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

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 Bessel-Fourier 
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 $p'(z)$.