Sarah Hamilton Kim Knudsen Samuli Siltanen Gunther Uhlmann
Complex Geometrical Optics (CGO) solutions have, for almost three decades, played a large role in the rigorous analysis of nonlinear inverse problems. They have the added bonus of also being useful in practical reconstruction algorithms. The main benefit of CGO solutions is to provide solutions in the form of almost-exponential functions that can be used in a variety of ways, for example for defining tailor-made nonlinear Fourier transforms to study the unique solvability of a nonlinear inverse problem.

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Unique continuation property for the elasticity with general residual stress
Gunther Uhlmann Jenn-Nan Wang
We prove the unique continuation property for the isotropic elasticity system with arbitrarily large residual stress. This work improves the result obtained in [10] where the residual stress is assumed to be small.
keywords: Unique continuation property residual stress.
Instability of the linearized problem in multiwave tomography of recovery both the source and the speed
Plamen Stefanov Gunther Uhlmann
In this paper we consider the linearized problem of recovering both the sound speed and the thermal absorption arising in thermoacoustic and photoacoustic tomography. We show that the problem is unstable in any scale of Sobolev spaces.
keywords: source and speed. instability multiwave tomography inverse problem Thermoacoustic tomography
Inverse problems with partial data in a slab
Xiaosheng Li Gunther Uhlmann
In this paper we consider several inverse boundary value problems with partial data on an infinite slab. We prove the unique determination results of the coefficients for the Schrödinger equation and the conductivity equation when the corresponding Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same single hyperplane.
keywords: Inverse problems Incomplete data Slab.
Reconstructing the metric and magnetic field from the scattering relation
Nurlan Dairbekov Gunther Uhlmann
We develop a method for reconstructing the conformal factor of a Riemannian metric and the magnetic field on a surface from the scattering relation associated to the corresponding magnetic flow. The scattering relation maps a starting point and direction of a magnetic geodesic into its end point and direction. The key point in the reconstruction is the interplay between the magnetic ray transform, the fiberwise Hilbert transform on the circle bundle of the surface, and the Laplace-Beltrami operator of the underlying Riemannian metric.
keywords: ray transform. Riemannian metric scattering relation magnetic flow
Reconstructions from boundary measurements on admissible manifolds
Carlos E. Kenig Mikko Salo Gunther Uhlmann
We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrödinger operator $-\Delta_g + q$ in a fixed admissible $3$-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $\Delta_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. [10] on admissible manifolds, and extends the reconstruction procedure of Nachman [31] in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.
keywords: Schrödinger equation conductivity equation anisotropic media. Inverse problem
Optimal three-ball inequalities and quantitative uniqueness for the Stokes system
Ching-Lung Lin Gunther Uhlmann Jenn-Nan Wang
We study the local behavior of a solution to the Stokes system with singular coefficients in $R^n$ with $n=2,3$. One of our main results is a bound on the vanishing order of a nontrivial solution $u$ satisfying the Stokes system, which is a quantitative version of the strong unique continuation property for $u$. Different from the previous known results, our strong unique continuation result only involves the velocity field $u$. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial optimal three-ball inequalities for $u$. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution $u$ to the Stokes system from those three-ball inequalities. As an application, we derive a minimal decaying rate at infinity of any nontrivial $u$ satisfying the Stokes equation under some a priori assumptions.
keywords: Optimal three-ball inequalities Carleman estimates Stokes system.
Reconstruction of obstacles immersed in an incompressible fluid
Horst Heck Gunther Uhlmann Jenn-Nan Wang
We consider the reconstruction of obstacles inside a bounded domain filled with an incompressible fluid. Our method relies on special complex geometrical optics solutions for the stationary Stokes equation with a variable viscosity.
keywords: inverse problem. complex spherical waves Stokes system
On the stable recovery of a metric from the hyperbolic DN map with incomplete data
Plamen Stefanov Gunther Uhlmann Andras Vasy
We show that given two hyperbolic Dirichlet to Neumann maps associated to two Riemannian metrics of a Riemannian manifold with boundary which coincide near the boundary are close then the lens data of the two metrics is the same. As a consequence, we prove uniqueness of recovery a conformal factor (sound speed) locally under some conditions on the latter.
keywords: metric. DN map Inverse problem

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