1,054 research outputs found
Simultaneous Identification of the Diffusion Coefficient and the Potential for the Schr\"odinger Operator with only one Observation
This article is devoted to prove a stability result for two independent
coefficients for a Schr\"odinger operator in an unbounded strip. The result is
obtained with only one observation on an unbounded subset of the boundary and
the data of the solution at a fixed time on the whole domain
Imaging of buried objects from experimental backscattering time dependent measurements using a globally convergent inverse algorithm
We consider the problem of imaging of objects buried under the ground using
backscattering experimental time dependent measurements generated by a single
point source or one incident plane wave. In particular, we estimate dielectric
constants of those objects using the globally convergent inverse algorithm of
Beilina and Klibanov. Our algorithm is tested on experimental data collected
using a microwave scattering facility at the University of North Carolina at
Charlotte. There are two main challenges working with this type of experimental
data: (i) there is a huge misfit between these data and computationally
simulated data, and (ii) the signals scattered from the targets may overlap
with and be dominated by the reflection from the ground's surface. To overcome
these two challenges, we propose new data preprocessing steps to make the
experimental data to be approximately the same as the simulated ones, as well
as to remove the reflection from the ground's surface. Results of total 25 data
sets of both non blind and blind targets indicate a good accuracy.Comment: 34 page
Reconstruction from blind experimental data for an inverse problem for a hyperbolic equation
We consider the problem of reconstruction of dielectrics from blind
backscattered experimental data. Experimental data were collected by a device,
which was built at University of North Carolina at Charlotte. This device sends
electrical pulses into the medium and collects the time resolved backscattered
data on a part of a plane. The spatially distributed dielectric constant
is the unknown
coefficient of a wave-like PDE. This coefficient is reconstructed from those
data in blind cases. To do this, a globally convergent numerical method is
used.Comment: 27 page
Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method
The problem to be studied in this work is within the context of coefficient
identification problems for the wave equation. More precisely, we consider the
problem of reconstruction of the refractive index (or equivalently, the
dielectric constant) of an inhomogeneous medium using one backscattering
boundary measurement. The goal of this paper is to analyze the performance of a
globally convergent algorithm of Beilina and Klibanov on experimental data
acquired in the Microwave Laboratory at University of North Carolina at
Charlotte. The main challenge working with experimental data is the the huge
misfit between these data and computationally simulated data. We present data
pre-processing steps to make the former somehow look similar to the latter.
Results of both non-blind and blind targets are shown indicating good
reconstructions even for high contrasts between the targets and the background
medium.Comment: 25 page
Thermoacoustic tomography with an arbitrary elliptic operator
Thermoacoustic tomography is a term for the inverse problem of determining of
one of initial conditions of a hyperbolic equation from boundary measurements.
In the past publications both stability estimates and convergent numerical
methods for this problem were obtained only under some restrictive conditions
imposed on the principal part of the elliptic operator. In this paper
logarithmic stability estimates are obatined for an arbitrary variable
principal part of that operator. Convergence of the Quasi-Reversibility Method
to the exact solution is also established for this case. Both complete and
incomplete data collection cases are considered.Comment: 16 page
A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem
We consider a transmission wave equation in two embedded domains in ,
where the speed is in the inner domain and in the outer
domain. We prove a global Carleman inequality for this problem under the
hypothesis that the inner domain is strictly convex and . As a
consequence of this inequality, uniqueness and Lip- schitz stability are
obtained for the inverse problem of retrieving a stationary potential for the
wave equation with Dirichlet data and discontinuous principal coefficient from
a single time-dependent Neumann boundary measurement
Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide
We consider an inverse problem of reconstructing the conductivity function in
a hyperbolic equation using single space-time domain noisy observations of the
solution on the backscattering boundary of the computational domain. We
formulate our inverse problem as an optimization problem and use Lagrangian
approach to minimize the corresponding Tikhonov functional. We present a
theorem of a local strong convexity of our functional and derive error
estimates between computed and regularized as well as exact solutions of this
functional, correspondingly. In numerical simulations we apply domain
decomposition finite element-finite difference method for minimization of the
Lagrangian. Our computational study shows efficiency of the proposed method in
the reconstruction of the conductivity function in three dimensions
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