. This field
satisfies
∇ ·
1
μδ
∇uδ(·, x0) + k2
εδuδ(·, x0) = −δx0
in R2
, (3)
with the same radiation condition (2) as ui
b, where
μδ(x) :=
μ if x ∈ R2
∪m
j=1Dδ
j
μj if x ∈ Dδ
j, j = 1, . . . , m,
(4)
εδ(x) :=
ε if x ∈ R2
∪m
j=1Dδ
j
εj if x ∈ Dδ
j, j = 1, . . . , m.
(5)
Let us
δ be the scattered field defined by
us
δ := uδ − ui
b.
Then, following [18, 3] it can be shown that this field is asymptotically close to δ2
˜us
with
˜us
(·, x0) :=
m
j=1
γ j
μM( j) μj
μb
∇ui
b(Sj, x0) · ∇ui
b(Sj, ·) + |Dj|k2
bγ j
ε ui
b(Sj, x0)ui
b(Sj, ·) (6)
where k2
b := k2
εbμb, γ j
μ :=
μj
μb
− 1, γ j
ε :=
εj
εb
− 1, and where Mj
(
μj
μb
) and |Dj| are respectively
the polarization tensor (see remark 1) and the size of Dj for j = 1, . . . , m. More precisely,
one can prove the following theorem.
Theorem 1. Let x0 ∈ and let us
δ and ˜us
be defined as previously. Then there exists a
function o(δ) independent of x0 such that limδ−→0 o(δ) = 0 and
us
δ(·, x0) − δ2
˜us
(·, x0) L2(∂ )
δ2
o(δ),
∂us
δ
∂ν
(·, x0) − δ2 ∂ ˜us
∂ν
(·, x0)
L2(∂ )
δ2
o(δ).
Remark 1. To define the matrix M( j)
we need to introduce some additional notation [14]. Let
γ j
:= 1
μj
, 1 j m and γ 0
= 1
μ
. For any fixed 1 j0 m, we denote the coefficient γ by
γ (x) :=
γ 0
if x ∈ R2
¯Dj0
γ j0
if x ∈ Dj0
.
(7)
3

By ϕl, l = 1, 2, we denote the solution to
∇y · γ (y)∇yϕl = 0 in R2
ϕl − yl −→ 0 as |y| −→ ∞
The polarization tensor M( j0 )
(c) = mj0
k,l (c) of the inclusion Dj0
, where c = γ 0
γ j0
, is a
symmetric positive definite matrix defined by
mj0
k,l (c) = c−1
Dj0
∂yk
ϕl dy
Note that if the inclusion Dj0
is a ball then its polarization tensor M( j0 )
has the following
explicit form:
M( j0 )
(c) =
2|Dj0
|
μj
μb
+ 1
I2
where I2 is the identity matrix of order 2 and |Dj0
| is the size of Dj0
.
3. RG-sampling method
In this section, to solve the inverse problem of monopolar and dipolar source identification,
we will adapt a method based on the linear sampling methods [10] and the reciprocity gap
concept [1]; this method was introduced by Colton and Haddar [9] for the inverse problem
of identification of an inclusion. This algorithm supposes that one is capable of measuring
u(·, x0)|∂ , ∂u
∂ν
(·, x0)|∂ , where x0 is located on the boundary and u is the total field associated
with a point source at x0. From this study, we solve the small inclusion identification problem.
To present this method, we need to introduce some notation.
We define the reciprocity gap functional R by
R(u, v) =
∂
u
∂v
∂ν
− v
∂u
∂ν
ds,
where u and v are two functions defined on a neighborhood of ∂ .
Let ˜S be the single-layer operator defined by
˜S : L2
( ) −→ V : g −→ ˜Sg,
where
˜Sg(x) = b(x, y)g(y) ds(y) for x ∈
and
b(x, y) =
i
4
H(1)
0 (kb|x − y|),
where H(1)
0 denotes the Hankel function of the first kind of order 0 and
V = v ∈ H1
loc(R2
)/ v + k2
bv = 0 in ∪m
j=1{Sj} .
We consider the problem: for z ∈ to find gz ∈ L2
( ) solution of
R(u(·, x0), ˜Sgz) = R(u(·, x0), b(z, ·)) ∀ x0 ∈ . (8)
Let F be the operator associated with equation (8) defined by
F : L2
( ) −→ L2
( )
g −→ Fg(x0) = (x0, y) g(y) ds(y); x0 ∈ ,
(9)
4

