## Journals

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### Open Access Journals

IPI

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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keywords:

IPI

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.
IPI

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

IPI

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.

IPI

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.

IPI

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.

DCDS

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.
IPI

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.

IPI

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.

## Year of publication

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