Convexification With Viscosity Term for an Inverse Problem of Tikhonov

TL;DR

This work tackles the 1D coefficient inverse problem of recovering the ground conductivity $\sigma(z)$ from backscattered measurements, an ill-posed nonlinear task. It develops a globally convergent convexification method with a viscosity term $\varepsilon$, transforming the CIP into a boundary-value problem and then into a Carleman-weighted Tikhonov functional, whose strong convexity guarantees global convergence of a gradient descent method. The authors prove quantitative convergence results (strong convexity, accuracy under noise, and global convergence) and provide a practical scheme to reconstruct $\sigma(z)$ by averaging over a frequency-like parameter $k$. The resulting framework offers a robust, theoretically justified approach for geophysical conductivity reconstruction and extends convexification techniques to viscosity-augmented inverse problems.

Abstract

In 1965 A.N. Tikhonov, the founder of the theory of Ill-Posed and Inverse Problems, has posed an coefficient inverse problem of the recovery of the unknown electric conductivity coefficient from measurements of the back reflected electrical signal. In the geophysical application targeted by Tikhonov, this coefficient depends only on the depth and characterizes the electrical conductivity of the ground. The goal of this paper is to construct for this problem a version of the globally convergent convexification numerical method for this problem. In this version, the viscosity term is introduced. A Carleman estimate allows to prove global convergence of this method.