Adrian Nachman
A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data.
Mon, 12/12/2011 - 15:01 — SomNambulAuthors: Adrian Nachman, Amir Moradifam, Alexandre Timonov
Subjects: Analysis of PDEs
AbstractWe consider the hybrid problem of reconstructing the isotropic electric
conductivity of a body $\Omega$ from interior Current Density Imaging data
obtainable using MRI measurements. We only require knowledge of the magnitude
$|J|$ of one current generated by a given voltage $f$ on the boundary
$\partial\Omega$. As previously shown, the corresponding voltage potential u in
$\Omega$ is a minimizer of the weighted least gradient problemConductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions.
Mon, 12/12/2011 - 15:01 — SomNambulAuthors: Adrian Nachman, Amir Moradifam, Alexandru Tamasan
Subjects: Analysis of PDEs
AbstractWe 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 $|J|$. 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.Convergence of the alternating split Bregman algorithm in infinite-dimensional Hilbert spaces.
Mon, 12/12/2011 - 15:01 — SomNambulAuthors: Adrian Nachman, Amir Moradifam
Subjects: Functional Analysis
AbstractWe prove results on weak convergence for the alternating split Bregman
algorithm in infinite dimensional Hilbert spaces. We also show convergence of
an approximate split Bregman algorithm, where errors are allowed at each step
of the computation. To be able to treat the infinite dimensional case, our
proofs focus mostly on the dual problem. We rely on Svaiter's theorem on weak
convergence of the Douglas-Rachford splitting algorithm and on the relation
between the alternating split Bregman and Douglas-Rachford splitting algorithms
discovered by Setzer.Reconstruction in the Calderon Problem with Partial Data.
Fri, 08/28/2009 - 23:56 — SomNambulAuthors: Adrian Nachman, Brian Street
Subjects: Analysis of PDEs
AbstractWe consider the problem of recovering the coefficient \sigma(x) of the
elliptic equation \grad \cdot(\sigma \grad u)=0 in a body from measurements of
the Cauchy data on possibly very small subsets of its surface. We give a
constructive proof of a uniqueness result by Kenig, Sj\"ostrand, and Uhlmann.
We construct a uniquely specified family of solutions such that their traces on
the boundary can be calculated by solving an integral equation which involves
only the given partial Cauchy data.
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