An inverse initial boundary value problem for the wave equation in the ...

c 2002 Society for Industrial and Applied Mathematics 

SIAM J. CONTROL OPTIM. Vol. 41, No. 4, pp. 1194–1211

AN INVERSE INITIAL BOUNDARY VALUE PROBLEM FOR THE WAVE EQUATION IN THE PRESENCE OF IMPERFECTIONS OF SMALL VOLUME∗ HABIB AMMARI†

Dedicated to Jean-Claude N´ed´elec for his 60th birthday Abstract. We consider for the wave equation the inverse problem of identifying locations and certain properties of the shapes of small conductivity inhomogeneities in a homogeneous background medium from dynamic boundary measurements on part of the boundary and for a finite interval in time. Using as weights particular background solutions constructed by a geometrical control method, we develop an asymptotic method based on appropriate averaging of the partial dynamic boundary measurements. Our approach is expected to lead to very effective computational identification algorithms. Key words. inverse problem, wave equation, reconstruction, geometric control AMS subject classifications. 35R30, 35B40, 35B37, 35L05 PII. S0363012901384247

1. The inverse initial boundary value problem. Let Ω be a bounded, smooth subdomain of R2 . For simplicity, we take ∂Ω to be C ∞ , but this condition could be considerably weakened. Let n denote the outward unit normal to ∂Ω. We suppose that Ω contains a finite number of inhomogeneities, each of the form zj + αBj , where Bj ⊂ R2 is a bounded, smooth domain containing the origin. The total collection of inhomogeneities thus takes the form Bα = ∪m j=1 (zj + αBj ). The points zj ∈ Ω, j = 1, . . . , m that determine the location of the inhomogeneities are assumed to satisfy |zj − zl | ≥ d0 > 0

∀ j = l

and dist(zj , ∂Ω) ≥ d0 > 0

∀ j.

As a consequence of this assumption, it follows immediately that m ≤ 4|Ω| . We πd20 also assume that α > 0, the common order of magnitude of the diameters of the inhomogeneities, is sufficiently small so that these are disjoint and their distance to R2 \ Ω is larger than d0 /2. Let γ0 denote the conductivity of the background medium; for simplicity, we shall assume in this paper that it is constant. Let γj denote the constant conductivity of the jth inhomogeneity, zj + αBj . Using this notation, we introduce the piecewise constant conductivity
γα (x) =

γ0 ,

x ∈ Ω \ Bα ,

γj ,

x ∈ zj + αBj , j = 1, . . . , m.

∗ Received by the editors January 29, 2001; accepted for publication (in revised form) April 12, 2002; published electronically October 29, 2002. This work is partially supported by ACI Jeunes Chercheurs (0693) from the Ministry of Education and Scientific Research, France. http://www.siam.org/journals/sicon/41-4/38424.html † Centre de Math´ ´ ematiques Appliqu´ees, CNRS UMR 7641 & Ecole Polytechnique, 91128 Palaiseau Cedex, France ([email protected]).

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1195

Consider the initial boundary value problem for the (scalar) wave equation  2   (∂t − div γα grad)uα = 0 in Ω × (0, T ),  (1) uα |t=0 = ϕ, ∂t uα |t=0 = ψ in Ω,    uα |∂Ω×(0,T ) = f. Define u to be the solution of the wave equation in the absence of any inhomogeneities. Thus u satisfies  2 (∂ − γ0 ∆)u = 0 in Ω × (0, T ),    t (2) u|t=0 = ϕ, ∂t u|t=0 = ψ in Ω,    u|∂Ω×(0,T ) = f. Here T > 0 is a final observation time, and the initial conditions ϕ, ψ ∈ C ∞ (Ω) and the boundary condition f ∈ C ∞ (0, T ; C ∞ (∂Ω)) are subject to the compatibility conditions ∂t2l f |t=0 = (γ0 )l (∆l ϕ)|∂Ω and ∂t2l+1 f |t=0 = (γ0 )l (∆l ψ)|∂Ω ,

l = 1, 2, . . . .

From the above compatibility conditions on ϕ, ψ and f , it follows that the initial boundary value problem (2) has a unique solution in C ∞ ([0, T ] × Ω); see [14]. It is also classical to prove that the transmission problem for the wave equation (1) has a unique weak solution uα ∈ C 0 (0, T ; H 1 (Ω)) ∩ C 1 (0, T ; L2 (Ω)); see, for example, [19]. α Indeed, Lions proved in [19, Chapter VI, Theorem 4.1, p. 369] that ∂u ∂n |∂Ω belongs 2 2 to L (0, T ; L (∂Ω)). His proof is based on an extension of the multiplier method. Throughout this paper, we shall use quite standard L2 -based Sobolev spaces to measure regularity. The notation H s is used to denote those functions which, along with all their derivatives of order less than or equal to s, are in L2 . H01 denotes the closure of C0∞ in the norm of H 1 . Sobolev spaces with negative indices are in general defined by duality, using an L2 -inner product. We shall only need two such spaces, namely, H −1 , which is defined as the dual of H01 , and H −2 , which is defined as the dual of H02 that is the closure of C0∞ in the norm of H 2 . Define νj to be the outward unit normal to ∂(zj + αBj ) for j = 1, . . . , m. Let Γ ⊂ Ω be a given part of the boundary ∂Ω. The aim of this paper is to identify the location and certain properties of the shapes of the inhomogeneities Bα from only knowledge of boundary measurements of ∂uα ∂n

on Γ × (0, T ),

i.e., on the part Γ of the boundary ∂Ω and on the finite interval in time (0, T ). For this purpose, we develop an asymptotic method based on appropriate averaging, using particular background solutions as weights. These particular solutions are constructed by a control method as was done in the original work [32]. The first fundamental step in the design of our reconstruction method is the α + derivation of an asymptotic formula for ∂u ∂νj |∂(zj +αBj ) in terms of the reference solution u, the location zj of the imperfection zj + αBj , and the geometry of Bj . The second step consists of the use of this asymptotic formula to derive integral boundary formulae with a convenient choice of test functions, which is based on a geometrical control method and solving Volterra-type integral equations. We expect that these