where ∈ L2
( × ) is defined by (x, y) := R(u(·, x), b(y, ·)).
Equation (8) is equivalent to: find gz ∈ L2
( ) such that
Fgz(x0) = R(u(·, x0), b(z, ·)), ∀ x0 ∈ .
We shall explain in the next sections how the solution to this equation enables us to determine
the position of the small inclusions and their equivalent properties. We consider first the cases
of monopolar and dipolar sources.
3.1. Application to monopolar sources
We assume in this section that contains m monopolar sources S1, . . . , Sm with respectively
intensities λ1, λ2, ..., λm ∈ R∗
+, the scattered field us
m(·, x0) has the following expression:
us
m(·, x0) =
m
j=1
λjui
b(Sj, x0) ui
b(Sj, ·). (10)
The expression of us
m(·, x0) is inspired from the asymptotic behavior of the scattered field
when contains small inclusions Dδ
j. When the inclusion has the same magnetic permeability
as , it behaves like a monopolar source zj with an equivalent intensity λj = δ2
|Dj|k2
bγ j
ε .
The inverse problem is to identify Sj and its intensity λj from the knowledge of the total
field um(·, x0) = ui
b(·, x0) + us
m(·, x0) and its normal derivative field ∂um
∂ν
measured on the
boundary ∂ .
Theorem 2. The equation
R(um(·, x0), ˜Sgz) = R(um(·, x0), b(z, ·)) ∀ x0 ∈ (11)
has a solution gz if and only if z ∈ {Sj, j = 1, . . . , m}.
If z = Sj and gz is a solution of (11), then the intensity λj is given by
λ−1
j = − b(z, y) gz(y) ds(y) − (ui
b − b)(z, z).
Proof. For all (x0, y) ∈ × , we have
(x0, y) = R (um(·, x0), b(y, ·))
= R(ui
b(·, x0), b(y, ·)) + R(us
m(·, x0), b(y, ·)).
The functions ui
b(·, x0) and b(y, ·) are solutions to the same Helmholtz equation inside .
Then
R ui
b(·, x0), b(y, ·) = 0
and
R us
m(·, x0), b(y, ·) =
m
j=1
λj ui
b(Sj, x0)R ui
b(Sj, ·), b(y, ·) .
For the same reason, since the function ui
b(Sj, ·) is the solution of
ui
b(Sj, ·) + k2
bui
b(Sj, ·) = −δSj
in ,
we obtain
R ui
b(Sj, ·), b(y, ·) = b(y, Sj ).
5

Then, we have
(x0, y) =
m
j=1
λj ui
b(Sj, x0) b(y, Sj ).
We now consider the right-hand side:
R (um(·, x0), b(z, ·)) = R ui
b(·, x0), b(z, ·) + R us
m(·, x0), b(z, ·) .
Since
b(z, ·) + k2
b b(z, ·) = −δz,
the Green representation theorem gives us
R ui
b(·, x0), b(z, ·) = −ui
b(z, x0)
From the expression of us
m we observe that
R us
m(·, x0), b(z, ·) =
m
j=1
λj ui
b(Sj, x0)R ui
b(Sj, ·), b(z, ·) .
The functions b(z, ·) and b(Sj, ·) are outgoing solutions of the same Helmholtz equation
outside . Then
R( b(Sj, ·), b(z, ·)) = 0.
Since the function (ui
b − b)(Sj, ·) ∈ H2
loc(R2
) is the solution of the Helmholtz equation
inside , using the same argument, we obtain
R us
m(·, x0), b(z, ·) = −
m
j=1
λj ui
b(Sj, x0) (ui
b − b)(Sj, z).
In conclusion, equation (11) is equivalent to
m
j=1
λjui
b(Sj, x0) b(Sj, y)gz(y) ds(y) + ui
b − b (Sj, z) = −ui
b(z, x0), ∀x0 ∈ . (12)
The functions ui
b(Sj, ·) and ui
b(z, ·) are the solutions of the Helmholtz equation in R2

