Stability estimates for the X-ray transform of tensor fields and boundary rigidity

Stefanov, Plamen; Uhlmann, Günther

doi:10.1215/s0012-7094-04-12332-2

Cited by 124 publications

(278 citation statements)

References 19 publications

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“…Thus, f is a potential field. Theorem 1.1 implies a stability estimate for the problem of recovering the solenoidal part of a tensor field f from the ray transform If , see [19].…”

Section: )mentioning

confidence: 99%

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Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

Sharafutdinov

2007

J Geom Anal

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The Dirichlet-to-Neumann (DN) map Λ g : C ∞ (∂M ) → C ∞ (∂M ) on a compact Riemannian manifold (M, g) with boundary is defined by Λ g h = ∂u/∂ν| ∂M , where u is the solution to the Dirichlet problem ∆u = 0, u| ∂M = h and ν is the unit normal to the boundary. If g t = g + tf is a variation of the metric g by a symmetric tensor field f , then Λ g t = Λ g + tΛ f + o(t). We study the question: how do tensor fields f look like for whichΛ f = 0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: for the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold.

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“…Thus, f is a potential field. Theorem 1.1 implies a stability estimate for the problem of recovering the solenoidal part of a tensor field f from the ray transform If , see [19].…”

Section: )mentioning

confidence: 99%

“…The problem for generic metrics is solved in [19,20]. In [21], the linear problem is considered under some assumption that is weaker than the simplicity.…”

Section: Introductionmentioning

confidence: 99%

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

Sharafutdinov

2007

J Geom Anal

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“…So, the first nontrivial case is m = 2. In the latter case, the theorem was recently proved in [1] and [11]. The analytical part of our proof follows these articles with some improvements.…”

Section: Introductionmentioning

confidence: 70%

“…In Section 3, we construct a parametrix for I * I on the space of solenoidal tensor fields. Our construction of the parametrix has many features in common with that of [11] but is done in more invariant terms. In Section 4, we recall some known facts about the transmission condition which are presented in the form that we need.…”

Section: Introductionmentioning

confidence: 99%

Regularity of ghosts in tensor tomography

2005

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We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the kernel of I and is L 2 -orthogonal to all potential fields. We prove that a nontrivial ghost is smooth in the case of a simple metric. This implies that the the wave front set of the solenoidal part of a field f can be recovered from the ray transform If . We give an explicit procedure for recovering the wave front set.

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“…A conditional and non-sharp stability estimate for metrics with small curvature is also established in [167]. In [179], stability estimates for s-injective metrics [see (99) below] were shown and sharp estimates about the recovery of a 1-form f = f j dx j and a function f from the associated I g f which is defined by…”

Section: Definition 214 We Say Thatmentioning

confidence: 94%

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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Stability estimates for the X-ray transform of tensor fields and boundary rigidity

Cited by 124 publications

References 19 publications

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

Regularity of ghosts in tensor tomography

Inverse problems: seeing the unseen

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