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HABIB AMMARI

boundary integral formulae will form the basis of very effective computational identifying algorithms. A similar approach may be applied to the full (time-dependent) Maxwell equations with small inhomogeneities of different electric permittivity or magnetic permeability (or both). This will be discussed in a forthcoming paper. The elastodynamic inverse problem will also be considered. Whereas the determination of conductivity profiles from knowledge of boundary measurements has received a great deal of attention (see, for example, [1], [4], [9], [12], [15], and [34]), the reconstruction of imperfections within dynamics is much less investigated. To the best of our knowledge, the present paper is the first attempt to design an effective method to determine the location and the size of small conductivity imperfections inside a homogeneous medium from the dynamical measurements on part of the boundary. The inverse problem considered in this paper is more complicated from the mathematical point of view and more interesting in applications than the one solved in [4] and [34] because, in many applications, one cannot get measurements for all t or on the whole boundary, and so one cannot, by taking a Fourier transform in the time variable, reduce our dynamic inverse problem to the inverse problem for the Helmholtz equation considered in [4] and [34]. The general approach we will take to recuperate the locations and shapes of the imperfections is to integrate the solution against special test functions. Our method is quite similar to the ideas used (in the time-independent case) by Calder´ on [11] in his proof of uniqueness of the linearized conductivity problem and later by Sylvester and Uhlmann in their important work [29] on uniqueness of the three-dimensional inverse conductivity problem (see Nachman [21] for the two-dimensional problem). It is also closely related to ideas used by Yamamoto in his original work [32] on inverse source hyperbolic problems and by Rakesh and Symes [27]. For discussions on other interesting inverse source hyperbolic problems, the reader is referred, for example, to Isakov [17], Belishev and Kurylev [8], Romanov and Kabanikhin [28], Yamamoto [31], [33], Puel and Yamamoto [23], [24], [25], [26], Grasselli and Yamamoto [16], Bruckner and Yamamoto [10], Nicaise [22], and Sun [30]. 2. An energy estimate. We start the derivation of the asymptotic formula for with the following energy estimate of uα − u. Proposition 2.1. There exist constants 0 < α0 and C such that, for 0 < α < α0 , the following energy estimate holds: ∂uα + ∂νj |∂(zj +αBj )

||∂t (uα − u)||L∞ (0,T ;H −1 (Ω)) + ||uα − u||L∞ (0,T ;L2 (Ω)) ≤ Cα.

(3)

The constants α0 and C depend on the domains {Bj }m j=1 , the domain Ω, d0 , T , γ0 , , the data ϕ, ψ, and f but are otherwise independent of the points {zj }m {γj }m j=1 j=1 . Proof. Since uα − u ∈ H01 (Ω), we have, for any v ∈ H01 (Ω),  (4)

Ω

∂t2 (uα − u)v +

 Ω

γα grad(uα − u) · grad v =

m 

 (γ0 − γj )

j=1

Let vα be defined by


vα ∈ H01 (Ω), div γα grad vα = ∂t (uα − u)

in Ω.

zj +αBj

grad u · grad v.

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RECONSTRUCTION OF DIELECTRIC IMPERFECTIONS

Then 



1 γα grad(uα − u) · grad vα = − ∂t (uα − u)(uα − u) = − ∂t 2 Ω Ω

and

 Ω

∂t2 (uα − u)vα

Ω

(uα − u)2

div γα grad ∂t vα vα

 Ω

γα grad ∂t vα · grad vα

1 = − ∂t 2

 Ω

γα | grad vα |2 .

Thus it follows that    m  2 2 ∂t γα | grad vα | + ∂t (uα − u) = −2 (γ0 − γj ) Next

Ω

 =

= −

Ω



Ω

zj +αBj

j=1

grad u · grad vα .

  m      (γ0 − γj )  grad u · grad v α  ≤ C|| grad u||L2 (Bα ) || grad vα ||L2 (Ω) .  zj +αBj  j=1 

Since u ∈ C ∞ ([0, T ] × Ω), we have || grad u||L2 (Bα )

  12 m  ≤ || grad u||L∞ (Bα ) α  |Bj | ≤ Cα, j=1

which gives

  m      (γ0 − γj ) grad u · grad vα  ≤ Cα|| grad vα ||L2 (Ω) ,  zj +αBj  j=1 

and so 1/2

    2 2 2 2 ∂t γα | grad vα | + ∂t (uα − u) ≤ Cα γα | grad vα | + (uα − u) . Ω

Ω

Ω

Ω

From the Gronwall lemma, it follows that 1/2  1/2

 γα | grad vα |2 + (uα − u)2 ≤ Cα. Ω

Ω

Combining this last estimate with the fact that ||∂t (uα − u)||L∞ (0,T ;H −1 (Ω)) ≤ C|| grad vα ||L∞ (0,T ;L2 (Ω)) , we obtain the desired estimate (3). We remark that, taking (at least formally) v = ∂t (uα − u) in (4), we arrive at   m    2 2 ∂t |∂t (uα − u)| + γα | grad(uα − u)| = 2 (γ0 −γj ) grad u·grad ∂t (uα −u). Ω

j=1

zj +αBj

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HABIB AMMARI

Using now the regularity of u in Ω and estimate (3) given above, we see that   m      (γ0 − γj ) grad u · grad ∂t (uα − u) ≤ C|| grad u||H 2 (Bα ) ||∂t (uα − u)||H −1 (Ω)  zj +αBj  j=1  ≤ Cα2 , where C is independent of t and α, and so we obtain    |∂t (uα − u)|2 + γα | grad(uα − u)|2 ≤ Cα2 , ∂t Ω

which yields the estimate ||∂t (uα − u)||L∞ (0,T ;L2 (Ω)) + ||uα − u||L∞ (0,T ;H01 (Ω)) ≤ Cα, where C is independent of α and the points {zj }m j=1 . 3. An asymptotic formula. Before formulating the main result of this section, we need to introduce some additional notation. For any 1 ≤ j ≤ m, let Φj denote the vector-valued solution to  ∆Φj = 0 in Bj , and R2 \ Bj ,         Φj is continuous across ∂Bj ,    γ0 ∂Φj  ∂Φj  (5)   γj ∂νj + − ∂νj  = −νj ,   −      lim |Φj (y)| = 0. |y|→+∞

