direct algorithms for solving some inverse source problems in 2d...
TRANSCRIPT
Direct algorithms for solving some inverse source
problems in 2D elliptic equations
Batoul Abdelaziz1, Abdellatif El Badia1 and Ahmad El Hajj1
February 9, 2017
AbstractThis paper deals with the resolution of some inverse source problems in the 2D elliptic equation∆u + µu = F from Cauchy data. Two types of sources are considered, pointwise sources andsources having compact support within a finite number of small subdomains. An identificationdirect algorithm, based on an algebraic approach, is proposed. This is a new result, as far as weknow, except in the case µ = 0 which is already considered in [14].
AMS Classification: 35R30, 35B25, 35B35, 35J05, 35J08, 65M32Key words: 2D Helmholtz’s equation, bioluminescence tomography, EEG/MEG, pointwisesources, inverse source problems, algebraic algorithm.
1 Introduction
Inverse problems (IP) are of increasing importance in several applied domains. Among these, theinverse source problems (ISP) have attracted great attention of many researchers, particularly overrecent years because of their role in many practical domains. Beside their applications to pollutionin the environment [15, 21] and dislocation problems [13], these inverse problems have been widelyused in several biomedical imaging techniques as the photo- and thermo-acoustic tomography [5,26], electroencephalography/magnetoencephalography (EEG/MEG) problems [17, 23] and opticaltomography [6] including bioluminescence tomography (BLT) [28] and fluorescence tomography(FT) [8].
Let Ω be an open bounded domain in R2 and µ a given real number. In this paper, we consider the
problem of determining a source F in the 2D elliptic equation
∆u+ µu = F in Ω (1.1)
from boundary measurements.
The inverse problem that we consider, here, can be motivated by several applications, nevertheless,we focus only on three of them: the interior Helmholtz’s equation, bioluminescence tomography(BLT) and fluorescence tomography (FT). Based on the underlying physical motivations, two typeof sources are considered:
1. Monopolar pointwise sources of the form
F =m∑
j=1
λjδSjSj ∈ Ω, λj 6= 0 (1.2)
where δS stands for the Dirac distribution at point S, m is a nonnegative integer and λj is a non-nullscalar quantity.
1Sorbonne University, Université de Technologie de Compiègne, Laboratoire de Mathématiuqes Ap-
pliquées de Compiègne, LMAC Laboratory of Applied Mathematics of Compiègne, 60205 Compiègne Cedex,
France.
[email protected], [email protected], [email protected]
1
2. Sources having compact support within a finite number of small subdomains Dj , namely,
F =m∑
j=1
hjχDj with Dj = Sj + εBj ⊂ Ω (1.3)
where hj is a non-null function belonging to the space L1(Ω), Bj is a bounded domain in R2 con-
taining the origin and ε is a positive real number sufficiently small representing the common orderof magnitude of the diameters of the subdomains Dj . In the following, we focus our work only onthe case 0 < ε < 1, which amounts to consider a domain of a size smaller than 1. However, this doesnot restrict the generality since we can always be brought back to this framework by using a suit-able rescaling argument. In both cases, the points Sj = (aj , bj) are assumed to be mutually distinct.
Our aim is to reconstruct the source term F from the Cauchy data (f, g) := (u|Γ ,∂u∂ν |Γ
) prescribed
on a sufficiently regular boundary Γ of Ω. Here, ν = (ν1, ν2) denotes the outward unit normal toΓ. However, as the used techniques in this paper are valid in both cases µ > 0 and µ < 0, andfor a lighter reading, although some results corresponding to a negative µ are also presented (thosecorresponding to BLT and FT inverse problems), our focus will be on the 2D Helmholtz case, wherewe denote µ = κ2. For this Helmholtz problem, two approaches will be considered, firstly having asingle high frequency κ and then using multiple frequencies.
The three-dimensional version of the equation under study has been widely discussed in the liter-ature using algebraic approaches for both cases, µ = 0 and µ 6= 0. The question of the monopolesidentification, using the so-called algebraic method, has been initially considered in [14] for the Pois-son equation (corresponding to the case µ = 0) because of their interest in the inverse EEG/MEGproblem. This work has been revisited and extended in [9, 24] for a combination of monopolesand dipoles and recently in [10, 25] for sources of general order poles. The proposed algorithms inthe latter papers are based on the invertibility of a Hankel-type matrix using the calculation of itsdeterminant. Later, the algebraic method proposed in [14] has been extended to the 3D Helmholtzequation (µ 6= 0) in [16] for monopoles. Then, in [1, 2], we generalized the work in [14] using asimpler and more direct algebraic method, considering a general type of sources. Note that thealgorithms presented in [1, 2] treat also sources of the form (1.3) in the 3D case, while previousworks have considered only sources supported by hollow and solid spheres. On the other hand, inthe 2D case, the authors in [14], have also identified the monopoles in the case µ = 0. However, forµ 6= 0, the reconstruction of monopoles and sources with small support has not yet been treatedusing a direct method such as the algebraic method.
Let us mention that, recently, Kress and Rundell have considered, in [22], the reconstruction ofmonopolar sources for the 2D Helmholtz problem. Their basic idea, for a κ small enough, is toapproach iteratively the 2D Helmholtz equation by a problem on the 2D Laplace equation, solvingthe latter as done in their former work in [18]. Moreover, one can mention the work of Ikehata in[20], where the author has considered the Helmholtz problem in two-dimensional space with sourcesF of either the form, χ
Bρ(x), where B is an open subset of Ω and χB its characteristic function,
or F = div[ρ(x)χB(x)a] with a a nonzero constant vector. Then, under additional conditions, theconvex hull of B was reconstructed iteratively using the Cauchy data. One can also refer, in thecase of the exterior Helmholtz problem, to the paper [4] concerning monopolar sources having aknown number where an iterative scheme was proposed to reconstruct these sources. Furthermoreand always in the Helmholtz equation schema, considering a source term F ∈ L2(Ω) and usingboundary multiple frequency measures, Bao et al have proved in [7] a uniqueness result and a sta-bility estimate for the source term taking frequencies that vary within a set with an accumulationpoint. Similar results accompanied by an identification algorithm are presented by Acosta et al in[3] in a heterogenous medium.
The remainder of this paper is organized as follows. First, in Section 2, we consider the inverseHelmholtz problem (1.1)-(2.1) at a single fixed wavenumber κ. Then, Section 3 is devoted tosolve the multiple-frequency inverse Helmholtz problem. Finally, numerical results are performedin Section 4 considering the algorithm employed for the latter case.
2
2 Identification algorithm using a single wavenumber (κ > 1)
In this section, we aim to reconstruct the source parameters from a single Cauchy data using asingle wavenumber. More precisely, Subsection 2.1 is devoted to the statement of the problem. InSubsection 2.2, we treat the case with monopolar sources (1.2) and then later in Subsection 2.3 wedeal with the case of sources with small supports (1.3).
2.1 Statement of the problem
Let F be a source of the form (1.2) or (1.3) and consider the following mapping in H12 (Γ)×H− 1
2 (Γ)
Λ : F → (u|Γ ,∂u
∂ν |Γ
).
Then, the inverse problem is formulated as follows:
Given (f, g) ∈ H12 (Γ)×H− 1
2 (Γ), determine F such that Λ(F ) = (f, g). (2.1)
Remark 2.1.