and satisfy the radiation condition; it follows from the analyticity of scattered ﬁelds that this
equality is valid for x0 ∈ R2
. From the unique continuation principle equation (12) is
also valid for x0 ∈ R2
∪{Sj, z}.
Case 1. If z = Sj ∀ 1 j m. When x0 tends to z in (12), we obtain a contradiction since
ui
b(z, ·) presents a singularity at z while ui
b(Sj, z) is bounded.
Case 2. We suppose that there exists j0 ∈ {1, ..., m} such that z = Sj0
. The equality
m
j=1
λj ui
b(Sj, x0) b(Sj, y) g(y) ds(y) + ui
b − b (Sj, Sj0
) = −ui
b(Sj0
, x0)
is equivalent to (since the family of functions { b(Sj, ·), j = 1, . . . , m} is linearly
independent)
⎧
⎪⎪⎪⎨
⎪⎪⎪⎩
b(Sj, y) g(y) ds(y) + ui
b − b (Sj, Sj0
) = 0 for j = j0
b(Sj0
, y) g(y) ds(y) + ui
b − b (Sj0
, Sj0
) = −
1
λj0
.
6

For the existence of solutions, it suffices to take g as a linear combination of
b(Sj, ·), j = 1, . . . , m such that
⎧
⎪⎪⎨
⎪⎪⎩
b(Sj, ·), g L2( ) = − ui
b − b (Sj, Sj0
) for j = j0
b(Sj0
, ·), g L2( ) = − ui
b − b (Sj0
, Sj0
) −
1
λj0
.
3.2. Application to dipolar sources
We suppose that contains m dipolar sources located at C1, . . . ,Cm. The scattered field us
d
has an expression of the form
us
d (·, x0) =
m
j=1
Mj∇ui
b(Cj, x0) · ∇ui
b(Cj, ·), (13)
where Mj is an inversible matrix of order 2. The expression of us
d is inspired from the asymptotic
behavior of the scattered field when the domain contains small inclusions Dδ
j. When those
inclusions have the same electric permittivities as , they behave as dipolar sources with
equivalent polarization tensors Mj := δ2
γ j
μM( j) μj
μ0
.
The inverse problem is to identify Cj and its matrix Mj from knowledge of the total field
ud (·, x0) = ui
b(·, x0) + us
d (·, x0) and its normal derivative ∂ud
∂ν
(·, x0) measured on the boundary
∂ . For that purpose, we consider the problem: find gz ∈ L2
( ) solution of
R(ud (·, x0), ˜Sgz) = R(ud (·, x0),
∂ b
∂xp
(z, ·)) ∀ x0 ∈ , (14)
where p = 1 or 2 and z ∈ . We summarize our result in the following theorem.
Theorem 3. For p = 1, 2, equation (14) has a solution if and only if z ∈ {Cj, j = 1, ..., m}.
If z = Cj and gp
z is a solution of (14), then the matrix Mj is the solution of the system
Mj∇ b(z, ·), gp
z L2( )
+ Mj∇
∂(ui
b − b)
∂xp
(z, z) = −˜ep, p = 1, 2 (15)
where (˜e1, ˜e2) is the canonical basis of R2
.
Proof. Using the same reasoning established in the proof of theorem 2, we obtain that for
(x0, y) ∈ × ,
(x0, y) = R(us
d (·, x0), b(y, ·)) = −
m
j=1
Mj∇ui
b(Cj, x0) · ∇ b(Cj, y). (16)
On the other hand,
R(ud (·, x0),
∂ b
∂xp
(z, ·)) = R ui
b(·, x0),
∂ b
∂xp
(z, ·) + R us
d (·, x0),
∂ b
∂xp
(z, ·) .
We first observe that
R ui
b(·, x0),
∂ b
∂xp
(z, ·) =
∂ui
b
∂xp
(z, x0).
Since the functions ∂ b
∂xp
(z, ·) and Mj∇ui
b(Cj, x0) · ∇ b(Cj, ·) are the outgoing solutions of the
same Helmholtz equation outside , then
R(Mj∇ui
b(Cj, x0) · ∇ b(Cj, ·),
∂ b
∂xp
(z, ·)) = 0. (17)
7

