 0924 Balgaisha Mukanova
 An inverse resistivity problem: 1. Frechet differentiability of the
cost functional and Lipschitz continuity of the gradient
(285K, .pdf)
Feb 13, 09

Abstract ,
Paper (src),
View paper
(auto. generated pdf),
Index
of related papers

Abstract. Mathematical model of
vertical electrical sounding (VES) over a medium with
continuously changing conductivity $\sigma(z)$ is studied by
using a resistivity method. The considered model leads to an
inverse problem of identification of the unknown leading
coefficient $\sigma(z)$ of the elliptic equation
$\frac{\partial}{\partial z}(\sigma(z)\frac{\partial u}{\partial
z})+\frac{\sigma(z)}{r}\frac{\partial}{\partial r}(r
\frac{\partial u}{\partial r})=0$ in the layer $\Omega=\{(r,z)\in
R^2:~0\leq r<\infty,~ 0<z<H\}$. The measured data
$\psi(r):=(\partial u/\partial r)_{z=0}$ is assumed to be given
on the upper boundary of the layer, in the form of the tangential
derivative. The proposed approach is based on transformation of
the inverse problem, by introducing the reflection function
$p(z)=(\ln \sigma(z))'$ and then using the BesselFourier
transformation with respect to the variable $r\geq 0$. As a
result the inverse problem is formulated in terms of the
transformed potential $V(\lambda,z)$ and the reflection function
$p(z)$. It is proved that the transformed cost functional is
Fr\'{e}chet differentiable with respect to the reflection
function $p(z)$. Moreover, an explicit formula for the Fr\'{e}chet
gradient of the cost functional is derived. Then Lipschitz
continuity of this gradient is proved in class of reflection
functions $p(z)$ with H\"{o}lder class of derivative $p'(z)$.
 Files:
0924.src(
0924.keywords ,
BalMukFrechDiff.pdf.mm )