1. For monopolar sources (1.2), the uniqueness issue can be easily attained. It can be obtainedby means of Holmgren’s theorem and the regularity of the direct problem, that is u ∈ H1−s(Ω)with s > 0, as it is done in the 3D case [16].
2. In the case of sources having compact support (1.3), the uniqueness is not guaranteed as itis shown in the counter-example given in [16, Subsection 5.1] with F = hχD. However,the algorithm proposed in this paper enables us to reconstruct uniquely, modulo a small error,some characteristics of the source F , particularly the number of the domains Dj , their centersand some quantities related to the densities hj and the domains Bj .
2.2 Pointwise sources
In this subsection, we seek to solve the inverse problem (2.1) associated to equation (1.1) with thesource term of the form (1.2). More precisely, it consists in identifying the sources number m, theintensities λj and the 2D locations Sj given a single Cauchy data (f, g). This section is dividedinto three subsections. First, in Subsection 2.2.1, we extend, by means of a change of variables, the2D Helmholtz equation (1.1) to a new three-dimensional Helmholtz equation defined over Ω × R.Then, Subsection 2.2.2 is devoted to establish the new relationships between the Cauchy data (f, g)and the sources parameters (m,λj ,Sj). Finally, Subsection 2.2.3 presents the identification methodemployed to reconstruct the sources using these algebraic relations.
2.2.1 Transformation of the 2D Helmholtz equation
Due to the complexity of the 2D Helmholtz equation and the absence of a direct method for thesources reconstruction in such a case, our basic idea is to extend the 2D problem to an equivalent3D Helmholtz problem via a suitable change of variable. Then, we solve the latter 3D problemusing the tools developed in [1, 2]. Before doing so, we need to introduce some notations.
For all ω ∈ R and η > 0, we set
ρ =√
ω2 + κ2 (2.2)
and, as seen in Fig. 1, we denote
Γ = Γ× [−η, η], Γ+ = Ω× η, Γ− = Ω× −η.
Then, using the change of variables
v(x, y, z) = u(x, y)e−iωz where i2 = −1 and z ∈ [−η, η], (2.3)
3
we can check that, if u is a solution of (1.1)-(1.2) with µ = κ2 then, the function v satisfies thesystem
∆v + ρ2v = e−iωzm∑
j=1
λjδSjin Ω× [−η, η]
(v, ∂v∂ν
) = (fe−iωz, ge−iωz) on Γ
(v, ∂v∂z
) = (ue−iωη,−iωue−iωη) on Γ+
(v,− ∂v∂z
) = (ueiωη, iωueiωη) on Γ−
(2.4)
where ν = (ν, 0) is the outward unit normal to Γ×R. Thus, the problem, now, is to establish rela-tionships between the Cauchy data pair (f, g) and the source parameters (m,λj ,Sj) from equation(2.4).
Fig. 1: The new 3D domain.
Remark 2.2. Note that the former methods employed to reconstruct 3D monopoles aren’t validanymore. This is due to the fact that the sources, that we aim to reconstruct in the 3D equation(2.4), are not pointwise sources, but sources supported by lines. This necessitates the use of a newresolution technique and forms the object of the following.
2.2.2 Reciprocity gap formulae
The principal aim of this section is to establish a reciprocity formula that permits to relate thegiven Cauchy data (f, g) to the characteristics (m,λj , Sj) of the source F . This is accomplished bymultiplying (2.4) by a test function, a solution of the homogenous equation (2.7), here,
ϕnω(x, y, z) = (y + iz)ne−ixρ, (2.5)
integrating by parts, using Green’s formula and then passing to the limit η → +∞. In this caseand as shown later in Theorem 2.3, the left-sided term of (2.4) leads, for all n ∈ N, to the operatorR(n, f, g) defined as:
4
R(n, f, g) :=n∑
α=0
(
n
α
)
(−1)α∫
Γ
yn−αg
(∫
R
e−ix√
ω2+κ2
δ(α)(ω)dω
)
ds
+i
n∑
α=0
(
n
α
)
(−1)α∫
Γ
ν1yn−αf
(∫
R
√
ω2 + κ2 e−ix√
ω2+κ2
δ(α)(ω)dω
)
ds
−nn−1∑
α=0
(
n− 1
α
)
(−1)α∫
Γ
ν2yn−1−αf
(∫
R
e−ix√
ω2+κ2
δ(α)(ω)dω
)
ds
(2.6)
where δ(α) indicates the αth derivative of Dirac distribution delta,
ν = (ν1(x, y), ν2(x, y)) := (ν1, ν2) and
(
n
α
)
=
n!α!(n−α)!
if n ≥ α
0 if n < α.
Thus, the reciprocity gap formulae, behind our algebraic identification method, are stated as follows.
Theorem 2.3. Let (f, g) ∈ H12 (Γ)×H− 1
2 (Γ) and let u be the corresponding solution of (1.1) withµ = κ2. Then,
R(n, f, g) =m∑
j=1
n∑
α=0
(
n
α
)
(−1)αλjbn−αj
∫
R
e−iaj
√ω2+κ2
δ(α)(ω)dω, ∀n ∈ N.
Note that, in the stated theorem and also in formula (2.6), the integrals with respect to ω are tobe understood in the duality sense.
Proof. The proof of Theorem 2.3 is done in two steps.Step 1. Since the test functions ϕn
ω defined in (2.5) satisfy the homogenous equation
∆v + ρ2v = 0 in Ω× R, (2.7)
then, multiplying (2.4) by ϕnω, integrating by parts and using Green’s formula, we obtain
Rη(ϕnω, f, g) =
m∑
j=1
λj
∫ η
−η
ϕnω(aj , bj , z)e
−iωz dz (2.8)
where
Rη(ϕnω, f, g) =
∫
Γ
∫ η
−η
(
g ϕnω − f
∂ϕnω
∂ν
)
e−iωzdzds−∫
Ω
(
iωu ϕnω + u
∂ϕnω
∂z
)
|z=η
e−iωηdxdy
+
∫
Ω
(
iωu ϕnω + u
∂ϕnω
∂z
)
|z=−η
eiωηdxdy.
Let θ ∈ C∞(R) be a function with a compact support (i.e. θ ∈ C∞c (R)) such that θ(ω) = 1 over
[
− η
2, η
2
]
. Then, multiplying (2.8) by θ(ω) and integrating, with respect to ω, over R lead to
∫
R
θ(ω)Rη(ϕnω, f, g)dω =
m∑
j=1
λj
∫
R
∫ η
−η
θ(ω)ϕnω(aj , bj , z)e
−iωzdz dω. (2.9)
Now, we desire to get the reciprocity gap formulae, given in Theorem 2.3, by passing to the limitη → +∞ in the previous equation. This will be the object of the following step.
Step 2. To justify the passage to the limit in (2.9), it is sufficient to examine the convergence,when η → +∞, of all the terms involved in (2.9), denoted by
Iη1 =
∫
R
∫
Γ
∫ η
−η
θ(ω)
(
g ϕnω − f
∂ϕnω
∂ν
)
e−iωzdzdsdω,
Iη2 =
∫ η
−η
∫
R
θ(ω)ϕnω(aj , bj , z)e
−iωz dω dz,
Iη+ =
∫
R
∫
Ω
θ(ω)
(
iωu ϕnω + u
∂ϕnω
∂z
)
|z=η
e−iωηdxdydω,
Iη− =
∫
R
∫
Ω
θ(ω)
(
iωu ϕnω + u
∂ϕnω
∂z
)
|z=−η
eiωηdxdydω.
