An inverse resistivity problem: 2. Unilateral convexity of the objective functional

Abstract

Mathematical model of vertical electrical sounding over a medium with continuously changing conductivity σ(z) is studied by using a resistivity method. In the first part [Mukanova, An inverse resistivity problem: 1. Frechet differentiability of cost functional and Lipschitz continuity of the gradient, Appl. Anal. 88 (2009), pp. 749–765] of the study, the Fréchet differentiability of the transformed cost functional with respect to the function p(z) = (ln σ(z))′ and the Lipschitz continuity of its gradient were proved. In this part of the study the unilateral convexity of the cost functional is derived. The obtained results permit one to construct two-steps numerical algorithm for the inverse resistivity problem. At the first step the logarithmic derivative of the coefficient p(z) = (ln σ(z))′ is recovered by using the obtained expilicit form of the gradient. Then the unknown conductivity coefficient σ(z) is defined by an analytical formula. Computational experiments are performed for some physically possible cases.

Keywords:

• vertical electrical sounding
• coefficient inverse problem
• unilateral convexity
• two-steps algorithm

AMS Subject Classifications:

• 34A55
• 35R30
• 65L08

Acknowledgement

The author gratefully thanks Prof. A. Hasanoglu (Hasanov) for the formulation of the problem and valuable suggestions.