The existence and uniqueness of this Φj can be established using single layer potentials with suitably chosen densities; see [12]. In terms of this function, we are able to prove α + the following result about the asymptotic behavior of ∂u ∂νj |∂(zj +αBj ) . Proposition 3.1. For y ∈ ∂Bj , we have, in the weak sense, 

  ∂uα γ0 ∂Φj (6) |∂(zj +αBj )+ (zj + αy, t) = νj + −1 |+ (y) · grad u(zj , t) + o(1). ∂νj γj ∂νj The term o(1) depends on the shapes of the domains {Bj }m j=1 and Ω, the constants , the data ϕ, ψ, and f but is otherwise independent of the points d0 , T , γ0 , {γj }m j=1 {zj }m . j=1 For simplicity, let us restrict our attention to the case of a single inhomogeneity, i.e., the case m = 1. The proof, for any fixed number m of well-separated inhomogeneities, follows by iteration of the argument that we will present for the case m = 1. In order to further simplify notation, we assume that the single inhomogeneity has the form αB; that is, we assume it is centered at the origin. We denote the conductivity inside αB by γ∗ and define Φ∗ to be the same as Φj , defined in (5), but with Bj and γj replaced by B and γ∗ , respectively. Define ν to be the outward unit normal to ∂B. Let Uα = grad uα (x, t) and U0 = grad u(x, t) in Ω × (0, T ). We start with a formal derivation of the asymptotic formula (6). Following a common practice in multiscale expansions, we introduce the local variable y = αx . We expect that Uα (x, t) will differ appreciably from U0 (x, t) for x near the origin,

RECONSTRUCTION OF DIELECTRIC IMPERFECTIONS

1199

but it will differ little from U0 (x, t) for x far from the origin. Therefore, in the spirit of matched asymptotic expansions, we shall represent Uα (x, t) by two different expansions: an inner expansion for x near the origin and an outer expansion for x far from the origin. The outer expansion must begin with U0 , so we write Uα (x, t) = U0 (x, t) + β1 (α)U1 (x, t) + β2 (α)U2 (x, t) + · · ·

for |x| >> O(α), t ∈ (0, T ),

where the gauge functions β1 (α), β2 (α), . . . and the functions U1 , U2 , . . . are to be found. We write the inner expansion as Uα (zj +αy, t) = V0 (y, t)+µ1 (α)V1 (y, t)+µ2 (α)V2 (y, t)+· · ·

for |y| = O(1), t ∈ (0, T ),

where the gauge functions µ1 (α), µ2 (α), . . . and the functions V0 , V1 , V2 , . . . are to be found. Here the gauge functions βi (α) and µi (α) satisfy βi (α) >> βi+1 (α) and µi (α) >> µi+1 (α) as α tends to 0. The inner and outer expansions must be asymptotically equal in some overlap domain within which the stretched variable |y| is large and |x| is small. In this domain, the matching condition is U0 (x, t) + β1 (α)U1 (x, t) + · · · ∼ V0 (y, t) + µ1 (α)V1 (y, t) + · · · . From the terms of order α0 , we obtain the first matching condition V0 (y, t) → U0 (0, t) as |y| → +∞ (for t ∈ (0, T )). Since (7)

∂t2 uα − div γα Uα = 0 and curl Uα = 0,

by substituting the inner and outer expansions into these equations and formally equating coefficients of α−1 , we get curl y V0 = 0, div y γ(y)V0 = 0 in R2 , where


γ(y) =

Therefore,

V0 (y) = grad

γ0 in R2 \ B, γ∗ in B.

  γ0 − 1 Φ∗ (y) + y · grad u(0, t), γ∗

and so, by multiplying by νj , we arrive at  

 γ0 ∂Φ∗  ∂uα  + (αy, t) = ν · grad u(0, t) + (8) −1 (y) · grad u(0, t) + o(1). ∂ν ∂(αB) γ∗ ∂ν + In the case of m (well-separated) inhomogeneities zj +αBj , j = 1, . . . , m, we (formally) obtain from (8) that the following asymptotic formula holds for any y ∈ ∂Bj :  

 γ0 ∂Φj  ∂uα  + (zj + αy, t) = νj · grad u(zj , t) + −1 (y)·grad u(zj , t)+o(1). ∂νj ∂(zj +αBj ) γj ∂νj +

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HABIB AMMARI

Proof of Proposition 3.1. Let θ be given in C0∞ (]0, T [). For any function v ∈ L (0, T ; L2 (Ω)), we define 1

 vˆ(x) = We remark that ∂ t v(x) = −

T 0

0

T

v(x, t) θ(t) dt ∈ L2 (Ω).

ˆα satisfies v(x, t)θ (t) dt. So we deduce from (7) that U

    div γα U ˆα =   ˆ0 = Analogously, U

T 0

T

0

uα θ (t) dt

in Ω,

ˆα = 0 in Ω. curl U

U0 (x, t) θ(t) dt satisfies     γ0 div U ˆ0 =  

0

T

u θ (t) dt

in Ω,

ˆ0 = 0 in Ω. curl U

ˆ0 × n = grad∂Ω fˆ × n on the boundary ∂Ω, where grad∂Ω ˆα × n = U Indeed, we have U is the tangential gradient. Following [6], we introduce qα∗ as the unique solution to the following problem:

  Ω  ∗   ∆q = 0 in Ω = B and in B,  α  α       qα∗ is continuous across ∂B,    ∂qα∗  ∂qα∗   ˆ0 (αy) · ν on ∂B,  γ − γ = −(γ0 − γ∗ )U 0 + ∗   ∂ν  ∂ν −      ∗  qα = 0 on ∂ Ω. The jump condition

  ∂qα∗  ∂qα∗  ˆ0 (αy) · ν = −(γ0 − γ∗ )U γ0 + − γ∗ ∂ν  ∂ν −

on ∂B

ˆ0 (x) − grady q ∗ ( x ) belongs to the functional space ˆα (x) − U guarantees that U α α Zα (Ω) = {v ∈ L2 (Ω), div (γα v) ∈ L2 (Ω), curl v ∈ L2 (Ω), v × n = 0 on ∂Ω}. Since

(9)

    ∗ x ˆ ˆ  div γ U − U − grad q α α 0  y α  α        T  γ∗    u − χ(Ω \ αB)u − χ(αB)u θ (t) dt = α  γ 0

