09-24 Balgaisha Mukanova

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

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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 $p'(z)$.

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