An inverse boundary value problem in electrodynamics

Ola, Petri; Päivärinta, Lassi; Somersalo, Erkki

doi:10.1215/s0012-7094-93-07014-7

Cited by 170 publications

(136 citation statements)

References 16 publications

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“…presented the boundary admittance map for time-harmonic fields at a fixed frequency for non-chiral bodies, and posed the inverse problem of whether or not the material parameters of a body could be determined from knowledge of this boundary map. In [9] it was shown that this is in fact the case assuming knowledge of the parameters near the boundary. In [8], the second author showed that the admittance map is well defined also in the case of a chiral body, and that knowledge of this map determines the material parameters of the body, including the chirality, throughout the body, assuming that the parameters are known to agree to infinite order at the boundary.…”

Section: Introductionmentioning

confidence: 99%

Total determination of material parameters from electromagnetic boundary information

Joshi¹,

McDowall²

2000

Pacific J. Math.

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Section: Introductionmentioning

confidence: 99%

Total determination of material parameters from electromagnetic boundary information

Joshi¹,

McDowall²

2000

Pacific J. Math.

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“…(4) In the case that β 1 = β 2 = 0, the result of [7] follows without any assumption on ε or µ at ∂Ω. The reason for this is that exponentially growing solutions are constructed without the need for intertwining operators, and so the parameters may be extended outside Ω in a non-smooth way.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…They showed that these parameters could be recovered approximately provided they differed only slightly from known constants. In [7] this assumption was dropped, and it was shown that the parameters are recoverable provided they are known in a small neighborhood of the boundary of the body.…”

Section: Introductionmentioning

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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“…For Maxwell's equations the analog of the DN map is the admittance map that maps the tangential component of the electric field to the tangential component of the magnetic field [175]. 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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An inverse boundary value problem in electrodynamics

Cited by 170 publications

References 16 publications

Total determination of material parameters from electromagnetic boundary information

Total determination of material parameters from electromagnetic boundary information

An electromagnetic inverse problem in chiral media

Inverse problems: seeing the unseen

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