To the reconstruction of a riemannian manifold via its spectral data (Bc–Method)

Belishev, M. I.; Kuryiev, Yarosiav V.

doi:10.1080/03605309208820863

Cited by 191 publications

(266 citation statements)

References 5 publications

Supporting

Mentioning

258

Contrasting

Unclassified

Order By: Relevance

“…This linear problem is also in the heart of the non-linear problem of recovery a metric or a sound speed (a conformal factor) from the lengths of the geodesics measured at the boundary (boundary rigidity) or from knowledge of the lens relation (lens rigidity), see, e.g., [7,6,8,25,28,36,34,32,29]; or from knowledge of the hyperbolic Dirichlet-to-Neumann (DN) map [3,4,23,27,33,30,2]. It is the linearization of the first two (f is a tensor field then); and the lens relation is directly related to the DN map and its canonical relation as an FIO.…”

Section: 1)mentioning

confidence: 99%

“…It is the linearization of the first two (f is a tensor field then); and the lens relation is directly related to the DN map and its canonical relation as an FIO. Although fully non-linear methods for uniqueness (up to isometry) exist, see, e.g., [3], stability is always derived from stability of the linearization. Very often, see for example [32], even uniqueness is derived from injectivity and stability of the linearization, see also [30] for an abstract treatment.…”

Section: 1)mentioning

confidence: 99%

See 1 more Smart Citation

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

Monard

Stefanov

Uhlmann

2015

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract. We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but still ill-posed, if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.

show abstract

Section: 1)mentioning

confidence: 99%

Section: 1)mentioning

confidence: 99%

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

Monard

Stefanov

Uhlmann

2015

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…If the source F in (1) can be controlled, that is, if we can measure the trace of u on (0, ∞)×∂M for all F ∈ C ∞ 0 ((0, ∞)×∂M), then the problem to determine (M, g) is equivalent with Gel'fand's inverse problem, whence it has unique solution [7,8]. Contrary to the problem with a single measument as considered in the present paper, Gel'fand's problem is overdetermined.…”

Section: Introductionmentioning

confidence: 99%

Inverse Problem for the Wave Equation with a White Noise Source

Helin

Lassas

Oksanen

2014

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract. We consider a smooth Riemannian metric tensor g on R n and study the stochastic wave equation for the LaplaceBeltrami operator ∂ 2 t u − ∆ g u = F . Here, F = F (t, x, ω) is a random source that has white noise distribution supported on the boundary of some smooth compact domain M ⊂ R n . We study the following formally posed inverse problem with only one measurement. Suppose that g is known only outside of a compact subset of M int and that a solution u(t, x, ω 0 ) is produced by a single realization of the source F (t, x, ω 0 ). We ask what information regarding g can be recovered by measuring u(t, x, ω 0 ) on R + × ∂M ? We prove that such measurement together with the realization of the source determine the scattering relation of the Riemannian manifold (M, g) with probability one. That is, for all geodesics passing through M , the travel times together with the entering and exit points and directions are determined. In particular, if (M, g) is a simple Riemannian manifold and g is conformally Euclidian in M , the measurement determines the metric g in M .

show abstract

“…It is known that g can be recovered uniquely from g , up to a diffeomorphism as above, see e.g. [24]. This result however relies on a unique continuation theorem by Tataru [194] and it is unlikely to provide Hölder type of stability estimate as above.…”

Section: Sketch Of the Proofmentioning

confidence: 93%

“…We note that, in contrast to the linear case, one cannot reduce the study of the inverse problem of the conductivity equation (25) to the Schrödinger equation with a non-linear potential. The main technical lemma in the proof of Theorem 1.7 is Lemma 1.8 Let γ (x, t) be as in (23) and (24). Let 1 < p < ∞, 0 < α < 1.…”

Section: Non-linear Conductivitiesmentioning

confidence: 99%

Inverse problems: seeing the unseen

2014

View full text Add to dashboard Cite

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.

show abstract

To the reconstruction of a riemannian manifold via its spectral data (Bc–Method)

Cited by 191 publications

References 5 publications

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

Inverse Problem for the Wave Equation with a White Noise Source

Inverse problems: seeing the unseen

Contact Info

Product

Resources

About