5
Indeed, using (2.5), the binomial formula and since the Fourier transform
∫
R1[−η,η](iz)
αe−iωzdz,
where 1[−η,η] is the characteristic function of [−η, η], converges when η → +∞ to 2π(−1)αδ(α), inthe distributions sense, one has
limη→+∞
Iη1 = 2π
n∑
α=0
(
n
α
)
(−1)α∫
Γ
yn−αg
(∫
R
e−ix√
ω2+κ2
δ(α)(ω)dω
)
ds
+2πin∑
α=0
(
n
α
)
(−1)α∫
Γ
ν1yn−αf
(∫
R
√
ω2 + κ2 e−ix√
ω2+κ2
δ(α)(ω)dω
)
ds
−2πn
n−1∑
α=0
(
n− 1
α
)
(−1)α∫
Γ
ν2yn−1−αf
(∫
R
e−ix√
ω2+κ2
δ(α)(ω)dω
)
ds.
(2.10)
Similarly, we obtain
limη→+∞
Iη2 = 2π
n∑
α=0
(
n
α
)
(−1)αbn−αj
∫
R
e−iaj
√ω2+κ2
δ(α)(ω)dω. (2.11)
To achieve the proof, one has to prove that
limη→+∞
Iη± = 0. (2.12)
It suffices to show the result (2.12) for Iη+ since the case of Iη− is proved analogously. First, we cansee that
(
iωu ϕnω + u
∂ϕnω
∂z
)
|z=η
= iue−ix√
ω2+κ2
[
ωn∑
α=0
(
n
α
)
yn−α(iη)α + n
n−1∑
α=0
(
n− 1
α
)
yn−1−α(iη)α]
.
Thus,
Iη+ =n∑
α=0
∫
Ω
∫
R
θ(ω)fα(x, y, ω)(iη)αe−iωηdωdxdy +
n−1∑
α=0
∫
Ω
∫
R
θ(ω)gα(x, y, ω)(iη)αe−iωηdωdxdy
where
fα(x, y, η, ω) = iωue−ix√
ω2+κ2
(
n
α
)
yn−α and gα(x, y, η, ω) = inue−ix√
ω2+κ2
(
n− 1
α
)
yn−1−α.
Furthermore, since θfα, θgα ∈ C∞c (R) with respect to ω, then using Fourier transform properties,
one has∫
R
θ(ω)fα(x, y, ω)(iη)αe−iωηdω = (θfα)(α)(η)
∫
R
θ(ω)gα(x, y, ω)(iη)αe−iωηdω = (θgα)(α)(η).
By Riemann-Lebesgue lemma and Lebesgue dominated convergence theorem, we get limη→+∞
Iη+ = 0.
Finally, passing to the limit η → +∞ in (2.9) and using (2.10), (2.11) and (2.12), we obtain thedesired result.
Remark 2.4. Theorem 2.3 is still valid when µ = −κ2. In fact, we start by replacing in (2.2)the parameter ρ by
√κ2 − ω2 and in (2.5) the test function ϕn
ω by (y + iz)ne−xρ. Moreover, weconsider a function γς(ω) belonging to C∞
c (R) such that γς(ω) = 1 over [−κ + ς, κ − ς], with ς asmall enough constant. Then, multiplying equation (2.8) by γς(ω), instead of θ(ω), and passing tothe limit η → +∞ lead to the corresponding results.
6
The relationships, behind the identification algorithm, given in Theorem 2.3, can be written as
R(n, f, g) =m∑
j=1
n∑
α=0
µαj
(
n
α
)
bn−αj (2.13)
where
µαj = λjIα,j with Iα,j = (−1)α
∫
R
e−iaj
√ω2+κ2
δ(α)(ω) dω. (2.14)
Here, the number Iα,j can be calculated explicitly: I0,j = e−iajκ and for α = 1, · · · , n, as
Iα,j =
(−1)ℓiajβℓ
κ2ℓ−1 θℓ−1(iajκ)e−ajκ if α = 2ℓ
0 if α = 2ℓ+ 1
(2.15)
where θℓ is the ℓth degree reverse Bessel polynomial defined by
θℓ(ξ) =ℓ∑
j=0
(2ℓ− j)!
(ℓ− j)!j!
ξj
2ℓ−j
and βℓ is a constant defined recursively by
β1 = 1βℓ = (2ℓ− 1)βℓ−1.
2.2.3 Identification method
The main objective of the following consists in establishing an identification method for solvingequations (2.13) in order to determine the parameters (m,λj , aj , bj). Since the number of unknownsis much greater than the number of equations, then the algebraic equations (2.13) can’t be solved forwhatever value of n. This necessitates us to truncate equations (2.13) from a non-negative integerconstant K. Namely, we set
cn :=
m∑
j=1
K∑
α=0
µαj (nα)b
n−αj , ∀n ∈ N. (2.16)
Then, according to (2.13), we can see that, for n ≤ K
R(n, f, g) =m∑
j=1
n∑
α=0
µαj (nα)b
n−αj =
m∑
j=1
K∑
α=0
µαj (nα)b
n−αj = cn.
Moreover, since we have Iα,j = O(
1κℓ
)
when α = 2ℓ and κ is taken large enough, we can checkthat, for n > K,
R(n, f, g) =m∑
j=1
n∑
α=0
µαj (nα)b
n−αj = cn +O
(
1
κs
)
where
O
(
1
κs
)
=m∑
j=1
n∑
α=K+1
µαj (nα)b
n−αj , with s =
K+22
if K is even
K+12
if K is odd.(2.17)
Therefore, for a κ greater than 1, we choose and fix a non-negative integer K such that 1κs is small
enough. Thanks to this truncation, we are able to approximate the coefficients cn by R(n, f, g) andconsequently we are capable of determining the quantities m, bj and µα
j by solving the algebraicequations (2.16) by means of the identification algorithm developed in [1, 2].
More precisely, we begin by defining the complex Hankel matrix
HJ,K =
c0 c1 · · · cJ−1
c1 c2 · · · cJ...
......
...cJ−1 cJ · · · c2J−2
for J ∈ N∗, (2.18)
whose rank is given by the following theorem.
7
Theorem 2.5. Let K be a given non-negative integer and HJ,K be the Hankel matrix defined in(2.18), where J is a known upper bound of
J =
(K + 1)m if K is even
Km if K is odd.(2.19)
Assume that the ordinates bj of Sj are distinct, then
rank(HJ,K) = J.
Proof. The proof is similar to that of [2, Theorem 2.8], taking into consideration the parity of K,as seen in (2.17).
As a second step and after computing the rank of the Hankel matrix, we introduce the companionmatrix
BK =
0 1 · · · 0 00 0 1 · · · 0...
.... . .
. . . 0...
.... . .
. . . 1d0 d1 · · · · · · dJ−1
(2.20)
where J is defined in (2.19) and D = (d0, ..., dJ−1)t is the unique solution to the linear system
HJ,KD = ξJ with ξJ = (cJ , · · · , c2J−1)t. Then, the ordinates bj are obtained by the following
theorem:
Theorem 2.6. Let K be a given non-negative integer, L = Jm
, BK be the companion matrix definedin (2.20) and assume that the ordinates bj of Sj , j = 1, · · · , J , are distinct. Then,
1. BK admits m eigenvalues of multiplicity L.
2. The m eigenvalues of multiplicity L are the ordinates bj of Sj , j = 1, · · · , J .
The proof of this theorem is similar to [2, Theorem 2.10].