0

      ˆα − U ˆ0 − grady q ∗ x  = 0 in Ω, curl U  α  α          U ˆα − U ˆ0 − grady q ∗ x × n = 0 on ∂Ω, α α

in Ω,

RECONSTRUCTION OF DIELECTRIC IMPERFECTIONS

1201

where χ(ω) is the characteristic function of the domain ω, we arrive, as a consequence of the energy estimate (3), at the following:     ˆα − U ˆ0 − grady q ∗ x  U ∈ Zα (Ω),  α  α          ˆα − U ˆ0 − grady qα∗ x  = 0(α) in Ω,  div γα U α     ∗ x  ˆ ˆ  curl U = 0 in Ω, − U − grad q α 0  y α  α          U ˆα − U ˆ0 − grady q ∗ x × n = 0 on ∂Ω. α α From [6], we know that this yields the estimate         ˆ ∗ x  ˆα − U ˆ0 − grady q ∗ x  ˆ + U − U − grad q ≤ Cα,  div γα U    α 0 y α α α α L2 (Ω) L2 (Ω) and so



  ˆ0 − grady q ∗ x ˆα − U · ν|+ = 0(α) U α α

on ∂(αB).

Let q∗ be the unique (scalar) solution to  2   ∆q∗ = 0 in R \ B and in B,     q∗ is continuous across ∂B,      ∂q∗  ∂q∗  ˆ0 (0) · ν γ − γ = −(γ0 − γ∗ )U  0 + ∗  ∂ν  ∂ν −        lim q∗ = 0.

on ∂B,

|y|→+∞

From [12, Theorem 1], it follows that   x    ≤ Cα1/2 ,  (grady q∗ − grady qα∗ ) α L2 (Ω) which yields



  ˆα − U ˆ0 − grady q∗ x · ν = o(1) U α

on ∂(αB).

Writing q∗ in terms of Φ∗ gives   

  T ∂uα  γ0 ∂Φ∗  + (αy) − ν · grad u(0, t) − −1 (y) · grad u(0, t) θ(t) dt = o(1) ∂ν ∂(αB) γ∗ ∂ν + 0 for any θ ∈ C0∞ (]0, T [). In view of (9), the remainder o(1) in the above asymptotic formula is bounded by Cα ||θ||H 2 (0,T ) , where the constant Cα is independent of θ and goes to zero as α → 0. Therefore,  

 ∂uα  γ0 ∂Φ∗  + (αy) − ν · grad u(0, t) − (y) · grad u(0, t) = o(1) −1 ∂ν ∂(αB) γ∗ ∂ν + holds in a weak sense, and so, by iterating the same argument for the case of m (well-separated) inhomogeneities zj + αBj , j = 1, . . . , m, we arrive at the promised asymptotic formula (6).

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HABIB AMMARI

4. The identification procedure. Let β(x) ∈ C0∞ (Ω) be a cutoff function such that β(x) ≡ 1 in a subdomain Ω of Ω that contains the inhomogeneities Bα . For an arbitrary η ∈ R2 , we assume that we are in possession of the boundary measurements of ∂uα ∂n for

on Γ × (0, T )

√ ϕ(x) = ϕη (x) = eiη·x , ψ(x) = ψη (x) = −i γ0 |η|eiη·x and f (x, t) = fη (x, t) = eiη·x−i

√

γ0 |η|t

.

This particular choice of data ϕ, ψ, and f implies that the background solution u of the wave equation (2) in the absence of any inhomogeneity is given by u(x, t) = uη (x, t) = eiη·x−i

√

γ0 |η|t

in Ω × (0, T ).

Suppose that T and the part Γ of the boundary ∂Ω are such that they geometrically control Ω, which roughly means that every geometrical optic ray, starting at any point x ∈ Ω at time t = 0, hits Γ before time T at a nondiffractive point; see [7]. Then, from [19, Theorem 6.4, p. 75] and [7], it follows that, for any η ∈ R2 , we can construct by the Hilbert uniqueness method a unique gη ∈ H01 (0, T ; L2 (Γ)) in such a way that the unique weak solution wη in C 0 (0, T ; L2 (Ω)) ∩ C 1 (0, T ; H −1 (Ω)) of the wave equation  2 (∂t − γ0 ∆)wη = 0 in Ω × (0, T ),       wη |t=0 = β(x)eiη·x ∈ H01 (Ω),    ∂t wη |t=0 = 0 in Ω, (10)     wη |Γ×(0,T ) = gη ,      w | η ∂Ω\Γ×(0,T ) = 0, satisfies wη (T ) = ∂t wη (T ) = 0. Let vα,η ∈ C 0 (0, T ; L2 (Ω)) ∩ C 1 (0, T ; H −1 (Ω)) be defined by  2 (∂t − γ0 ∆)vα,η = 0 in Ω × (0, T ),       vα,η |t=0 = 0 in Ω,     

 m  γ0 γ0 ∂Φj  η · νj + ∂t vα,η |t=0 = i 1− −1 |+ eiη·zj δ∂(zj +αBj ) in Ω,   γ γ ∂ν  j j j  j=1     vα,η |∂Ω×(0,T ) = 0. ∂Φ

Since ∂νjj |+ (y)δ∂(zj +αBj ) ∈ H −1 (Ω) for j = 1, . . . , m, the existence and uniqueness of a solution vα,η can be established by transposition; see [20] and [19, Theorem 4.2, p. ∂vα,η 46]. Indeed, we can prove that ∂n |Γ ∈ H −1 (0, T ; L2 (Γ)). To do so, let θ be defined as  θ ∈ H01 (Ω),     

 m  γ0 γ0 ∂Φj  η · νj + i 1− −1 |+ eiη·zj δ∂(zj +αBj ) ∈ H −1 (Ω) in Ω,  γ0 ∆θ = γ γ ∂ν j j j j=1

RECONSTRUCTION OF DIELECTRIC IMPERFECTIONS

and introduce

 z(x, t) =

0

t

1203

vα,η (x, s) ds + θ(x) ∈ L2 (Ω).