The function ˜Hj := Mj∇ui
b(Cj, x0) · ∇(ui
b − b)(Cj, ·) ∈ H2
loc(R2
) is the solution of the
Helmholtz equation ˜Hj + k2
b
˜Hj = 0 in . Therefore, using (17) we have
R us
d (·, x0),
∂ b
∂xp
(z, ·) =
m
j=1
R Mj∇ui
b(Cj, x0) · ∇(ui
b − b)(Cj, ·),
∂ b
∂xp
(z, ·)
=
m
j=1
Mj∇ui
b(Cj, x0) · ∇
∂(ui
b − b)
∂xp
(Cj, z). (18)
Combining (16) and (18) shows that (14) is equivalent to
m
j=1
Mj∇ui
b(Cj, ·) · ∇ b(Cj, y)g(y) ds(y) + ∇
∂(ui
b − b)
∂xp
(Cj, z)
= −
∂ui
b
∂xp
(z, ·) on . (19)
The functions
∂ui
b
∂xp
(z, ·) and
∂ui
b
∂xp
(Cj, ·), for p = 1, 2, are the solutions of the Helmholtz equation
in R2
¯ and satisfy the radiation condition. From the analyticity of the scattered field and
unique continuation principle, equation (19) is valid in R2
∪{Cj, z}.
Case 1. If z = Cj; 1 j m. We obtain a contradiction in (19) since
∂ui
b
∂xp
(z, ·) presents a
singularity at z, while
∂ui
b
∂xp
(Cj, z) is bounded for j = 1, . . . , m.
Case 2. Assume that there exists j0 ∈ {1, . . . , m} such that z = Cj0
. To obtain a solution to
equation (19), it suffices to find gp
z such that
⎧
⎪⎪⎪⎨
⎪⎪⎪⎩
Mj∇ b(Cj, ·), gp
Cj0 L2( )
= −Mj∇
∂(ui
b − b)
∂xp
(Cj,Cj0
) for j = j0
Mj0
∇ b(Cj0
, ·), gp
Cj0 L2( )
= −˜ep − Mj0
∇
∂(ui
b − b)
∂xp
(Cj0
,Cj0
).
This is possible, since Mj∇ b(Cj, ·) , j = 1, . . . , m, for = 1, 2, are linearly
independent functions of L2
( ) (we used here the fact that Mj is inversible).
Using this fact, reciprocally we also obtain that for any solution gp
z , for p = 1, 2,
to (19), the matrix Mj0
is a solution of the system
Mj0
∇ b(Cj0
, ·), gp
Cj0 L2( )
= −˜ep − Mj0
∇
∂(ui
b − b)
∂xp
(Cj0
,Cj0
).
3.3. The case of monopolar and dipolar sources
We suppose that contains m1 monopolar sources S1, . . . , Sm1
with respective intensities
λ1, λ2, . . . , λm1
and m2 dipolar sources C1, . . . ,Cm2
with respective polarization tensors
M1, . . . , Mm2
. The scattered field us
(·, x0) has the following expression:
us
(·, x0) =
m1
j=1
λjui
b(Sj, x0) ui
b(Sj, ·) +
m2
=1
M ∇ui
b(C , x0) · ∇ui
b(C , ·). (20)
The inverse problem is to identify Sj and its intensity λj, C and its polarization tensor M .
We can find the set of point sources in two steps. In the first step, we can find the set
of monopolar sources by considering equation (11). In the second step, to find the dipolar
sources, we consider equation (14). To prove this result, we proceed as in theorems 2 and 3,
using the fact that the functions ui
b(Sj, ·),
∂ui
b
∂x1
(C , ·), and
∂ui
b
∂x2
(C , ·) are linearly independent
in L2
( ). We summarize this result in the following theorem.
8

