Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data
This work provides a globally convergent numerical method for a specific 1-D inverse scattering problem, which is an incremental improvement over existing methods.
The paper proposes a new numerical method for a 1-D inverse medium scattering problem using multi-frequency data, constructing a weighted cost functional with a Carleman Weight Function to achieve strict convexity. The method converges to the exact solution from any starting point in a ball, and computational results on simulated and experimental data show good accuracy.
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.