Remark 2.7. In practice, for a given positive constant κ > 1, we choose the integer K such that1κs is small enough, where s is defined in (2.17). Then, we estimate the coefficients cn definedin (2.16) by R(n, f, g). This introduces an accuracy error O
(
1κs
)
in our identification algorithm,precisely, in determining of the rank of Hankel matrix HJ,K and the eigenvalues of companionmatrix BK (see [27, p. 321-322] for estimating results on SVD). Therefore, through theorems 2.5and 2.6 respectively, we can find, modulo a small error, the number of sources and the ordinates oftheir locations. To determine the positions of the point sources, in particular the coordinates aj , weproceed in the same way, considering the test functions
ψnω(x, y, z) = (x+ iz)ne−iy
√ω2+κ2
.
Remark 2.8. In the previous theorems, we have assumed that the projected points onto the x- andthe y-axis of the point sources Sj are distinct. Henceforth, we were able to identify the points Sj
through these projection points. However, if by bad luck, one of the projected points onto the x-or y-axis coincide, we can do the same thing by choosing another basis in the xy-plane where theprojected points are distinct. This is possible since for all orthonormal basis (~u,~v) in the xy-plane,the following functions
ϕnω(S) = (~v.S+ iz)ne−i~u.S
√ω2+κ2
ψnω(S) = (~u.S+ iz)ne−i~v.S
√ω2+κ2
∣
∣
∣
∣
∣
∣
∣
with S = (x, y, z)
remain solutions of equation (2.7), for all n ∈ N. Let us mention that, to reach a better identi-fication of the point sources, it is desirable to project the point sources in a basis (~u,~v) where theabsolute gap between the singular values of the corresponding Hankel matrix is the largest possible.In practice, to attain such a basis, we can assume, for example, that ~u = (cos(θ), sin(θ), 0) and~v = (− sin(θ), cos(θ), 0) and then use the angle θ ∈ [0, 2π] that realizes the largest gap between thesingular values of the Hankel matrix (2.18).
8
Remark 2.9. Note that, based on the techniques used in our former works [1, 2], the identificationprocess employed in the mono-frequency case can be easily extended to multipolar sources of the form
F =L∑
ℓ=1
Nℓ∑
j=1
Kℓ∑
α=0
λα1,α2j,ℓ
∂α
∂α1x ∂α2
y
δSℓj
(2.21)
where the quantities L, N ℓ, Kℓ are integers, the coefficients λα1,α2j,ℓ are scalar quantities and
α = α1 +α2 with (α1, α2) ∈ N2. Therefore, we skip this case and only show (in Subsection 3.4) the
generalization in the multiple frequencies framework.
2.3 Sources with small supports
In this subsection, we consider the case of sources having compact support within a finite number ofsmall subdomains, given in (1.3). Our aim consists in establishing relationships between the sourceF and the Cauchy data (f, g) in order to identify, using a single wavenumber κ, some characteristicsof the source F , particularly, the number of the domain Dj , their centers Sj and some quantitiesrelated to the densities hj and domains Bj .
Using the change of variables v defined in (2.3), we obtain the following theorem.
Theorem 2.10. Let (f, g) ∈ H12 (Γ)×H− 1
2 (Γ) and let u be the corresponding solution of (1.1) withµ = κ2 and F given by (1.3). Then,
R(n, f, g) = ε2m∑
j=1
n∑
α=0
(
n
α
)
bn−αj
α∑
β=0
(−1)β∫
Bj
∫
R
Φj(α, β, ω, t)δ(β)(ω) dωdt,
where R is the operator defined in (2.6) and Φj is the following function
Φj(α, β, ω, t) =
(
α
β
)
εα−βhj(Sj + εt)tα−β2 e−i(aj+εt1)
√ω2+κ2
with t = (t1, t2).
Proof. The proof is similar to that of Theorem 2.3.
The relationships, behind the identification algorithm, given in Theorem 2.10, can be written as
R(n, f, g) =
m∑
j=1
n∑
α=0
ναj
(
n
α
)
bn−αj ∀n ∈ N (2.22)
where
ναj = ε2α∑
β=0
(
α
β
)
εα−β
∫
Bj
hj(t)tα−β2 Itβ,jdt (2.23)
with
Itβ,j = (−1)β∫
R
e−i(aj+εt1)√
ω2+κ2
δ(β)(ω) dω.
To solve equation (2.22), as in Subsection 2.2.3, we need to truncate it, from a non-negative integerconstant K. First, we set
cn :=m∑
j=1
K∑
α=0
ναj (nα)bn−αj , ∀n ∈ N. (2.24)
According to (2.22), we can see that, for n ≤ K,
R(n, f, g) =
m∑
j=1
n∑
α=0
ναj (nα)bn−αj =
m∑
j=1
K∑
α=0
ναj (nα)bn−αj = cn.
Moreover, via a simple calculation as in (2.15), replacing aj by aj + εt1, we can prove that, for alarge κ,
9
Itβ,j =
0 if β = 2ℓ+ 1
O(
1κℓ
)
if β = 2ℓ.(2.25)
Now, set
τ = max
(
ε,1√κ
)
.
Then, from (2.23) and (2.25), we can check that, for K > n,
R(n, f, g) =m∑
j=1
n∑
α=0
ναj (nα)bn−αj = cn +O (τr)
where
O (τr) =m∑
j=1
n∑
α=K+1
ναj (nα)bn−αj , with r =
K + 4 if K is even
K + 3 if K is odd.
Finally, for κ > 1 and ε < 1, choosing an integer K such that τr is small enough, we approximatethe coefficients cn by R(n, f, g) modulo O (τr). Then, we determine the quantities m, bj , ν
αj by
solving the algebraic equations (2.24) by means of the identification algorithm developed in [1, 2]that we recalled in Subsection 2.2.3.
3 Identification algorithm using multiple wavenumbers
In this section, we aim to reconstruct sources F of the form (1.2), (2.21) or (1.3), from Cauchy data(fκ, gκ) := (uκ|Γ ,
∂uκ
∂ν |Γ), for all κ ∈ κ1, ..., κM, where uκ is the solution of (1.1). Note that, in
such a case, to reconstruct the source parameters using a direct algebraic method, one needs neitherto transform the equation under study (1.1) nor to pass to the three-dimensional space.
Remark 3.1. Although our paper deals with the 2D Helmholtz’s equation, it is rather interestingto note that, in the three-dimensional case, it is unnecessary to consider multi-frequency measures.Indeed, the resolution of this problem with a single frequency, and as seen in [2], permits us to takeas much test functions, (x+ iy)neiκz, n ∈ N, as we desire and consequently leads to similar resultsas the inverse multi-frequency problem.On the other hand, if one has in disposition multiple frequencies, it is better to treat the mono-frequency inverse problem with the frequency that leads to the best results, which is normally thesmallest frequency as seen in [2, Section 4.1.6].
This section is divided into five subsections. After stating the inverse source problem in Subsection3.1, we treat in Subsection 3.2 the case with monopolar sources (1.2) and then we propose inSubsection 3.3 an algebraic algorithm allowing to solve the corresponding inverse source problem.In Subsection 3.4, we extend this algorithm to the case of multipolar sources. Finally, Subsection3.5 deals with sources having small supports (1.3).
