Convexification-based globally convergent numerical method for a 1D coefficient inverse problem with experimental data

TL;DR

This paper introduces a convexification-based numerical method for solving a 1D coefficient inverse problem using experimental data, ensuring global convergence and robustness against initial guesses.

Contribution

The paper develops a new convexification approach with a Carleman estimate, guaranteeing global convergence for a 1D inverse problem with experimental data.

Findings

The method achieves accurate reconstruction of the dielectric constant from backscattering data.

Numerical results demonstrate successful application to both simulated and experimental data.

The approach guarantees convergence without requiring an initial guess.

Abstract

To compute the spatially distributed dielectric constant from the backscattering data, we study a coefficient inverse problem for a 1D hyperbolic equation. To solve the inverse problem, we establish a new version of Carleman estimate and then employ this estimate to construct a cost functional which is strictly convex on a convex bounded set with an arbitrary diameter in a Hilbert space. The strict convexity property is rigorously proved. This result is called the convexification theorem and is considered as the central analytical result of this paper. Minimizing this convex functional by the gradient descent method, we obtain the desired numerical solution to the coefficient inverse problems. We prove that the gradient descent method generates a sequence converging to the minimizer and we also establish a theorem confirming that the minimizer converges to the true solution as the noise in the measured data and the regularization parameter tend to zero. Unlike the methods that are based on optimization, our convexification method converges globally in the sense that it delivers a good approximation of the exact solution without requiring any initial guess. Results of numerical studies of both computationally simulated and experimental data are presented.