Inverse problems: seeing the unseen

By Gunther Uhlmann. 2014

158 citations recorded by the source corpus.

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. 65–73, 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?

Subfields: Biomedical Engineering, Electrical and Electronic Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/13cf5ad59d00fda894dce1d8dc32722d86dafb97

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In Biomedical Engineering

In Electrical and Electronic Engineering

In Mathematical Physics

Citations in the corpus

The Calderón problem for variable coefficients nonlocal elliptic operators (2017, background): “We refer readers to [50] for detailed survey on this inverse problem.”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, background): “d in terms of the normal derivative qf= @ uj @ for regular enough boundaries. For more details on the classical Calderon problem and its applications to medical, seismic and industrial imaging, see [72, 78]. 1.2. Fractional Schrodinger equation. In this article, we focus on the fractional Schrodinge”
The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, background): “This is a nonlocal (fractional) counterpart of the classical Calderón problem arising in electrical impedance tomography where one obtains information about the electrical properties of some bounded domain by doing voltage and current measurements on the boundary [39, 40].”
The Calderón problem for the fractional Schrödinger equation (2016, unknown): “We refer to the survey [Uhlmann 2014] for more details.”
Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities (2019, methodology): “This result is motivated by the Calderón inverse problem with partial data, see [24] and [40] for review, and it provides the solution of the linearized version of the partial data problem at the zero potential.”
SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “The study of the Dirichlet-to-Neumann operator (also known as the voltageto-current map) is essential for applications to electrical impedance tomography, which is used in medical and geophysical imaging (see [104] for a recent survey).”
The fractional Calderón problem: Low regularity and stability (2017, methodology): “] 12 May 2019 THE FRACTIONAL CALDERON PROBLEM: LOW REGULARITY AND STABILITY 2 n= 2 typically requires dierent methods, and corresponding results may be found in [Bu08,NS10,BIY15,AFR16]. We refer to [Uh14] for further references. In this work we continue the line of research initiated in [GSU16], whi”
Fractional anisotropic Calderón problem on closed Riemannian manifolds (2021, background): “The anisotropic Calderón problem consists of recovering the electrical conductivity matrix of a medium, up to a change of variables, from the current and voltage measurements performed along the boundary of the medium, see [3], [37].”