Theorem 4. Equations (11) and (14) have the solutions gz and gp
z , respectively, if and only if
z ∈ {Sj, j = 1, . . . , m1} and z ∈ {C , = 1, . . . , m2}, respectively.
If z = Sj and gz is a solution of (11), then the intensity λj is given by
λ−1
j = − b(z, y) gz(y) ds(y) − (ui
b − b)(z, z).
If z = C and gp
z is a solution of (14), then the matrix M is a solution of the system
M ∇ b(z, ·), gp
z L2( ) + M ∇
∂(ui
b − b)
∂xp
(z, z) = −˜ep, p = 1, 2.
where (˜e1, ˜e2) is the canonical basis of R2
.
3.4. Small inclusions identification
We suppose now that contains m small inclusions and we have the situation described in
section 1. We recall the field ˜us
(·, x0),
˜us
(·, x0) :=
m
j=1
γ j
μM( j) μj
μb
∇ui
b(Sj, x0) · ∇ui
b(Sj, ·) + |Dj|k2
bγ j
ε ui
b(Sj, x0)ui
b(Sj, ·).
Following theorem 1, the field δ2
˜us
(·, x0) is an approximation of the scattered field us
δ(·, x0)
when δ is small.
We denote by ˜uδ(·, x0) := ui
b(., x0)+δ2
˜us
(·, x0), and by Fδ and ˜Fδ the following operators:
Fδ, ˜Fδ : L2
( ) −→ L2
( ), (21)
where for g ∈ L2
( ) and x0 ∈ :
Fδ g(x0) = R(uδ(·, x0), b(y, ·)) g(y) ds(y), (22)
and
˜Fδ g(x0) = R( ˜uδ(·, x0), b(y, ·)) g(y) ds(y). (23)
We recall that when these inclusions have the same electric permittivities as , they behave as
dipolar sources with equivalent polarization tensors Mj = δ2
γ j
μM( j)
(
μj
μb
) and when they have
the same magnetic permeabilities as , they behave as monopolar sources with equivalent
intensities λj = δ2
|Dj|k2
bγ j
ε . We denote in the following by b(z, ·), for z ∈ , the function
b(z, ·) or ∂ b
∂xp
(z, ·).
Let η > 0, gη
δ,z and ˜gη
δ,z solutions in L2
( ) respectively of the following equations:
(ηI + F∗
δ Fδ )gη
δ,z = F∗
δ R(uδ(·, x0), b(z, ·)) (24)
(ηI + ˜F∗
δ
˜Fδ )˜gη
δ,z = ˜F∗
δ R( ˜uδ(·, x0), b(z, ·)). (25)
We obtain the following result.
Theorem 5. If gη
δ,z and ˜gη
δ,z are respectively the solutions of (24) and (25), then
(i)
lim
δ−→0
1
δ2
gη
δ,z − ˜gη
δ,z L2( ) = 0.
9

(ii) Assume that z = Sj and b = b in (24). Then
lim
δ−→0
|λ−1
j + b(z, y) gη
δ,z(y) ds(y) − ui
b − b (z, z)| = 0 (26)
for η = η(δ) Cδ2
, with C being a constant independent of δ and such that η(δ) goes to
0 as δ −→ 0.
(iii) Assume that z = Sj and gη,p
δ,z the solution of (24) with b = ∂ b
∂xp
. Then
lim
δ−→0
Mj∇ b(z, ·), gη,p
δ,z L2( )
+ Mj∇
∂(ui
b − b)
∂xp
(z, z) + ˜ep = 0 (27)
for η = η(δ) as in point (ii).
Proof. To prove point (i) of the theorem, we first need to prove that
Fδ − ˜Fδ L(L2( )) Cδ2
o(δ),
where C is a constant independent of δ. This is a direct consequence of the fact that
(Fδ − ˜Fδ ) (g)(x0) = R us
δ(·, x0) − δ2
˜us
(·, x0), b(y, ·) g(y) ds(y)
and theorem 1.
We also observe that
Fδ L(L2( )) C δ2
, ˜Fδ L(L2( )) C δ2
and consequently
F∗
δ Fδ − ˜F∗
δ
˜Fδ L(L2( )) C δ4
,
for some constant independent from δ. Taking the difference between equations (24) and (25),
and then performing the scalar product in L2
( ) with gη
δ,z − ˜gη
δ,z, we obtain
η gη
δ,z − ˜gη
δ,z
2
L2( )
+ ˜Fδ gη
δ,z − ˜gη
δ,z
2
L2( )
+ (F∗
δ Fδ − ˜F∗
δ
˜Fδ )gη
δ,z, gη
δ,z − ˜gη
δ,z L2( )
= ( ˜F∗
δ − F∗
)R(uδ(·, x0), b(z, ·)), gη
δ,z − ˜gη
δ,z L2( )
+ ˜F∗
δ R (us
δ − δ2
˜us
δ )(·, x0), b(z, ·) , gη
δ,z − ˜gη
δ,z L2( )
.
Then
gη
δ,z − ˜gη
δ,z L2( )
1
η
Fδ − ˜Fδ L(L2( ))[ Fδ L(L2( )) + ˜Fδ L(L2( ))] gη
δ,z L2( )
+
1
η
Fδ − ˜Fδ L(L2( )) R(uδ(·, x0), b(z, ·)) L2( )
+
1
η
˜Fδ L(L2( )) R us
δ − δ2
˜us
δ (·, x0), b(z, ·) L2( )
From the first step of the proof and theorem 1 we obtain
1
δ2
gη
δ,z − ˜gη
δ,z L2( )
C
o(δ)
η
which ends the proof of the first point.
From theorem 2:
λ−1
j = − b(z, y) ˜gδ,z(y) ds(y) − (ui
b − b)(z, z),
10

