Convexification With Viscosity Term for an Inverse Problem of Tikhonov

TL;DR

This paper introduces a globally convergent convexification method with a viscosity term for an inverse problem of recovering ground conductivity from electrical signals, utilizing Carleman estimates for proof.

Contribution

It develops a new convexification approach incorporating viscosity for a classical inverse problem, ensuring global convergence through Carleman estimates.

Findings

Proves global convergence of the method.

Constructs a convexification algorithm with viscosity.

Provides theoretical guarantees for the inverse problem solution.

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.