It is easy to see that z satisfies the initial boundary value problem  2  (∂t − γ0 ∆)z = 0 in Ω, z| = θ ∈ H01 (Ω), ∂t z|t=0 = 0 in Ω,  t=0 z|∂Ω×(0,T ) = 0. Classical regularity results (see [19, Theorem 4.1, p. 44]) yield ∂z |Γ ∈ L2 (0, T ; L2 (Γ)), ∂n ∂v

α,η ∂z and so ∂n |Γ = ∂t ( ∂n |Γ ) ∈ H −1 (0, T ; L2 (Γ)). The following holds. Proposition 4.1. Suppose that Γ and T geometrically control Ω. For any η ∈ R2 , we have 

   m  γ0 γ0 ∂Φj e2iη·zj η · νj + α i 1− −1 |+ (y) eiαη·y dsj (y) γj γj ∂νj ∂Bj j=1

 = −γ0 T 

0

T

 Γ

gη

∂vα,η . ∂n

∂v

α,η Here 0 Γ gη ∂n is in the sense of the duality pairing between H01 (0, T ) and H (0, T ). Proposition 4.1 is obtained by multiplying (∂t2 − γ0 ∆)vα,η = 0 by wη and integrating by parts over (0, T ) × Ω. In fact, we have

−1

 −γ0

T



0

gη

Γ

 m  ∂vα,η γ0 =α eiη·zj η i 1− ∂n γ j 

 j=1 x − zj γ0 ∂Φj δ∂(zj +αBj ) eiη·x β(x) dx, · νj + −1 |+ γj ∂νj α Ω

where the integral on the right-hand side is in the sense of the duality pairing between H01 (Ω) and H −1 (Ω). Thus  −γ0

0

T

 m  ∂vα,η γ0 =α eiη·zj η gη i 1− ∂n γ j Γ 

 j=1 x − zj γ0 ∂Φj eiη·x dsj (x) · νj + −1 |+ γ ∂ν α j j ∂(zj +αBj )



since β(x) ≡ 1 in a subdomain Ω of Ω that contains the inhomogeneities Bα . By a change of variables, the above identity leads to the desired formula. Taking now Taylor expansion of eiαη·y and having in mind that [12]

   γ0 ∂Φj νj + −1 |+ (y) dsj (y) = 0, γj ∂νj ∂Bj

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HABIB AMMARI

we obtain the more convenient asymptotic formula. Proposition 4.2. Suppose that Γ and T geometrically control Ω. For any η ∈ R2 , we have α2

m  j=1

1−

γ0 γj



e2iη·zj η ·

 ∂Bj



= γ0

νj +

0

T

 Γ

gη



γ0 −1 γj

 ∂Φj |+ (y) η · y dsj (y) ∂νj

∂vα,η + o(α2 ). ∂n

Next, for any η ∈ R2 , let θη denote the solution to the Volterra equation of the second kind:   T √ √    ∂ θ (x, t) + e−i γ0 |η|(s−t) (θη (x, s) − i γ0 |η|∂t θη (x, s)) ds = gη (x, t) t η  t (11)    for x ∈ Γ, t ∈ (0, T ),  θη (x, 0) = 0 for x ∈ Γ. The existence and uniqueness of this θη in H 1 (0, T ; L2 (Γ)) for any η ∈ R2 can be established using the resolvent kernel. Since gη ∈ H01 (0, T ; L2 (Γ)), the solution θη belongs, in fact, to H 2 (0, T ; L2 (Γ)). Note that it was Yamamoto [32] who first conceived the idea of using such a Volterra equation to apply the geometrical control for solving inverse source problems. We also note from differentiation of (11) with respect to t that θη is the unique solution of the ODE √ √  2 ∂t θη − θη = ei γ0 |η|t ∂t (e−i γ0 |η|t gη ) for x ∈ Γ, t ∈ (0, T ), (12) θη (x, 0) = 0, ∂t θη (x, T ) = 0 for x ∈ Γ. Therefore, the function θη may be found in practice explicitly with variation of parameters. It also immediately follows from this observation that θη belongs to H 2 (0, T ; L2 (Γ)) since gη ∈ H01 (0, T ; L2 (Γ)). To identify the locations and certain properties of the small inhomogeneities Bα , α let us view the averaging of the boundary measurements ∂u ∂n |Γ×(0,T ) , using the solution θη to the Volterra equation (11) or, equivalently, the ODE (12) as a function of η. The following holds. Theorem 4.3. Let η ∈ R2 . Let uα be the unique solution in C 0 (0, T ; H 1 (Ω)) ∩ C 1 (0, T ; L2 (Ω)) to the wave equation (1) with √ √ ϕ(x) = eiη·x , ψ(x) = −i γ0 |η|eiη·x , and f (x, t) = eiη·x−i γ0 |η|t .

Suppose that Γ and T geometrically control Ω; then we have  

 T  ∂u ∂u ∂uα ∂uα − + ∂t θη ∂t − θη ∂n ∂n ∂n ∂n 0 Γ 

 T √ √ ∂u ∂uα − =− ei γ0 |η|t ∂t (e−i γ0 |η|t gη ) ∂n ∂n 0 Γ (13)

 m  γ0   = α2 − 1 e2iη·zj Mj (η) · η − |η|2 |Bj | γj j=1 + o(α2 ),

RECONSTRUCTION OF DIELECTRIC IMPERFECTIONS

1205

where θη is the unique solution to the ODE (12), with gη defined as the boundary control in (10), and Mj is the polarization tensor of Bj , defined by  

  γ0 ∂Φj νj + (14) −1 |+ (y) y · el dsj (y) . (Mj )k,l = ek · γj ∂νj ∂Bj Here (e1 , e2 ) is an orthonormal basis of R2 . Proof. The first identity in (13) follows from integration by parts and use of the fact that θη is the solution to the ODE (12). T  ∂u ∂u α ∂ θ ∂ ( ∂uα − ∂n ) has From ∂t θη (T ) = 0 and ( ∂u ∂n − ∂n )|t=0 = 0, the term 0 Γ t η t ∂n to be interpreted as follows:  

 T  T ∂u ∂u ∂uα ∂uα 2 − =− (15) − . ∂t θη ∂t ∂t θη ∂n ∂n ∂n ∂n 0 0 Γ Γ Next, introducing u α,η (x, t) = u(x, t) − γ0