where ˜gδ,z is a solution for equation (25) with η = 0 constructed as the limit as η goes to 0 of
˜gη
δ,z. To prove the second point of the theorem, it suffices to show that
lim
δ−→0
b(z, y)gη
δ,z(y) ds(y) − b(z, y)˜gη
δ,z(y) ds(y) = 0
for η = η(δ) tends to 0 as δ tends to 0. This result is true since
b(z, y)gη
δ,z(y) ds(y) − b(z, y)gz(y) ds(y) C b(z, ·) L2( )
δ2
o(δ)
η
.
For the last point of the theorem, we first observe that
1
δ4
δ,z − ˜gη,p
δ,z L2( )
Mj
δ2
∇ b(z, ·) L2( )
1
δ2
gη,p
δ,z − ˜gη,p
δ,z L2( ).
Then
lim
δ−→0
1
δ4
δ,z − ˜gη,p
δ,z L2( )
= 0. (28)
From theorem 3, Mj is the solution of the system
Mj∇ b(z, ·), ˜gp
δ,z L2( )
= −˜ep − Mj∇
∂(ui
b − b)
∂xp
(z, z), p = 1, 2, (29)
where ˜gp
δ,z is a solution for equation (25) with η = 0 constructed as the limit as η tends to 0 of
˜gη,p
δ,z . Identity (27) follows directly from (28) and (29).
4. Numerical results
4.1. Description of the algorithm
This section is dedicated to some numerical experiments that test the validity of the inversion
procedure suggested by theorem 5. This procedure is based on determining the function gη
δ,z,
solution of equation (24). Our algorithm consists of the following steps.
(i) Select a grid of sampling points in a region known to contain the small inclusions.
(ii) Compute for each sampling point z the solutions gη
δ,z of (24) with the three possible choices
of right-hand side. The solutions are computed using a singular value decomposition of
the operator Fδ. We also use the Morozov discrepancy principle to determine η assuming
some noise level on the operator Fδ.
(iii) Plot the contours of the functions
z −→
1
gη,0
δ,z L2( )
and z −→
1
gη,1
δ,z L2( )
+
1
gη,2
δ,z L2( )
(30)
to identify the locations of the small inclusions. The latter correspond with the peaks
of these functions. The function gη,0
δ,z corresponds with b = b and serves for the
identification of monopolar sources. The functions gη,p
δ,z , for p = 1, 2, correspond with
b = ∂ b/∂xp and serve for the identification of dipolar sources (see theorem 5).
(iv) Evaluate λj or Mj using respectively formulas (26) or (27).
From the implementation point of view let us quote that there exists an issue with the
numerical evaluation of Fδ. For the case of small inclusions, the contribution of the scattered
field is in general small compared to the incident field. Therefore, evaluating
R(u(·, x0), v)
11

inclusions
z
Γ
Ω
Σ
x0
Σ
ui
b(·, x0)
Figure 2. Configuration of the experiments.
for some test function v requires a very precise quadrature rule in order to take into account
the contribution of us
(·, x0). To overcome this difficulty, we split R(u(·, x0), v) into
R(u(·, x0) − b(·, x0), v) + R( b(·, x0), v),
where the second term can be computed analytically and in the first term the quantity
u(·, x0) − b(·, x0) is in general of comparable magnitude as us
(·, x0).
4.2. Numerical tests
All examples use synthetic data obtained by solving the forward scattering problem (with
the small inclusions). These data are then corrupted with pointwise relative random noise.
The noise magnitude is 1% except in one example where the magnitude is 20%, showing the
robustness of our method with respect to this type of noise. The domain is a square of the
center (0, 0) and a length = 4λ, with λ = 1 being the wavelength outside . The boundary
is the union of four segments of length = 3λ distant by λ/2 from ∂ , and on which 16 × 4
point sources x0 are uniformly distributed (see figure 2). We shall also give some examples
where the aperture of the sources is reduced. We use a 40×40 sampling points in the sampling
domain.
In all subsequent figures, the square with the solid line represents the boundary of ,
i.e. the measurements location. The dash–dotted line represents the boundary of the sampling
domain. We also indicate in these figures the wavelength λ in the vacuum and the index
n :=
√
εbμb of the medium inside .
4.2.1. The case of small monopolar inclusions. In the first example, contains two circular
inclusions having the same radius r1 = r2 = λ
14
and the same magnetic permeability as ,
μb = μj = 1, j = 1, 2. The electric permittivities inside and inside the inclusions are,
respectively, εb = 2 and εj = 4, j = 1, 2. In this experiment, we shall vary the distance
between the two inclusions in order to test the resolution of the identification procedure. The
numerical results show that if the distance between the two inclusions is larger than 0.45λ√
2
, then
their positions can be identified; see figures 3 and 4. For smaller distances, only the region that
contains both scatterers can be identified; see figure 5.
In the following numerical experiment (figure 6), we shall test the accuracy of the
reconstruction of the (equivalent) intensities in terms of the size of the small scatterer. The
12