3.1 Statement of the inverse problem
Let K be a set of M wavenumbers
K = κ1, κ2, · · · , κM, M ∈ N∗.
We consider, for n = 1, · · · ,M , the elliptic problems ∆uκn+κ2nuκn = F and we define the operators
Λκn(F ) = (uκn |Γ,∂uκn
∂ν |Γ). Then, the inverse source problem considered, here, is formulated as:
Given M Cauchy data (fκn , gκn) ∈ H12 (Γ)×H− 1
2 (Γ), determine F such that
Λκn(F ) = (fκn , gκn) for all n = 1, · · · ,M.
In the following, based on the former work [2, 12], we propose an algebraic method allowing to solvethis inverse problem in the case of monopolar sources (1.2), multipolar sources (2.21) and sourceswith small supports (1.3).
10
3.2 Pointwise sources
First, we begin by establishing algebraic relationships between (m,λj ,Sj) and the Cauchy data. Todo so, we need, first, to introduce, for any real κ ≥ 0, the following space
Hκ = v ∈ H1(Ω) : ∆v + κ2v = 0
and then, define, for all (f, g) ∈ H12 (Γ)×H− 1
2 (Γ) and v ∈ Hκ, the operator
R(v, f, g) =
∫
Γ
(
gv − f∂v
∂ν
)
ds. (3.1)
Then, multiplying equation (1.1)-(1.2) by v, an element of Hκ, integrating by parts and usingGreen’s formula lead to
R(v, fκ, gκ) =m∑
j=1
λjv(Sj) for all v ∈ Hκ. (3.2)
Here, (fκ, gκ) denote the corresponding Cauchy data to the solution uκ of (1.1).
Now, considering, for each κ ∈ K, the function
vdκ(x, y) = eiκd·X (3.3)
where X = (x, y) and d = (d1, d2) with d21 + d22 = 1 and replacing v by vdκ in (3.2), one obtains
R(vdκ, fκ, gκ) =m∑
j=1
λjeiκd.Sj ∀κ ∈ K. (3.4)
Then, the latter equation is solved in order to determine m, λj and Sj . Indeed, set the number Mas
M = 2m,
with m being a known upper bound of the number of sources, fix a real number κ0 > 0 and choosethe wavenumbers in K as
κn = nκ0, n = 1, · · · ,M.
Thus, equations (3.4) can be written as
cn := R(vdκn, fκn , gκn) =
m∑
j=1
λj(eiκ0d.Sj )n for n = 1, · · · , 2m. (3.5)
Therefore, the identification process is attained in two steps. The first step consists in determiningthe number of sources through the rank of the Hankel matrix:
Hdm =
c1 c2 · · · cmc2 c3 · · · cm+1
......
......
cm cm+1 · · · c2m−1
. (3.6)
More precisely, assuming that the points eiκ0d.Sj , for j = 1, · · · ,M , are mutually distinct, namelythe direction d = (d1, d2) satisfies
(H) d · (Sj − Sl) 6= 2qπκ0, ∀ j 6= l, q ∈ Z,
we have the following result.
Theorem 3.2. Let Hdm be the Hankel matrix defined in (3.6) where m is a known upper bound of
m. Under hypothesis (H), we have
rank(
Hdm
)
= m.
11
Proof. The proof is similar to that done in [12, Proposition 1]. First, we rewrite the algebraicequations (3.5) in the matrix form
ξn = AnΛ, for all n = 1, · · · , m,where
ξn = (cn, · · · , cm+n−1)t, Λ = (λ1, · · · , λm)t (3.7)
and An is the m×m Vandermonde matrices
An =
(eiκ0d.S1)n · · · (eiκ0d.Sm)n
(eiκ0d.S1)n+1 · · · (eiκ0d.Sm)n+1
.... . .
...
(eiκ0d.S1)m+n−1 · · · (eiκ0d.Sm)m+n−1
, n ∈ N. (3.8)
Furthermore, if we denote by D the diagonal matrix
D = diag(eiκ0d.S1 , · · · , eiκ0d.Sm),
one gets, for all n ∈ N,
An+1 = AnD = A1Dn
and therefore
ξn = A1Dn−1Λ for all n = 1, · · · , m. (3.9)
Consequently, one obtains the decomposition of the Hankel matrix Hdm as
Hdm = A1[Λ, DΛ, ..., Dm−1Λ] = A1T (A0)
t
where (A0)t is the transpose matrix of A0 and T = diag(λ1, · · · , λm). Since λj 6= 0, for
j = 1, · · · ,m, T is a nonsingular matrix, which implies that T (A0)t is surjective. Therefore, one
has rank(A1T (A0)t) = rank(A1) = m (under (H)), ending the proof of the theorem.
The second step consists in determining the positions of the monopolar sources by means of theeigenvalues of the companion matrix:
Bd =
0 1 · · · 0 00 0 1 · · · 0...
.... . .
. . . 0...
.... . .
. . . 1q1 q2 · · · · · · qm
, (3.10)
where the vector Q = (q1, ..., qm)t is obtained by solving the linear system HdmQ = ξm+1, with ξm+1
defined in (3.7) replacing m by m. More precisely, we have the following theorem.
Theorem 3.3. Let Bd be the companion matrix defined in (3.10). Then, Bd admits m simpleeigenvalues represented by eiκ0(d1aj+d2bj), for j = 1, · · · ,m, where aj and bj are the coordinates ofthe positions Sj .
Proof. The proof of this theorem is a direct consequence of (3.9).
Remark 3.4. Note that, in order to obtain the 2D location of the monopoles, we use the previoustheorem, taking consecutively in (3.3) the directions d = (1, 0) and d = (0, 1) that give us the x− andthe y−coordinates of Sj . In the case where these two directions do not verify (H), we can choosetwo other directions, denoted by d = (d1, d2) and e = (e1, e2), to determine the source positions bysolving the corresponding system of 2m equations with 2m unknowns aj and bj .
Theorems 3.2 and 3.3 suggest that if one knows an upper bound m of the number of sources, one canestablish an algorithm to identify the coefficients m and eiκ0d.Sj , for j = 1, · · · ,m. Moreover, λj
can be determined by solving the linear system A1Λ = ξ1. This allows us to obtain the coefficientsm, aj , bj and λj , as suggested in the following algorithm.
12
3.3 Algebraic algorithm
Step 1. Let m be a known upper bound of the number of sources and κ0 > 0 a fixed wavenumber.For each wavenumber κn = nκ0, n = 1, · · · , 2m and using a single given Cauchy data (fκn , gκn),we compute c1, c2, · · · , c2m taking consecutively the directions as da = (1, 0) and db = (0, 1).Then, the number m is determined by the rank of one of the two Hankel matrices Hdr
m related todr, r = a, b. This rank is estimated using the Singular Value Decomposition method with an appro-priate threshold, following [19], see Section 4 for more details concerning the choice of the threshold.
Step 2. We start by solving the linear system Hdrm Q = ξm+1, for r = a, b. Then, considering the
companion matrices Bdr , (3.10), r = a, b, the coordinates aj and bj of the m monopolar sourcesare obtained as
aj =1
iκ0log (βj,a) +
2qaπ
κ0, with qa ∈ Z (3.11)
and
bj =1
iκ0log (βj,b) +
2qbπ
κ0, with qb ∈ Z (3.12)
where βj,a and βj,b, j = 1, · · · ,m are the m simple eigenvalues of the matrices Bda and Bdb respec-tively.
