9,763 research outputs found
On an inverse source problem for the full radiative transfer equation with incomplete data
A new numerical method to solve an inverse source problem for the radiative
transfer equation involving the absorption and scattering terms, with
incomplete data, is proposed. No restrictive assumption on those absorption and
scattering coefficients is imposed. The original inverse source problem is
reduced to boundary value problem for a system of coupled partial differential
equations of the first order. The unknown source function is not a part of this
system. Next, we write this system in the fully discrete form of finite
differences. That discrete problem is solved via the quasi-reversibility
method. We prove the existence and uniqueness of the regularized solution.
Especially, we prove the convergence of regularized solutions to the exact one
as the noise level in the data tends to zero via a new discrete Carleman
estimate. Numerical simulations demonstrate good performance of this method
even when the data is highly noisy.Comment: 22 pages, 4 figures, to be published in SIAM Journal on Applied
Mathematic
PDE-based numerical method for a limited angle X-ray tomography
A new numerical method for X-ray tomography for a specific case of incomplete
Radon data is proposed. Potential applications are in checking out bulky
luggage in airports. This method is based on the analysis of the transport PDE
governing the X-ray tomography rather than on the conventional integral
formulation. The quasi-reversibility method is applied. Convergence analysis is
performed using a new Carleman estimate. Numerical results are presented and
compared with the inversion of the Radon transform using the well-known
filtered back projection algorithm. In addition, it is shown how to use our
method to study the inversion of the attenuated X-ray transform for the same
case of incomplete data
Mathematics of Hybrid Imaging. A Brief Review
The article provides a brief survey of the mathematics of some of the newly
being developed so called "hybrid" (also called "multi-physics" or
"multi-wave") imaging techniques.Comment: Dedicated to the memory of Professor Leon Ehrenprei
Inversion formulas for the linearized impedance tomography problem
We consider the linearized electrical impedance tomography problem in two
dimensions on the unit disk. By a linearization around constant coefficients
and using a trigonometric basis, we calculate the linearized
Dirichlet-to-Neumann operator in terms of moments of the conduction coefficient
of the problem. By expanding this coefficient into angular trigonometric
functions and Legendre-M\"untz polynomials in radial coordinates, we can find a
lower-triangular representation of the parameter to data mapping. As a
consequence, we find an explicit solution formula for the corresponding inverse
problem. Furthermore, we also consider the problem with boundary data given
only on parts of the boundary while setting homogeneous Dirichlet values on the
rest. We show that the conduction coefficient is uniquely determined from
incomplete data of the linearized Dirichlet-to-Neumann operator with an
explicit solution formula provided
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
An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernel
An integro-differential Dirac system with an integral term in the form of
convolution is considered. We suppose that the convolution kernel is known a
priori on a part of the interval, and recover it on the remaining part, using a
part of the spectrum. We prove the uniqueness theorem, provide an algorithm for
the solution of the inverse problem together with necessary and sufficient
conditions for its solvability
Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography
The paper contains a simple approach to reconstruction in Thermoacoustic and
Photoacoustic Tomography. The technique works for any geometry of point
detectors placement and for variable sound speed satisfying a non-trapping
condition. A uniqueness of reconstruction result is also obtained
Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions
We consider the problem of recovering an isotropic conductivity outside some
perfectly conducting or insulating inclusions from the interior measurement of
the magnitude of one current density field . We prove that the
conductivity outside the inclusions, and the shape and position of the
perfectly conducting and insulating inclusions are uniquely determined (except
in an exceptional case) by the magnitude of the current generated by imposing a
given boundary voltage. We have found an extension of the notion of
admissibility to the case of possible presence of perfectly conducting and
insulating inclusions. This also makes it possible to extend the results on
uniqueness of the minimizers of the least gradient problem
with to cases where
has flat regions (is constant on open sets)
Maxwell's Equations with Scalar Impedance: Direct and Inverse Problems
The article deals with electrodynamics in the presence of anisotropic
materials having scalar wave impedance. Maxwell's equations written for
differential forms over a 3-manifold are analysed. The system is extended to a
Dirac type first order elliptic system on the Grassmannian bundle over the
manifold. The second part of the article deals with the dynamical inverse
boundary value problem of determining the electromagnetic material parameters
from boundary measurements. By using the boundary control method, it is proved
that the dynamical boundary data determines the electromagnetic travel time
metric as well as the scalar wave impedance on the manifold. This invariant
result leads also to a complete characterization of the non-uniqueness of the
corresponding inverse problem in bounded domains of R^3.
AMS-classifications: 35R30, 35L20, 58J4
A regularized weighted least gradient problem for conductivity imaging
We propose and study a regularization method for recovering an approximate
electrical conductivity solely from the magnitude of one interior current
density field. Without some minimal knowledge of the boundary voltage
potential, the problem has been recently shown to have nonunique solutions,
thus recovering the exact conductivity is impossible. The method is based on
solving a weighted least gradient problem in the subspace of functions of
bounded variations with square integrable traces. The computational
effectiveness of this method is demonstrated in numerical experiments
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