# Solving a 1-D inverse medium scattering problem using a new multi-frequency globally strictly convex objective functional

## TL;DR

This paper introduces a new multi-frequency convex functional for 1-D inverse scattering, enabling accurate reconstructions without initial guesses, proven globally convex and convergent.

## Contribution

It develops a novel globally strictly convex objective functional for 1-D inverse scattering using multi-frequency data, with proven convergence and error estimates.

## Key findings

- Proven global convexity of the new functional
- Guaranteed convergence of the gradient projection algorithm
- Numerical results demonstrate accurate reconstructions

## Abstract

We propose a new approach to constructing globally strictly convex objective functional in a 1-D inverse medium scattering problem using multi-frequency backscattering data. The global convexity of the proposed objective functional is proved using a Carleman estimate. Due to its convexity, no good first guess is required in minimizing this objective functional. We also prove the global convergence of the gradient projection algorithm and derive an error estimate for the reconstructed coefficient. Numerical results show reasonable reconstruction accuracy for simulated data.