Step 3. The vector Λ can be obtained by solving the system A1Λ = ξ1.
Remark 3.5. Note that, in Step 2, the eigenvalues of the matrix Bdr , r = a, b, allow us to identifyonly the mesh points
(
1
iκ0log (βj,a) +
2pπ
κ0,
1
iκ0log (βj,b) +
2qπ
κ0
)
, (p, q) ∈ Z2.
To find the parameters qa, qb, satisfying the equalities (3.11) and (3.12), we, first, choose the meshpoints belonging in Ω and then select, among those, the ones verifying the 2m underlying equationssatisfied by aj and bj considering other directions of d.
3.4 Extension to multipolar sources
The proposed algorithm in the previous subsection can be extended to multipolar sources (2.21).Indeed, multiplying (1.1)-(2.21) by the test function vdκ, defined in (3.3), assuming that κn = nκ0,for n = 1, · · · , 2J , where κ0 is a fixed positive wavenumber and J is a known positive integer, oneobtains the following algebraic relationships:
cn := R(vκn , fκn , gκn) =
L∑
ℓ=1
Nℓ∑
j=1
Kℓ∑
α=0
ναj,ℓnα(Pℓ
j)n (3.13)
where R is the operator defined in (3.1),
ναj,ℓ = (−1)α(iκ)α∑
α=α1+α2
dα1
1 dα2
2 λα1,α2j,ℓ and Pℓ
j = eiκ0d.Sℓj .
The main objective of the following consists in establishing a general algebraic method for solvingequations (3.13) generalizing Theorem 3.2 and Theorem 3.3. Indeed, assume that J is a knownupper bound of the number
J =L∑
ℓ=1
(Kℓ + 1)N ℓ.
As in Subsection 3.2, we assume that the points Sℓj satisfy a condition similar to (H), we propose
an identification process in two steps. The first step consists in determining the number of sourcesby means of the following theorem.
13
Theorem 3.6. Let HdJ be the Hankel matrix defined in (3.6), corresponding to the coefficients cn
defined in (3.13), with J is a known upper bound of J . Then, we have
rank(
HdJ
)
=L∑
ℓ=1
(Kℓ + 1)N ℓ if and only if νKℓ
j,ℓ 6= 0 for j = 1, ..., N ℓ, ℓ = 1, ..., L.
As a second step, we introduce the companion matrix
B =
0 1 · · · 0 00 0 1 · · · 0...
.... . .
. . . 0...
.... . .
. . . 1q1 q2 · · · · · · qJ
(3.14)
where the vector Q = (q1, ..., qJ)t is obtained by solving the linear system HJQ = ξJ+1, where
ξJ+1 = (cJ+1, · · · , c2J)t. Thus, the points Pℓj are given by the following theorem.
Theorem 3.7. Let B be the companion matrix defined in (3.14). Assume that νKℓ
j,ℓ 6= 0 for
j = 1, ..., N ℓ and ℓ = 1, ..., L . Then,
1. B admits N ℓ eigenvalues of multiplicity Kℓ + 1 for ℓ = 1, ..., L.
2. The N ℓ eigenvalues of multiplicity Kℓ + 1 are the points Pℓj .
Sketches of the proofs.As in [2, Theorem 2.8], the proof of this Theorem 3.6 is mainly based on the decomposition of theHankel matrix Hd
J into
HdJ = A1I(A0)
t (3.15)
where, for all n ∈ N, An are the J × J complex matrices defined as
An = (Vn,1, · · · , Vn,L) (3.16)
with Vn,ℓ = (U0n,ℓ, · · · , UKℓ
n,ℓ ), where Uαn,ℓ are the confluent J ×N ℓ Vandermonde matrices
Uαn,ℓ =
nα(Pℓ1)
n · · · nα(PℓNℓ)
n
(n+ 1)α(Pℓ1)
n+1 · · · (n+ 1)α(PℓNℓ)
n+1
.... . .
...
(J + n− 1)α(Pℓ1)
n+J−1 · · · (J + n− 1)α(PℓNℓ)
n+J−1
, α = 0, · · · ,Kℓ
and I is the block matrixI = diag(I1, · · · , IL) (3.17)
where IL are the multi-diagonal matrices defined as
Iℓ =
(00)ν0ℓ (10)ν
1ℓ · · · (K
ℓ
0 )νKℓ
ℓ
(11)ν1ℓ (21)ν
1ℓ ... (K
ℓ
1 )νKℓ
ℓ 0...
... . .. ...
(Kℓ−1
Kℓ−1)ν
Kℓ−1ℓ ( Kℓ
Kℓ−1)ν
Kℓ
ℓ · · · 0
(Kℓ
Kℓ)νKℓ
ℓ 0 · · · 0
for ℓ = 1, · · · , L
with ναℓ = diag(να1,ℓ, · · · , ναNℓ,ℓ) α = 0, · · · ,Kℓ.
Moreover, the proof of Theorem 3.7 is similar to [2, Theorem 2.10].
Thanks to Theorem 3.6 and Theorem 3.7 and using the same algorithm detailed in Subsection3.3, we can identify the locations Sℓ
j . Furthermore, in order to determine ναj,ℓ, it is sufficient, forexample, to solve the linear systems A1Λ = ξ1.
14
3.5 Sources with small support
In this subsection, we focus on the reconstruction of sources having compact support within a finitenumber of small subdomains (1.3), using multiple frequencies. The proposed method allows to solvethe inverse source problem stated in Subsection 3.1 whose aim is to find the number m, the pointsSj and some characteristics of the domains Dj .
We proceed as in Subsection 3.2. Multiplying equation (1.1)-(1.3) by the test function vκ (definedin (3.3)), integrating by parts and using Green’s formula and the change of variable X = Sj + εt,with t = (t1, t2), one has
R(vκ, fκ, gκ) =
m∑
j=1
µε,κj eiκd.Sj
where R is the operator defined in (3.1) and
µε,κj = ε2
∫
Bj
hj(t)eiεκd.t dt with hj(t) = hj(Sj + εt).
As in Subsection 3.4, replacing κ by κn, where κn = nκ0, for n = 1, · · · , 2J , with κ0 is a fixedpositive constant and J is a positive integer, we obtain the algebraic relationships related to thesesource parameters and the Cauchy data
R(vκn , fκn , gκn) =m∑
j=1
µε,κnj
(
eiκ0d.Sj
)n
. (3.18)
Using the Taylor development of the exponential with respect to ε, we know that for 0 < ε < 1 andnon-negative integer K, we have
µε,κnj =
K∑
α=0
ναj nα +O(εK+3)
where ναj = ε2+α (iκ0)α
α!
∫
Bj
(d.t)αhj(t) dt.
Replacing µε,κnj by its Taylor development in (3.18), we get
R(vκn , fκn , gκn) =m∑
j=1
K∑
α=0
ναj nα(
eiκ0d.Sj
)n
+O(εK+3).
Now, for a given positive ε < 1, we choose a fixed integer K such that εK+3 is small enough and weassume that we know an upper bound J of (m + 1)K. Then, we approximate, for n = 1, · · · , 2J ,the coefficients cn by
cn := R(vκn , fκn , gκn) =m∑
j=1
K∑
α=0
ναj nα(
eiκ0d.Sj
)n
. (3.19)
Finally, we solve the algebraic relationships (3.19) using the same algorithm developed in the pre-vious subsection in order to recover (modulo εK+3) the quantities m, Sj and ναj .