−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z1
z2
λ n =2
Figure 3. Reconstruction of two circular inclusions distant by 0.6λ/
√
2.
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z1
z
2
λ n =2
√
2.
Table 1. Reconstruction of the equivalent intensities related to the small inclusions of figure 6.
Radius λ/100 λ/10 λ/7
λexact
1 1.21 0.01 2.48
λnumer
j 1.09 0.01 1.63
electric permittivity and the magnetic permeability inside are respectively εb = 2 and
μb = 1. We consider a circular inclusion centered at S1(−0.5, 0.6), with electric permittivity
and magnetic permeability ε1 = 4 and μ1 = 1 and consider the cases of three different
values of the radius: r1 = λ
100
, λ
10
and λ
7
. We observe in figure 6 that the location of the
small scatterer is well reconstructed in the three cases. However, as shown in table 1, the
accuracy of the identified equivalent intensity λ1 is not the same: the identified intensity λnum
1
is calculated using the formula of theorem 2 and the exact intensity λexact
1 is given by the
13

−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z1
z2
λ n =2
√
2.
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z
1
z
2
λ n =2
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z
1
z
2
λ n =2
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z
1
z
2
λ n =2
Figure 6. Reconstruction of small circular inclusions of radius r1 = λ
100 (left), r1 = λ
10 (middle)
and r1 = λ
7 (right).
Table 2. Equivalent intensities for the inclusions reconstructed in figure 7.
Center Radius λexact
j λnumer
j
(0.5, 0.2) λ/14 0.62 0.61
(−0.7, 0.5) λ/17 0.39 0.37
(0.5, 0.5) λ/24 0.22 0.22
(−0.5, 0.6) λ/36 0.09 0.09
asymptotic expansion (6). We explain the loss of precision in the case r1 = λ
7
by the fact that
the asymptotic expansion (6) would not be sufficiently accurate and in the case r1 = λ
100
by
the fact that the inclusion is too small and therefore the data are very sensitive to noise.
In the last example, we consider the case of multiple small inclusions. Figure 7 shows
the reconstruction of four circular inclusions centered at S1 = (0.5, 0.2), S2 = (−0.7, 0.5),
S3 = (0.5, 0.5) and S4 = (−0.5, −0.6) and with respective radii, r1 = λ
14
, r2 = λ
17
, r3 = λ
24
and r4 = λ
36
. These inclusions have the same magnetic permeability as the background: μb = 1
and the same electric permittivity ε1 = ε2 = ε3 = ε4 = 4. The electric permittivity inside
is εb = 2. We observe that a very good identification of the location is obtained. Table 2
indicates the identified intensities are also very accurate.
14

−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z
1
z
2
λ n =2
Figure 7. Numerical reconstruction of four small circular (monopolar) inclusions centered at
S1 = (0.5, 0.2), S2 = (−0.7, 0.5), S3 = (0.5, 0.5) and S4 = (−0.5, −0.6).
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z
1
z2
λ n =2
Figure 8. Similar experiment as figure 7 but with 20% pointwise relative random noise.
We end this subsection with two experiments. The first, illustrated by figure 8, shows the
results obtained with data corrupted with 20% pointwise relative random noise. One clearly
observes that our method is very robust with respect to this type of noise. This is due to the
fact that this noise is filtered by the reciprocity gap functional. This has also been observed in
previous works on this type of method.
The following example shows the results obtained when we reduce the aperture of sources
used. In figure 9 (left), the source lines on the left of the domain only are used (see figure 2).
We observe how one cannot avoid the echo between the two inclusions that are on the same
line perpendicular to the source location. In figure 9 (right), we use only two source lines, the
one on the left and the other on the bottom of the domain (see figure 2). We observe that these
two lines are sufficient to obtain satisfactory results.
15