 0

t

e−i

√

γ0 |η|s

vα,η (x, t − s) ds, x ∈ Ω, t ∈ (0, T ),

we rewrite  

 T  ∂u ∂u ∂uα ∂uα − + ∂t θη ∂t − θη ∂n ∂n ∂n ∂n 0 Γ  

 T  ∂ uα,η ∂ uα,η ∂uα ∂uα − + ∂t θη ∂t − θη = ∂n ∂n ∂n ∂n 0 Γ   T   t  t √ √ −i γ0 |η|s ∂vα,η −i γ0 |η|s ∂vα,η (x, t − s) ds + ∂t θη ∂t (x, t − s) ds . θη −γ0 e e ∂n ∂n 0 0 0 Γ Since θη satisfies the Volterra equation (11) and  

 t

 t √ √ √ −i γ0 |η|s ∂vα,η −i γ0 |η|t i γ0 |η|s ∂vα,η (x, t − s) ds = ∂t e (x, s) ds ∂t e e ∂n ∂n 0 0  t √ √ ∂vα,η ∂vα,η √ −i γ0 |η|t (x, s) ds + (x, t), = −i γ0 |η|e ei γ0 |η|s ∂n ∂n 0 we obtain by integrating by parts over (0, T ) that   t  T   t √ √ −i γ0 |η|s ∂vα,η −i γ0 |η|s ∂vα,η (x, t − s) ds + ∂t θη ∂t (x, t − s) ds θη e e ∂n ∂n 0 Γ 0 0    T  T √ ∂vα,η i γ0 |η|(t−s) = θη (s)e ds (x, t) ∂t θη + 0 t Γ ∂n  t √ √ ∂vα,η √ (x, s) ds dt ei γ0 |η|s −i γ0 |η|(e−i γ0 |η|t ∂t θη (t)) ∂n 0    T  T √ ∂vα,η √ i γ0 |η|(t−s) = (x, t) ∂t θη + (θη (s) − i γ0 |η|∂t θη (s))e ds dt 0 t Γ ∂n  T ∂vα,η (x, t) dt, gη (x, t) = ∂n 0 Γ

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HABIB AMMARI

and so, from Proposition 4.2, we obtain that  

  ∂uα ∂uα ∂u ∂u + ∂t θη ∂t θη − − ∂n ∂n ∂n ∂n 0 Γ



   m  γ0 γ0 ∂Φj 2 2iη·zj = −α 1 − e ν η · + − 1 | (y) η · y dsj (y) (16) j + γj γj ∂νj ∂Bj j=1 

T



+

T

0

 

  ∂uα ∂uα ∂ uα,η ∂ uα,η + ∂t θη ∂t + o(α2 ). θη − − ∂n ∂n ∂n ∂n Γ

In order to prove Theorem 4.1, it suffices then to find the leading order term in the asymptotic expansion of  0

T

 

  ∂ uα,η ∂ uα,η ∂uα ∂uα − + ∂t θη ∂t − . θη ∂n ∂n ∂n ∂n Γ

Let hα,η ∈ C 0 (0, T ; H01 (Ω)) ∩ C 1 (0, T ; L2 (Ω)) be the solution to  2 (∂t − γ0 ∆)hα,η = 0 in Ω × (0, T ),       hα,η |t=0 = 0 in Ω,     m  γ0 2  1 − eiη·x χ(zj + αBj ) ∂ h | = −γ |η| t α,η t=0 0   γ  j  j=1     hα,η |∂Ω×(0,T ) = 0,

in Ω,

where χ(zj + αBj ) denotes the characteristic function of the inhomogeneity zj + αBj . Since

 t   √  2 −i γ0 |η|s  (∂t − γ0 ∆) e vα,η (x, t − s) ds    0    

  m    √ γ0 γ0 ∂Φj    η · νj + = i 1− −1 |+ (y) eiη·zj δ∂(zj +αBj ) e−i γ0 |η|t   γ γ ∂ν j j j   j=1    in Ω × (0, T ),  

 t  √ −i γ0 |η|s  e vα,η (x, t − s) ds |t=0 = 0,    0   

 t    √  −i γ0 |η|s   e v (x, t − s) ds |t=0 = 0 in Ω, ∂ t α,η    0   

 t   √   −i γ |η|s 0  e vα,η (x, t − s) ds |∂Ω×(0,T ) = 0 0

1207

RECONSTRUCTION OF DIELECTRIC IMPERFECTIONS

and

 t   √  2 −i γ0 |η|s  (∂t − γ0 ∆) e hα,η (x, t − s) ds    0     m    √ γ0  2   = −γ0 |η| 1− eiη·x χ(zj + αBj )e−i γ0 |η|t   γj   j=1     t  √ −i γ0 |η|s e h (x, t − s) ds |t=0 = 0, α,η   0   

 t    √  −i γ0 |η|s   e h (x, t − s) ds |t=0 = 0 in Ω, ∂ t α,η    0   

 t   √   −i γ0 |η|s  e hα,η (x, t − s) ds |∂Ω×(0,T ) = 0,

in Ω × (0, T ),

0

setting  hα,η =

t 0

e−i

√

γ0 |η|s

hα,η (x, t − s) ds, it follows that

(∂t2 − γ0 ∆)(uα − u α,η −  hα,η ) = γ0

m  j=1

+

m  j=1

γ0 i 1− γj

1−

γ0 γj





∂uα − |+ + η · ∂n



νj +

√ γ0 |η|t

(∂t2 uα + |η|2 γ0 eiη·x−i



γ0 −1 γj





√ ∂Φj |+ (y) eiη·zj e−i γ0 |η|t δ∂(zj +αBj ) ∂νj

)χ(zj + αBj ),

and, therefore, by Propositions 3.1 and 2.1, we readily get that

(17)

 2 (∂ − γ0 ∆)(uα − u α,η −  hα,η ) = o(α2 ) in Ω × (0, T ),    t (uα − u α,η −  hα,η )|t=0 = 0, ∂t (uα − u α,η −  hα,η )|t=0 = 0    (uα − u α,η −  hα,η )|∂Ω×(0,T ) = 0,

in Ω,

where the right-hand side in the first equation in (17) is of order o(α2 ) in the H −2 (0, T ; H −1 (Ω)) norm. Following the proof of Proposition 2.1, we immediately obtain that ||uα − u α,η −  hα,η ||L2 (Ω) = o(α2 ),

t ∈ (0, T ), x ∈ Ω,

where the remainder o(α2 ) is independent of the points {zj }m j=1 . We now show that the estimate    ∂    = o(α2 ) (18) α,η − hα,η )  ∂n (uα − u  2 L (0,T ;L2 (Γ)) holds, which will immediately imply that  

 T  ∂ uα,η ∂ uα,η ∂uα ∂uα − + ∂t θη ∂t − θη ∂n ∂n ∂n ∂n 0 Γ (19)  T ∂ hα,η ∂ hα,η + ∂t θη ∂t + o(α2 ). θη = ∂n ∂n 0 Γ

1208

HABIB AMMARI

Let θ be given in C0∞ (]0, T [), and define  T ˆ u α,η (x, t)θ(t) dt, u α,η (x) = 0

ˆ  hα,η (x) = and

 0

 u ˆα (x) =

We have

T

0

T

 hα,η (x, t)θ(t) dt,

uα (x, t)θ(t) dt.

