# Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data

## TL;DR

This paper introduces a novel 1-D inverse scattering method using a Carleman-weighted cost functional that guarantees strict convexity, ensuring reliable convergence and demonstrating high accuracy with both simulated and experimental data.

## Contribution

The paper develops a new convex cost functional based on Carleman weights for 1-D inverse scattering, enabling guaranteed convergence from any initial guess.

## Key findings

- Method achieves high accuracy with simulated data.
- Method effectively reconstructs from experimental data.
- Gradient minimization converges globally within the specified ball.

## Abstract

A new numerical method is proposed for a 1-D inverse medium scattering problem with multi-frequency data. This method is based on the construction of a weighted cost functional. The weight is a Carleman Weight Function (CWF). In other words, this is the function, which is present in the Carleman estimate for the undelying differential operator. The presence of the CWF makes this functional strictly convex on any a priori chosen ball with the center at $\left\{ 0\right\} $ in an appropriate Hilbert space. Convergence of the gradient minimization method to the exact solution starting from any point of that ball is proven. Computational results for both computationally simulated and experimental data show a good accuracy of this method.