11 research outputs found
Nonlinear Inversion from Partial EIT Data: Computational Experiments
Electrical impedance tomography (EIT) is a non-invasive imaging method in
which an unknown physical body is probed with electric currents applied on the
boundary, and the internal conductivity distribution is recovered from the
measured boundary voltage data. The reconstruction task is a nonlinear and
ill-posed inverse problem, whose solution calls for special regularized
algorithms, such as D-bar methods which are based on complex geometrical optics
solutions (CGOs). In many applications of EIT, such as monitoring the heart and
lungs of unconscious intensive care patients or locating the focus of an
epileptic seizure, data acquisition on the entire boundary of the body is
impractical, restricting the boundary area available for EIT measurements. An
extension of the D-bar method to the case when data is collected only on a
subset of the boundary is studied by computational simulation. The approach is
based on solving a boundary integral equation for the traces of the CGOs using
localized basis functions (Haar wavelets). The numerical evidence suggests that
the D-bar method can be applied to partial-boundary data in dimension two and
that the traces of the partial data CGOs approximate the full data CGO
solutions on the available portion of the boundary, for the necessary small
frequencies.Comment: 24 pages, 12 figure
Inverse obstacle problem for the non-stationary wave equation with an unknown background
We consider boundary measurements for the wave equation on a bounded domain
or on a compact Riemannian surface, and introduce a method to
locate a discontinuity in the wave speed. Assuming that the wave speed consist
of an inclusion in a known smooth background, the method can determine the
distance from any boundary point to the inclusion. In the case of a known
constant background wave speed, the method reconstructs a set contained in the
convex hull of the inclusion and containing the inclusion. Even if the
background wave speed is unknown, the method can reconstruct the distance from
each boundary point to the inclusion assuming that the Riemannian metric tensor
determined by the wave speed gives simple geometry in . The method is based
on reconstruction of volumes of domains of influence by solving a sequence of
linear equations. For \tau \in C(\p M) the domain of influence is
the set of those points on the manifold from which the distance to some
boundary point is less than .Comment: 4 figure
Stable determination of an inclusion in an elastic body by boundary measurements (unabridged)
We consider the inverse problem of identifying an unknown inclusion contained
in an elastic body by the Dirichlet-to-Neumann map. The body is made by
linearly elastic, homogeneous and isotropic material. The Lam\'e moduli of the
inclusion are constant and different from those of the surrounding material.
Under mild a-priori regularity assumptions on the unknown defect, we establish
a logarithmic stability estimate. For the proof, we extend the approach used
for electrical and thermal conductors in a novel way. Main tools are
propagation of smallness arguments based on three-spheres inequality for
solutions to the Lam\'e system and refined local approximation of the
fundamental solution of the Lam\'e system in presence of an inclusion.Comment: 58 pages, 4 figures. This is the extended, and revised, version of a
paper submitted for publication in abridged for
Reconstruction of interfaces from the elastic farfield measurements using CGO solutions
In this work, we are concerned with the inverse scattering by interfaces for
the linearized and isotropic elastic model at a fixed frequency. First, we
derive complex geometrical optic solutions with linear or spherical phases
having a computable dominant part and an -decaying remainder term
with , where is the classical Sobolev space. Second,
based on these properties, we estimate the convex hull as well as non convex
parts of the interface using the farfields of only one of the two reflected
body waves (pressure waves or shear waves) as measurements. The results are
given for both the impenetrable obstacles, with traction boundary conditions,
and the penetrable obstacles. In the analysis, we require the surfaces of the
obstacles to be Lipschitz regular and, for the penetrable obstacles, the Lam\'e
coefficients to be measurable and bounded with the usual jump conditions across
the interface.Comment: 32 page