 ˆ ˆ  div γα grad(ˆ uα − u α,η −  hα,η ) = o(α2 ) ∈ L2 (Ω \ Ω ) ∩ H −1 (Ω), 

ˆ ˆ (ˆ uα − u α,η −  hα,η ) = 0 on ∂Ω,

which implies from [34] that     ∂ ˆ 2 ˆ   (ˆ α,η − hα,η )  2 = o(α )  ∂n uα − u L (Γ) for all θ ∈ C0∞ (]0, T [), whence    ∂  2   (uα − u α,η − hα,η )  ∂n  2 = o(α ) a.e. in t ∈ (0, T ), L (Γ) and so the desired estimate (18) holds. On the other hand, analogously to Proposition 4.1, by integration by parts and taking the Taylor expansion of eiη·x in zj + αBj , the following holds:  (20)

0

T

 m  γ0 ∂hα,η 2 1− e2iη·zj |η|2 |Bj | + o(α2 ). gη (x, t) (x, t) dt = α ∂n γ j Γ j=1



However,  (21)

0

T



∂hα,η (x, t) dt = gη (x, t) ∂n Γ

 0

T

 Γ

θη

∂ hα,η ∂ hα,η + ∂t θη ∂t , ∂n ∂n

and so, combining (16), (19), (20), and (21), we arrive at our promised asymptotic formula (13). The proof of Theorem 4.1 is then over. We are now in position to describe our identification procedure, which is based on Theorem 4.1. Let us neglect the asymptotically small remainder in the asymptotic formula (13) and define Λα (η) by  

 T  ∂u ∂u ∂uα ∂uα − + ∂t θη ∂t − . θη Λα (η) = ∂n ∂n ∂n ∂n 0 Γ The function Λα (η) is computed in the following way. First, we construct the control gη in (10) for given η ∈ R2 . Then we solve the ODE (12) to find the auxiliary test

RECONSTRUCTION OF DIELECTRIC IMPERFECTIONS

1209

α function θη . From the boundary measurements ∂u ∂n |Γ×(0,T ) , we form the integrals that come in the expression of Λα (η). Recall that the function e2iη·zj is exactly the Fourier transform (up to a multiplicative constant) of the Dirac function δ−2zj (a point mass located at −2zj ). From Theorem 4.3, it follows that the function Λα (η) is (approximately) the Fourier transform of a linear combination of derivatives of point masses, or

˘ α (η) ≈ α2 Λ

m 

Lj δ−2zj ,

j=1

where Lj is a second order constant coefficient, differential operator whose coefficients depend on the polarization tensor Mj defined by (14) (see [12] for its properties) and ˘ α (η) represents the inverse Fourier transform of Λα (η). The reader is referred to Λ [12] for properties of the tensor polarization Mj . The method of reconstruction we propose here consists, as in [4], in sampling ˘ α (η) at some discrete set of points and then calculating the corresponding values of Λ discrete inverse Fourier transform. After a rescaling by − 12 , the support of this discrete inverse Fourier transform yields the location of the small inhomogeneities Bα . This procedure generalizes the approach that we developed in [4] for the two-dimensional (time-independent) inverse conductivity problem. On other terms, once Λα (η) is computed from dynamic boundary measurements on Γ, we calculate its inverse Fourier transform. The asymptotic formula (13) in Theorem 4.1 asserts that this inverse Fourier transform is a distribution supported at the locations (zj )m j=1 . Once the locations are known, we may calculate the polarization tensors (Mj )m j=1 by solving an appropriate linear system arising from (13). These polarization tensors give ideas on the orientation and relative size of the inhomogeneities [18]. We wish to point out that, from the leading order term of Λα (η) given by (13), we cannot reconstruct more details of the shapes of the domains Bj . Higher order terms in the asymptotic expansion of Λα (η), with respect to α, are needed to reconstruct the domains Bj with high resolution. The number of data (sampling) points needed for an accurate discrete Fourier inversion of Λα (η) follows from Shannon’s sampling theorem [13]. We need (conservatively) order ( hδ )2 sampled values of Λα (η) to reconstruct, with resolution δ, a collection of inhomogeneities that lie inside a square of side h. In order to simulate α errors in the measurements of ∂u ∂n on Γ × (0, T ), as well as the errors inherent in the approximation (13) and in the calculations of gη , θη , and Λα (η) (by some quadrature rules), we should add random noise to the values of Λα (η). Numerical experiments in [4] for the two-dimensional (time-independent) inverse conductivity problem seem to suggest that the method is quite stable with respect to noise in measurements and errors in the different approximations. We are convinced that the use of approximate formulae such as (13) represents a very promising approach to the dynamical identification of small inhomogeneities that are embedded in a homogeneous medium. In particular, our method can be extended to solve the dynamical identification problem of small incompressible or rigid inclusions. Formally, we can recover these two cases by letting γj tend to +∞ or 0 in (5) and the asymptotic formula (13). Rigorously, to assert that (13) is still valid for incompressible or rigid inclusions, we should prove that the term o(α2 ) is uniform in γj as γj → +∞ or 0. We also believe that our method yields a good approximation to small amplitude perturbations in the conductivity (γα (x) = γ0 + αγ1 (x)) from the α measurements of ∂u ∂n on Γ × (0, T ). Our method may yield the Fourier transform of

