An inverse resistivity problem: 1. Lipschitz continuity of the gradient of the objective functional

Abstract

A mathematical model of vertical electrical sounding over a medium with continuously changing conductivity σ(z) is studied by using a resistivity method. The considered model leads to an inverse problem of identification of the unknown leading coefficient σ(z) of the elliptic equation in the layer Ω = {(r, z) ∈ R 2 : 0 ≤ r < ∞, 0 < z < H}. The measured data 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 σ(z))′ and then using the Bessel–Fourier transformation with respect to the variable r ≥ 0. As a result the inverse problem is formulated in terms of the transformed potential V(λ, z) and the reflection function p(z). It is proved that the transformed cost functional is Fréchet differentiable with respect to the function p(z). Moreover, an explicit formula for the Fréchet gradient of the cost functional is derived. Then Lipschitz continuity of this gradient is proved in class of reflection functions p(z) with Hölder class of derivative p′(z).