Carleman estimate for Maxwell's equations in anisotropic media and the observability inequality

Li, Shumin; Yamamoto, Masahiro

doi:10.1088/1742-6596/12/1/011

Cited by 8 publications

(13 citation statements)

References 12 publications

(16 reference statements)

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“…Since observability estimates have been proven for the systems of elasticity and electromagnetism (cf. [3] and [26], respectively), it is to be expected that stability and convergence estimates similar to the ones derived here can be obtained.…”

Section: Discussionsupporting

confidence: 79%

The Quasi-Reversibility Method for Thermoacoustic Tomography in a Heterogeneous Medium

Clason¹,

Klibanov²

2008

SIAM J. Sci. Comput.

102

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In this paper we consider thermoacoustic tomography as the inverse problem of determining from lateral Cauchy data the unknown initial conditions in a wave equation with spatially varying coefficients. This problem also occurs in several applications in the area of medical imaging and non-destructive testing. Using the method of quasi-reversibility, the original ill-posed problem is replaced with a boundary value problem for a fourth order partial differential equation. We find a weak H 2 solution of this problem and show that it is a well-posed elliptic problem. Error estimates and convergence of the approximation follow from observability estimates for the wave equation, which are proven using a Carleman estimate. We derive a numerical scheme for the solution of the quasi-reversibility problem by a B-spline Galerkin method, for which we give error estimates. Finally, we present numerical results supporting the robustness of this method for the reconstruction of initial conditions from full and limited boundary data.

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Section: Discussionsupporting

confidence: 79%

The Quasi-Reversibility Method for Thermoacoustic Tomography in a Heterogeneous Medium

Clason¹,

Klibanov²

2008

SIAM J. Sci. Comput.

102

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“…For an overview of inverse problems for the Maxwell system, see the monograph [39] by Romanov and Kabanikhin. For actual examples of inverse problems for the dynamical Maxwell system involving infinitely many boundary observations (this is the case when the identification of the electromagnetic coefficients is made from the Dirichlet-to-Neumann map), we refer to Beleshev and Isakov [3], Caro [12], Caro, Ola and Salo [13], Kurylev, Lassas and Somersalo [31], Ola, Paivarinta and Somersalo [35] and Salo, Kenig and Uhlmann [40]. It turns out that a small number of uniqueness and stability results for the inverse problem of determining the electromagnetic parameters of the Maxwell system with a finite number of measurements are available, such as [33,34]. In both cases, their proof is based on the methodology of [10] or [23], which is by means of a Carleman estimate.…”

Section: Inverse Problemmentioning

confidence: 99%

“…By the methodology by [10] or [23] with such Carleman estimates, several uniqueness and stability results are available for the inverse problem for the Maxwell's system (1.1). That is, in [33]- [34] Li and Yamomoto established the uniqueness in determining three coefficients, using finite number of measurements.…”

Section: Inverse Problemmentioning

confidence: 99%

Inverse boundary value problem for the dynamical heterogeneous Maxwell's system

2012

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We consider the inverse problem of determining the isotropic inhomogeneous electromagnetic coefficients of the non-stationary Maxwell equations in a bounded domain of R 3 , from a finite number of boundary measurements. Our main result is a Hölder stability estimate for the inverse problem, where the measurements are exerted only in some boundary components. For it, we prove a global Carleman estimate for the heterogeneous Maxwell's system with boundary conditions. for some λ 0 > 0 and µ 0 > 0. Next, in view of deriving existence and uniqueness results for (1.1), introduce the following functional spaceand denote by γ τ the unique linear continuous application from H(curl ,we see that the operator iA, whereFurther, in light of the last line of (1.1), set H(div 0, Ω) := {u ∈ L 2 (Ω) 3 , div u = 0} and H 0 (div 0, Ω) := {u ∈ H(div 0, Ω), γ n u = 0}, where γ n is the unique linear continuous mapping from H(div , Ω) := {u ∈ L 2 (Ω) 3 , div u ∈ L 2 (Ω)} onto H −1/2 (Γ), such that γ n u = u · ν when u ∈ C ∞ 0 (Ω) (see [16][Chap. IX A, Theorem 1]). Since H 0 := H(div 0; Ω) × H 0 (div 0; Ω) is a closed subspace of H and that H ⊥ 0 ⊂ ker A, the restriction A 0 Φ = A H 0 Φ := AΦ, Φ ∈ Dom(A 0 ) = Dom(A) ∩ H 0 := V, is, by Stone's Theorem [17][Chap. XVII A, §4, Theorem 3], the infinitesimal generator of a unitary group of class C 0 in H 0 . Thus, by rewriting (1.1)-(1.2) into the equivalent form Φ ′ = A 0 Φ Φ(0) = Φ 0 , with Φ = (D, B) T and Φ 0 = (D 0 , B 0 ) T , we get that: Lemma 1.1 Given (D 0 , B 0 ) ∈ V there exists a unique strong solution (D, B) to (1.1) starting from (D 0 , B 0 ) within the following class (D, B) ∈ C 0 (R; V) ∩ C 1 (R; H). (1.4) Moreover it holds true from [16][Chap. IX A, Remark 1] that V = H τ,0 (curl , div 0; Ω) × H n,0 (curl , div 0; Ω),where H * ,0 (curl , div 0; Ω) = {u ∈ H 1 (Ω) 3 , div u = 0 and γ * u = 0}, * = τ, n.

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“…As for the method of applying Carleman estimate to derive observability inequalities, see [22,27]. As for observability inequalities by Carleman estimates, see further [7,8,17,21,[28][29][30][31]33].…”

Section: Remark 11mentioning

confidence: 99%

Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality

Yuan

Yamamoto

2010

Chin. Ann. Math. Ser. B

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Carleman estimate for Maxwell's equations in anisotropic media and the observability inequality

Cited by 8 publications

References 12 publications

The Quasi-Reversibility Method for Thermoacoustic Tomography in a Heterogeneous Medium

The Quasi-Reversibility Method for Thermoacoustic Tomography in a Heterogeneous Medium

Inverse boundary value problem for the dynamical heterogeneous Maxwell's system

Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality

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