1210

HABIB AMMARI

the perturbation γ1 (x). This inverse problem is considered in [2]. Finally, we wish to emphasize the fact that, in the algorithm described in this paper, the locations zj , j = 1, . . . , m, of the inhomogeneities are found with an error O(α), and only the polarization tensors of the domains Bj can be reconstructed. α + Making use of higher order terms in the asymptotic expansion of ∂u ∂νj |∂(zj +αBj ) , we certainly would be able to reconstruct the small inhomogeneities with higher resolution from dynamical boundary measurements on part of the boundary and capture more details of the geometries of the domains Bj . Perhaps, more importantly, this would also allow us to identify quite general conductivity inhomogeneities without restrictions on their sizes. Results in this direction are now available for the conductivity problem. In [3], based on layer potential techniques, high order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of a finite number of diametrically small inhomogeneities with conductivities different from the background conductivity are rigorously derived. In [5], similar accurate asymptotic formulae are applied for the purpose of identifying the location and certain properties of the shape of the conductivity inhomogeneities. A real-time algorithm with a very high resolution and accuracy that makes use of constant current sources is designed. We believe that the results and techniques of [3] and [5] could be combined with the approach developed in this paper for recovering the small electromagnetic inhomogeneities from dynamic boundary measurements with higher resolution and accuracy. This very important issue will be considered in a forthcoming work. Acknowledgments. The author expresses his thanks to M. Vogelius for various interesting discussions. He is also very grateful to the referees for their comments, which enabled him to make many improvements to the presentation. REFERENCES [1] C. Alves and H. Ammari, Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium, SIAM J. Appl. Math., 62 (2001), pp. 94–106. [2] H. Ammari, Identification of small amplitude perturbations in the electromagnetic parameters from partial dynamic boundary measurements, J. Math. Anal. Appl., submitted; also available online from http://www.cmap.polytechnique/˜ammari/˜preprints. [3] H. Ammari and H. Kang, High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter, SIAM J. Math. Anal., submitted; also available online from http://www.cmap.polytechnique/˜ammari/˜preprints. [4] H. Ammari, S. Moskow, and M. Vogelius, Boundary integral formulas for the reconstruction of electromagnetic imperfections of small diameter, ESAIM Control Optim. Calc. Var., to appear; also available online from http://www.cmap.polytechnique/˜ammari/˜preprints. [5] H. Ammari and J. K. Seo, A new algorithm for the reconstruction of conductivity inhomogeneities, J. Amer. Math. Soc., submitted; also available online from http://www.cmap.polytechnique/˜ammari/˜preprints. [6] H. Ammari, M. Vogelius, and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. The full Maxwell equations, J. Math. Pures Appl. (9), 80 (2001), pp. 769–814. [7] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), pp. 1024–1065. [8] M. I. Belishev and Ya Kurylev, Boundary control, wave field continuation and inverse problems for the wave equation, Comput. Math. Appl., 22 (1991), pp. 27–52. [9] E. Beretta, A. Mukherjee, and M. Vogelius, Asymptotic formuli for steady state voltage potentials in the presence of conductivity imperfection of small area, Z. Angew. Math. Phys., 52 (2001), pp. 543–572.

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[10] G. Bruckner and M. Yamamoto, Determination of point wave sources by pointwise observations: Stability and reconstruction, Inverse Problems, 16 (2000), pp. 723–748. ´ n, On an inverse boundary value problem, in Proceedings of a Seminar on [11] A. P. Caldero Numerical Analysis and its Applications to Continuum Physics, Sociedade Brasileira de Matem´ atica, Rio de Janeiro, Brazil, 1980, pp. 65–73. [12] D. J. Cedio-Fengya, S. Moskow, and M. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computional reconstruction, Inverse Problems, 14 (1998), pp. 553–595. [13] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 61, SIAM, Philadelphia, 1992. [14] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, AMS, Providence, RI, 1998. [15] A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Ration. Mech. Anal., 105 (1989), pp. 299–326. [16] M. Grasselli and M. Yamamoto, Identifying a spatial body force in linear elastodynamics via traction measurements, SIAM J. Control Optim., 36 (1998), pp. 1190–1206. [17] V. Isakov, Inverse Source Problems, AMS, Providence, RI, 1990. [18] R. E. Kleinman and T. B. A. Senior, Rayleigh scattering, in Low and High Frequency Asymptotics, V. K. Varadan and V. V. Varadan, eds., North–Holland, Amsterdam, 1986, pp. 1–70. [19] J.-L. Lions, Contrˆ olabilit´ e exacte, Perturbations et Stabilisation de Syst` emes Distribu´ es, Tome 1, Contrˆ olabilit´ e Exacte, Masson, Paris, 1988. [20] J.-L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, 1972. [21] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), pp. 71–96. [22] S. Nicaise, Exact boundary controllability of Maxwell’s equations in heteregeneous media and an application to an inverse source problem, SIAM J. Control Optim., 38 (2000), pp. 1145–1170. [23] J.-P. Puel and M. Yamamoto, Applications de la contrˆ olabilit´ e exacte a ` quelques prob`lemes inverses hyperboliques, C. R. Acad. Sci. Paris S´er. I Math., 320 (1995), pp. 1171–1176. [24] J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems, 12 (1996), pp. 995–1002. [25] J.-P. Puel and M. Yamamoto, Smoothing property in multidimensional inverse hyperbolic problems: Applications to uniqueness and stability, J. Inverse Ill-Posed Prob., 4 (1996), pp. 283–296. [26] J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional inverse hyperbolic problem, J. Inverse Ill-Posed Prob., 5 (1997), pp. 55–83. [27] Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), pp. 87–96. [28] V. G. Romanov and S. I. Kabanikhin, Inverse Problems for Maxwell’s Equations, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. [29] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), pp. 153–169. [30] Z. Sun, On the continuous dependence for an inverse initial boundary value problem for the wave equation, J. Math. Anal. Appl., 150 (1990), pp. 188–204. [31] M. Yamamoto, Well-posedness of some inverse hyperbolic problems by the Hilbert uniqueness method, J. Inverse Ill-Posed Prob., 2 (1994), pp. 349–368. [32] M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), pp. 481–496. [33] M. Yamamoto, Determination of forces in vibrations of beams and plates by pointwise and line observations, J. Inverse Ill-Posed Prob., 4 (1996), pp. 437–457. [34] M. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 723–748.