Searcharxiv⌕ Search

arXiv subjects

Gunther Uhlmann

Publications and source records attributed to Gunther Uhlmann.

At least 145 records · Page 8Linked to original sources
arXiv2014

The geodesic ray transform on Riemannian surfaces with conjugate points

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.

math.DG↗
arXiv2014

The point source inverse back-scattering problem

We consider the inverse problem of recovering a potential by measuring the response at a point to a source located at the same point and then varying the point on the surface of a sphere. This is a similar to the inverse back-scattering problem. We show that if the angular derivatives of the difference of two potentials having the same data is controlled by the L^2 norm of the difference of the potentials they must be equal. In particular this shows injectivity of the inverse problem for radial potentials.

math.AP↗
arXiv2014

Uniqueness for the inverse backscattering problem for angularly controlled potentials

We consider the problem of recovering a smooth, compactly supported potential on R^3 from its backscattering data. We show that if two such potentials have the same backscattering data and the difference of the two potentials has controlled angular derivatives then the two potentials are identical. In particular, if two potentials differ by a finite linear combination of spherical harmonics with radial coefficinets and have the same backscattering data then the two potentials are identical.

math.AP↗
arXiv2014

Inverse boundary value problem for the Stokes and the Navier-Stokes equations in the plane

In this paper, we prove in two dimensions global identifiability of the viscosity in an incompressible fluid by making boundary measurements. The main contribution of this work is to use more natural boundary measurements, the Cauchy forces, than the Dirichlet-to-Neumann map previously considered in \cite{IY} to prove the uniqueness of the viscosity for the Stokes equations and for the Navier-Stokes equations.

math.AP↗
arXiv2013

Photoacoustic and thermoacoustic tomography with an uncertain wave speed

We consider the mathematical model of photoacoustic and thermoacoustic tomography in media with a variable sound speed. When the sound speed is known, the explicit reconstruction formula by P. Stefanov and G. Uhlmann (Inverse Problems, 25(7):075011, 16, 2009) can be used. We study how a modelling error in the sound speed affects the reconstruction formula and quantify the effect in terms of a stability estimate.

math.AP↗
arXiv2013

Determination of structures in the space-time from local measurements: a detailed exposition

We consider inverse problems for the Einstein equation with a time-depending metric on a 4-dimensional globally hyperbolic Lorentzian manifold $(M,g)$. We formulate the concept of active measurements for relativistic models. We do this by coupling the Einstein equation with equations for scalar fields and study the system $Ein(g)=T$, $T=T(g,ϕ)+F_1$, and $\square_g ϕ=F_2+S(g,ϕ,F_1,F_2)$. Here $F=(F_1,F_2)$ correspond to the perturbations of the physical fields which we control and $S$ is a secondary source corresponding to the adaptation of the system to the perturbation so that the conservation law $div_g(T)=0$ will be satisfied. The inverse problem we study is the question, do the observation of the solutions $(g,ϕ)$ in an open subset $U\subset M$ of the space-time corresponding to sources $F$ supported in $U$ determine the properties of the metric in a larger domain $W\subset M$ containing $U$. To study this problem we define the concept of light observation sets and show that these sets determine the conformal class of the metric. This corresponds to passive observations from a distant area of the space which is filled by light sources (e.g. we see light from stars varying in time). One can apply the obtained result to solve inverse problems encountered in general relativity and in various practical imaging problems.

math.AP↗
arXiv2013

On $L^p$ resolvent estimates for elliptic operators on compact manifolds

We prove uniform $L^p$ estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding resul of [3] in the case of Laplace-- Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1] very closely. We also show that spectral regions in our $L^p$ resolvent estimates are optimal.

math.AP↗
arXiv2013

Multi-scale discrete approximations of Fourier integral operators associated with canonical transformations and caustics

We develop an algorithm for the computation of general Fourier integral operators associated with canonical graphs. The algorithm is based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics. The procedure consists in the construction of a universal operator representation through the introduction of locally singularity-resolving diffeomorphisms, enabling the application of wave packet driven computation, and in the construction of the associated pseudo-differential joint-partition of unity on the canonical graphs. We apply the method to a parametrix of the wave equation in the vicinity of a cusp singularity.

math.NA↗
arXiv2012

Tensor tomography on surfaces

We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.

math.DG↗
arXiv2012

The inverse problem for the local geodesic ray transform

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces the geodesic X-ray transform is globally injective. In addition we prove stability estimates and propose a layer stripping type algorithm for reconstruction.

math.DG↗
arXiv2012

Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential

We show that the knowledge of the set of the Cauchy data on the boundary of a bounded open set in $\R^n$, $n\ge 3$, for the magnetic Schrödinger operator with $L^\infty$ magnetic and electric potentials determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives.

math.AP↗
arXiv2012

Determining a magnetic Schrödinger operator with a continuous magnetic potential from boundary measurements

We show that the knowledge of the set of the Cauchy data on the boundary of a $C^1$ bounded open set in $\R^n$, $n\ge 3$, for the Schrödinger operator with continuous magnetic and bounded electric potentials determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives.

math.AP↗
arXiv2012

Is a curved flight path in SAR better than a straight one?

In the plane, we study the transform $R_γf$ of integrating a unknown function $f$ over circles centered at a given curve $γ$. This is a simplified model of SAR, when the radar is not directed but has other applications, like thermoacoustic tomography, for example. We study the problem of recovering the wave front set $\WF(f)$. If the visible singularities of $f$ hit $γ$ once, we show that the "artifacts" cannot be resolved. If $γ$ is a closed curve, we show that this is still true. On the other hand, if $f$ is known a priori to have singularities in a compact set, then we show that one can recover $\WF(f)$, and moreover, this can be done in a simple explicit way, using backpropagation for the wave equation.

math.AP↗