195 research outputs found
Singular value decomposition for the 2D fan-beam Radon transform of tensor fields
In this article we study the fan-beam Radon transform of
symmetrical solenoidal 2D tensor fields of arbitrary rank in a unit disc
as the operator, acting from the object space to the data space
The orthogonal polynomial basis of solenoidal tensor
fields on the disc was built with the help of Zernike polynomials
and then a singular value decomposition (SVD) for the operator
was obtained. The inversion formula for the fan-beam tensor transform follows from this decomposition. Thus obtained inversion formula can be
used as a tomographic filter for splitting a known tensor field into potential
and solenoidal parts. Numerical results are presented.Comment: LaTeX, 37 pages with 5 figure
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
Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions
We consider a second-order selfadjoint elliptic operator with an anisotropic
diffusion matrix having a jump across a smooth hypersurface. We prove the
existence of a weight-function such that a Carleman estimate holds true. We
moreover prove that the conditions imposed on the weight function are
necessary.Comment: 43 page
Global stability for the multi-channel Gel'fand-Calder\'on inverse problem in two dimensions
We prove a global logarithmic stability estimate for the multi-channel
Gel'fand-Calder\'on inverse problem on a two-dimensional bounded domain, i.e.
the inverse boundary value problem for the equation on , where is a smooth matrix-valued potential defined on a bounded
planar domain
Increasing stability for the inverse problem for the Schr\"odinger equation
In this article, we study the increasing stability property for the
determination of the potential in the Schr\"odinger equation from partial data.
We shall assume that the inaccessible part of the boundary is flat and
homogeneous boundary condition is prescribed on this part. In contrast to
earlier works, we are able to deal with the case when potentials have some
Sobolev regularity and also need not be compactly supported inside the domain
CALDER 3N'S INVERSE PROBLEM with A FINITE NUMBER of MEASUREMENTS
We prove that an L 1e potential in the Schr\uf6dinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace W. As a corollary, we obtain a similar result for Calder\uf3n's inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces W, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of W
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
Inverse problems with partial data for a magnetic Schr\"odinger operator in an infinite slab and on a bounded domain
In this paper we study inverse boundary value problems with partial data for
the magnetic Schr\"odinger operator. In the case of an infinite slab in ,
, we establish that the magnetic field and the electric potential can
be determined uniquely, when the Dirichlet and Neumann data are given either on
the different boundary hyperplanes of the slab or on the same hyperplane. This
is a generalization of the results of [41], obtained for the Schr\"odinger
operator without magnetic potentials. In the case of a bounded domain in ,
, extending the results of [2], we show the unique determination of the
magnetic field and electric potential from the Dirichlet and Neumann data,
given on two arbitrary open subsets of the boundary, provided that the magnetic
and electric potentials are known in a neighborhood of the boundary.
Generalizing the results of [31], we also obtain uniqueness results for the
magnetic Schr\"odinger operator, when the Dirichlet and Neumann data are known
on the same part of the boundary, assuming that the inaccessible part of the
boundary is a part of a hyperplane
Inverse scattering at fixed energy on surfaces with Euclidean ends
On a fixed Riemann surface with Euclidean ends and genus ,
we show that, under a topological condition, the scattering matrix S_V(\la)
at frequency \la > 0 for the operator determines the potential
if for all
and for some , where denotes the distance
from to a fixed point . The topological condition is given by
for and by if . In \rr^2 this
implies that the operator S_V(\la) determines any potential
such that for all .Comment: 21 page
Determination of the characteristic directions of lossless linear optical elements
We show that the problem of finding the primary and secondary characteristic
directions of a linear lossless optical element can be reformulated in terms of
an eigenvalue problem related to the unimodular factor of the transfer matrix
of the optical device. This formulation makes any actual computation of the
characteristic directions amenable to pre-implemented numerical routines,
thereby facilitating the decomposition of the transfer matrix into equivalent
linear retarders and rotators according to the related Poincare equivalence
theorem. The method is expected to be useful whenever the inverse problem of
reconstruction of the internal state of a transparent medium from optical data
obtained by tomographical methods is an issue.Comment: Replaced with extended version as published in JM
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