Note that, the coefficients ναj contain some information over the domain Bj . Then, one can obtain,modulo εK+3, for instance, certain quantities related to the mass or the moment of Bj .
Remark 3.8. In the particular case where Dj = Sj +Bj , with the domains Bj are hollow or solidballs of center (0, 0) and radii rj0, r
j1, namely
Bj = (x, y) ∈ R2 : 0 ≤ rj0 <
√
x2 + y2 ≤ rj1and taking the terms hj as scalars, the points Sj , their number and quantities related to hj areeasily recovered. This is done as in [16, Theorem 2], using the following mean value relation givenin [11, Page 289], valid on all functions v solution of ∆v + µv = 0 in Ω
v(S0)J0(rõ) =
1
2πr2
∫
C
vdC
15
where Jυ(x) is the υth Bessel function and C is the circle of center S0 and radius r entirely containedin Ω.
4 Numerical simulations
This section studies numerically the robustness of the algebraic algorithm only in the multi-frequency case with respect to the different parameters interfering in the reconstruction process.The case considering a single frequency is not dealt with in this section. Indeed, in such a case, onecan hope to recover at best one or two sources. As a matter of fact, the first step of the proposedidentification method is to calculate the number of sources which is obtained, as seen in Theorem2.5, from the rank of the Hankel matrix HJ,K , defined in (2.18). The rank of HJ,K depends baselyon J and on κ. As tested numerically in [2, Section 4], with a realistic number of sensors, the rankis well-calculated when the size of the Hankel matrix (J × J) is about 6× 6 and when κ is not very
large. However, the coefficients of HJ,K , in here, are approximated as O
(
1
κK+1
2
)
, where K is the
order of approximation and the size of HJ,K is J × J with J ≥ (K + 1)m. In other words, for abetter approximation of HJ,K , both K and κ must be taken large enough which leads, hence, to adifficult numerical calculation of its rank. In order to overcome this difficulty, a work in progress isbeing performed.Therefore, considering the multi-frequency framework in this numerical study, the base wavenumberκ0 is fixed at 1.85 m−1 and Γ is assumed to be a unit circle whose center is the origin O. TheCauchy data (fκn , gκn) on the boundary Γ are obtained by means of the fundamental solution ofHelmholtz equation in R
2. In fact, fκn and gκn are respectively the trace and the normal trace ofwκn on Γ, where wκn is the fundamental solution corresponding to F (given by (1.2)), defined inthe free space as:
wκn(X) =
m∑
j=1
λjw0κn
(X − Sj), n = 1, ..., m
where
w0κ(X) =
1
4iH
(1)0 (κρ) and X = (x, y)
withH(1)0 the Hankel function of first kind of order zero and ρ =
√
x2 + y2.Moreover, the coefficientscn, defined in (3.5), are numerically computed using polar coordinates over a uniform meshing ofdistributed points on the unit circle.The reconstruction of the number of sources is the major step in the identification process. The-oretically, their number is the rank of the Hankel matrices Hd
m, which is numerically determinedusing SVD method with an appropriate threshold. However, since Hd
m is an ill-conditioned ma-trix, a regularization approach is employed. In fact, the (m + 1)th singular value, σm+1, of Hd
m istheoretically zero, whereas when the perturbed Hd
m + δHdm is given, one obtains a non zero σm+1.
Therefore, based on the classical estimate, [19],
|σm+1| ≤ ‖δHdm‖F , (4.1)
we truncate beyond a threshold inferior to ‖δHdm‖F . Here, ‖ · ‖F is the corresponding Frobenius
norm and δHdm is the related perturbation matrix of Hd
m that originates from the noise in data aswell as from the numerical quadrature error using a finite number of sensors on Γ.
Remark 4.1. We draw the attention of the reader to the fact that in the case of N sensors, thenumerical error can be seen as noise equivalent to (2π/N) perturbation. That is why, apart fromthe Subsection 4.1.5 dedicated to study the noise effect, we use the Cauchy data as non-noisy onesto observe the identification process in an approximately ideal framework.
Remark 4.2. The calculation of ‖δHdm‖F is related to the numerical quadrature error. In here,
this computation is not exact since we take into consideration just the numerical error (2π/N).Nevertheless, in reality, δHd
m is computed as
‖δHdm‖F ≃ m
√
2π
Nβ(κ0, sources)
16
where β is the error related to the wavenumber and to the source positions. Therefore, in thefollowing, we aren’t reasonably capable of using the truncation threshold ‖δHd
m‖F in the analysis ofthe impact of the wavenumber and the closeness of the sources over the identification process due tothe unprecise knowledge of β. Consequently, the estimation 4.1 will be used in the analysis of theimpacts of the number of sensors and of the number upper bound.
4.1 Determining number and position of monopole sources
In the following subsections, unless mentioned otherwise, we fix the number of monopoles at 5 havingfixed intensities λj = 1 with positions taken as in Table 1. Moreover, due to reasons explained later
j (location ♯) 1 2 3 4 5Sj (-0.7,0.3) (0.6,-0.3) (0.3,0.5) (-0.5,-0.4) (-0.1,0.0)
Table 1: The source positions.
in Subsection 4.1.3, the reconstruction method is employed on the x-axis rather than over the y-axis.
4.1.1 Impact of the number of sensors
The mesh level, represented by the number of sensors on the boundary, has a great impact onthe identification process. Refining the mesh gives more accessibility to the Cauchy data pair andthus more specificity in the reconstruction process, where the sources are obtained with an averageCPU time of about 1.59 seconds. Indeed, varying the number of sensors from 25 to 100 sensorsand fixing m at m = 8 (i.e. M = 2m = 16 frequency levels), we note that, as seen in Fig. 2,their increase enhances the identification of both the number and the position of the monopoles.Moreover, we remark, based on the SVD truncation level ‖δHda
8 ‖F (see Table 2), that 5 monopolescan’t be reconstructed with less than 50 sensors.
Number of sensors 25 35 50 100
‖δHda
8‖F ≃ 4.01 3.39 2.84 2.01
Table 2: The Frobenius norm of δHda
8with respect to the number of sensors.
0 1 2 3 4 5 6 7 8
10−5
10−4
10−3
10−2
10−1
100
101
102
singular value index
sing
ular
val
ue
effect of the mesh level (Ox−axis)
25 sensors35 sensors50 sensors100 sensors
25 35 50 100
10−5
10−4
10−3
10−2
10−1
100
number of sensors
loca
lizat
ion
erro
r
effect of the mesh level (Ox−axis)
Fig. 2: Singular value of Hda8 (left) and the localization error(right) for m = 5 on the x− axis with
respect to the number of sensors.
From now on, we fix our study to 50 sensors that enable us to recover precisely the number and thelocation of up to 5 monopoles.
17
4.1.2 Impact of the supposed number upper bound
The supposed upper bound of the number of sources has an effect on the identification process.Indeed, as seen in Fig. 3, as the supposed upper bound increases, the gap between the mth and(m + 1)th singular value decreases (taking m = 8, 10, 12, 14). This leads, based on the truncationthreshold (4.1) (see Table 3) to a wrong estimation of the rank of δHda
m when m exceeds a certainlevel (m = 12 as seen in Fig. 3 in the case of 5 monopoles) and consequently causing a wrong sourcesnumber reconstruction. This result is expected since the theoretical rank is fixed at m for whatevervalue of m. Therefore, when increasing m, we accumulate more and more error on the correspondingHankel matrix causing it to become more and more ill-conditioned and consequently obtaining awrong numerical rank estimation. Therefore, it is crucial to have a good a priori knowledge onthe upper bound which must not be so far than the exact needed number of sources for a betteridentification process.
m 8 10 12 14
‖δHdam ‖F ≃ 2.84 3.54 4.25 4.96
Table 3: The Frobenius norm of δHdam with respect to m.
