Electromagnetic Inverse Problems and Generalized Sommerfeld Potentials

Ola, Petri; Somersalo, Erkki

doi:10.1137/s0036139995283948

Cited by 93 publications

(132 citation statements)

References 7 publications

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“…Rather than construct solutions to (2.1) directly, we follow the idea of Ola and Somersalo in [8] and introduce a new 8 × 8 system…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…In [8], where chirality was not taken into account (β = 0), the above system included no first order term N , and so the authors were able to use the methods of [12] to construct exponentially growing solutions to a Schrödinger equation. When β = 0, such a reduction does not seem possible, and so here we must construct solutions to a first order perturbation of the Laplacian.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…A natural question to ask is, by rescaling the fields (E, H) can a system be found that has no first order term, in which case we would have a Schrödinger equation? Such a system was achieved in [8] for a non-chiral body by rescaling the fields. For a chiral body, however, the answer to this appears to be no; suppose that we write (A, B) = R(E, H) for some invertible matrix R of the form R = r 11 I 3 r 12 I 3 r 21 I 3 r 22 I 3 , and we set…”

Section: B)mentioning

confidence: 99%

“…In [8] Ola and Somersalo simplified the proof of interior identifiability in [7] by constructing a second order system of differential equations, which has as its principal part the Laplacian, in such a way that solutions to this system yields solutions to Maxwell's equations. They were able to construct a system with no first order part, that is a Schrödinger equation, and then use the results of [12] to construct exponentially growing solutions.…”

Section: Introductionmentioning

confidence: 99%

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An electromagnetic inverse problem in chiral media

McDowall

2000

Trans. Amer. Math. Soc.

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Abstract. We consider the inverse boundary value problem for Maxwell's equations that takes into account the chirality of a body in R 3 . More precisely, we show that knowledge of a boundary map for the electromagnetic fields determines the electromagnetic parameters, namely the conductivity, electric permittivity, magnetic permeability and chirality, in the interior. We rewrite Maxwell's equations as a first order perturbation of the Laplacian and construct exponentially growing solutions, and obtain the result in the spirit of complex geometrical optics.

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“…Rather than construct solutions to (2.1) directly, we follow the idea of Ola and Somersalo in [8] and introduce a new 8 × 8 system…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Section: Statement Of the Resultsmentioning

confidence: 99%

Section: B)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An electromagnetic inverse problem in chiral media

McDowall

2000

Trans. Amer. Math. Soc.

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“…The admittance map for isotropic Maxwell's equations determines uniquely the isotropic electric permittivity, magnetic permeability and conductivity [147]. This system can in fact be reduced to the Schrödinger equation − Q with Q an 8 × 8 system and the Laplacian times the identity matrix [148].…”

Section: Other Applicationsmentioning

confidence: 99%

Inverse problems: seeing the unseen

2014

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This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderón's problem since the famous analyst proposed it in the mathematical literature (Calderón in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. [65][66][67][68][69][70][71][72][73] 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?Communicated by Ari Laptev.

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