−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z1
z
2
λ n =2
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z1
z
2
λ n =2
Figure 9. Similar experiment as figure 7 but with limited aperture for the sources used. Left: the
source lines on the left of the domain only are used (see figure 2). Right: only two source lines are
used, the one on the left and the one at the bottom of the domain (see figure 2).
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z1
z
2
λ n =2
Figure 10. Reconstruction of four dipolar circular small inclusions of the same radius and centered
at S1 = (0, 0.2), S2 = (0, −0.1), S3 = (0.6, −0.7) and S4 = (−0.5, −0.5).
4.2.2. The case of small dipolar inclusions. Figure 10 illustrates the reconstruction of
four circular inclusions centered at S1 = (0, 0.2), S2 = (0, −0.1), S3 = (0.6, −0.7) and
S4 = (−0.5, −0.5), having the same radius r = λ
35
, the same electrical permittivity as the
background εb = 1 and the same magnetic permeability μj = 3, j = 1, . . . , 4. The magnetic
permeability inside is μb = 1. The algorithm gives an accurate estimate for the location and
is able to separate inclusions that are distant of λ/2. The polarization tensors calculated from
the system given in theorem 3 associated with these inclusions are respectively
M1 =
0.0013 0
0 0.0012
, M2 =
0.0013 0
0 0.0012
,
M3 =
0.0013 0
0 0.0013
, M4 =
0.0011 0
0 0.0012
.
16

−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z
1
z
2
λ n =2
Figure 11. Reconstruction of four dipolar circular small inclusions centered at S1 = (0, 0.2), S2 =
(0, −0.1), S3 = (0.6, −0.7) and S4 = (−0.5, −0.5) and of respective radius r1 = λ
17 , r2 = λ
23
and r3 = r4 = λ
35 .
The exact polarization tensor given by the asymptotic expansion (6) is equal to 0.0013 I2,
where I2 is the identity matrix of R2
. In figure 11, we illustrate the sensitivity of the numerical
algorithm with respect to the size of the inclusion. We consider the same configuration as
in figure 10 but change the radii to r1 = λ
17
, r2 = λ
23
and r3 = r4 = λ
35
. The identification
of the position is shown in figure 11 where one observes a similar accuracy as in figure
10. The corresponding polarization tensors calculated from the system given in theorem
3 are
M1 =
0.0048 0
0 0.0046
, M2 =
0.0028 0
0 0.0028
,
M3 =
0.0015 0
0 0.0013
, M4 =
0.0012 0
0 0.0012
,
while the exact polarization tensors given by (6) are respectively
M1 = 0.005 I2, M2 = 0.0028 I2, M3 = M4 = 0.0013 I2.
4.2.3. The case of a mixture between small dipolar and monopolar inclusions. The
last numerical experiment focuses on the case of the presence of dipolar and monopolar
inclusions. The domain has an electric permittivity εb = 2 and a magnetic permeability
μb = 1 and contains four circular inclusions of the same radius r = λ
33
. Two of these
inclusions, respectively, centered at S1 = (0.3, 0) and S2 = (−0.4, 0) have the same electric
permittivity ε1 = ε2 = 3 and the same magnetic permeability as . The two other inclusions
with respective centers S3 = (0.5, 0.5) and S4 = (−0.5, −0.6) have the same magnetic
permeability μ3 = μ4 = 3 and have the same electric permittivity as . Figure 12 (left) shows
the reconstruction of the monopolar inclusions performed by the contours of
z −→
1
gη,0
δ,z L2( )
,
17

−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z
1
z2
λ n =2
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
z
1
z2
λ n =2
Figure 12. Identification of a mixture between monopolar and dipolar small inclusions.
Left: contours of z −→ 1
g
η,0
δ,z L2( )
. Right: z −→ 1
g
η,1
δ,z L2( )
+ 1
g
η,2
δ,z L2( )
.
while figure 12 (right) shows the reconstruction of the dipolar inclusions performed by the
contours of
z −→
1
gη,1
δ,z L2( )
+
1
gη,2
δ,z L2( )
.
We observe in both cases very accurate reconstructions.
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