0 2 4 6 8 10 12 14
10−3
10−2
10−1
100
101
singular value index
sing
ular
val
ue
effect of the number upper bound
m=8m=10m=12m=14
Fig. 3: Singular values of Hdam for 5 monopoles with respect to m.
From now on, we fix the upper bound of the number of sources at m = 8 for the reconstruction of5 monopoles.
4.1.3 Obtaining the source positions and the separability effect
To obtain the source positions, one must use consecutively, as mentioned before, both the directionsda = (1, 0) and d2 = (0, 1). However, when projecting onto the x−axis and the y−axis, we see thatnumerically, as seen in the left and the right panels of Fig. 4, the identified number of sources isnot the same on whatever axes where the projection is performed. This is due to the fact that theseparability coefficient between the sources plays an important role in the identification process.Indeed, as seen in our example, in Fig. 4 (right), the number of sources is ill-estimated in they-projection since the sources projections are close (see Fig. 5). Therefore, to recover the sourcesnumber, one considers the numerical rank of the two Hankel matrices Hdr
m , r = a, b, obtainedrespecting the truncation threshold defined in (4.1), and then we take the maximum of these ranksas the desired number of sources that is then used in the position reconstruction.Note that the better separability coefficient between the sources over the x-axis is the reason behindthe use of this axis in the reconstruction method in the other subsections.As seen just above, the reconstruction of the sources depends on the separability coefficient betweenthe projected locations. Therefore, an important factor in the identification process is the choiceof a good projection axis that would yield to a good separability of the sources and consequently amore precise localization. To do so, in a practical point of view, a strategy that could be utilized is
18
0 1 2 3 4 5 6 7 8
10−3
10−2
10−1
100
101
singular value index
sing
ular
val
ue
m=5 (Ox−axis)
0 1 2 3 4 5 6 7 8
10−3
10−2
10−1
100
101
singular value index
sing
ular
val
ue
m=5 (Oy−axis)
Fig. 4: Singular values of Hdr8 , r = a, b for m = 5 with 50 sensors.
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
error=1.62*10−5
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
error=3.11*10−4
Fig. 5: The source positions projected on the x− and the y− axis for m = 5 with 50 sensors.
the following. For N discretization points (θj) over [0, 2π], we consider the corresponding directionsda = (cos θj , sin θj) and db = (− sin θj , cos θj). Then, for these directions, we calculate, for all1 ≤ j ≤ N , the numerical rank of the two Hankel matrices (Hdr
m )j , r = a, b, always respectingthe truncation threshold (4.1). Consequently, the number of sources is obtained as the maximumbetween these two ranks for all 1 ≤ j ≤ N . Now, to recover most precisely the sources locations,few steps should be done. First, one should choose only the axes having rank(Hdr
m )j = m forr = a, b. The existence of these axes is possible due to the natural hypothesis that these sources arewell-separated. Next, we calculate the condition numbers αj and βj of the corresponding Hankelmatrices (Hdr
m )j . Finally, to obtain the axes with the best location estimation, we take those whosatisfy the best conditionement of (Hdr
m )j which corresponds, as mentioned before, to the bestconditionement of (A0)j and consequently the highest separability coefficient. Technically, the axescontaining the matrices (Hdr
m )j with the best condition numbers are obtained in the sense of havingthe smallest Euclidean distance between (αj , βj) and the vector (1, 1).
4.1.4 Impact of the base wavenumber
The left and the right panels of Fig. 6 show singular values of Hda8 and the localization error
when changing the base wavenumber κ0. We observe that as we enlarge the base wavenumber, thenumber is estimated worse and the localization error increases. This result could be explained sincethe number of points per wavelength, defined by
p ≈ number of sensors
mκ0(4.2)
19
decreases as κ0 increases. We observe that when κ0 is higher than 3m−1, we don’t obtain thedesired results anymore. This is normal since the number of points per wavelength, as seen in therelation (4.2), are relatively small (∼ 2 points). Nevertheless, it is possible to improve the results byincreasing the number of sensors and consequently that of p. Indeed, as tested numerically, in thecase of κ0 = 3m−1, 100 sensors are sufficient to recover precisely the number and the location ofthe sources. However, an even higher number of sensors becomes "unrealistic" since we are limitedby the number of observations.
0 1 2 3 4 5 6 7 8
10−3
10−2
10−1
100
101
singular value index
sing
ular
val
ue
wave number effect (Ox−axis)
κ0=1.85
κ0=3
κ0=5
κ0=7
1.5 3 5 7
10−5
10−4
10−3
10−2
10−1
100
κ
loca
lizat
ion
erro
r
wave number effect (Ox−axis)
Fig. 6: Singular values of Hda8 (left) and the localization errors (right) projected on the x−axis for
m = 5 with 50 sensors.
4.1.5 Impact of the noise
Reconstruction stability on the x − axis projection with respect to the noise level is examined inthis subsubsection. In fact, Gaussian noise is added to fκn (and gκn) with a standard deviation thatvaries from 10−2 to 100 % (see Fig. 7). We have noted studying the SVD of the Hankel matrix Hda
m
that the number of monopoles are badly-estimated when the percentage of noise exceeds 100%.Moreover, we note that the localization error increases gradually as the percentage of the noiseadded increases. Indeed, the error is of order 10−1 whereas in a noise free framework, we had anerror of order 10−4.
1 2 3 4 5 6 7 8
10−3
10−2
10−1
100
101
singular value index
sing
ular
val
ue
effect of the noise
10−2 %
10−1 %
100 %
101 %
10−2
10−1
100
10−2
10−1
100
Gaussian noise %
loca
lizat
ion
erro
r
noise effect
Fig. 7: Singular values of Hda8 (left) and the localization errors (right) projected on the x−axis
with 50 sensors.
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4.2 Determining number and position of a combination of monopole
and dipole sources: impact of the number of sensors
In this subsection, we aim to reconstruct a combination of monopoles and dipoles. Applying themethodology proposed in Subsection 3.4, we study, in Fig. 8 the effect of the mesh level on theidentification process for 1 monopole and 2 dipoles with fixed intensities and moments. We note thesame results as before where both the variation of the singular values of Hda
8 and the localizationprecision enhance as we refine the mesh. Note that we have decided not to present the impact ofthe upper bound J , the separability between the sources, the wavenumber and the noise since theyare similar as those shown in the case of monopoles in Subsection 4.1.
0 1 2 3 4 5 6 7 8
10−5
10−4
10−3
10−2
10−1
100
101
singular value index
sing
ular
val
ue
effect of the mesh level (Ox−axis)
25 sensors35 sensors50 sensors100 sensors
25 35 50 100
10−5
10−4
10−3
10−2
10−1
100
number of sensors
loca
lizat
ion
erro
r
effect of the mesh level (Ox−axis)
Fig. 8: Singular value of Hda8 (left) and the localization error(right) for 2 dipoles with 1 monopole
on the x− axis with respect to the number of sensors.
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