Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide

TL;DR

This paper develops a Lagrangian-based optimization method for reconstructing conductivity in a hyperbolic waveguide problem, demonstrating its effectiveness through numerical simulations in three dimensions.

Contribution

It introduces a novel Lagrangian approach with proven local strong convexity for the inverse conductivity problem in waveguides, including error estimates and 3D numerical validation.

Findings

Method efficiently reconstructs conductivity in 3D.

Lagrangian functional exhibits local strong convexity.

Numerical results confirm accuracy and robustness.

Abstract

We consider an inverse problem of reconstructing the conductivity function in a hyperbolic equation using single space-time domain noisy observations of the solution on the backscattering boundary of the computational domain. We formulate our inverse problem as an optimization problem and use Lagrangian approach to minimize the corresponding Tikhonov functional. We present a theorem of a local strong convexity of our functional and derive error estimates between computed and regularized as well as exact solutions of this functional, correspondingly. In numerical simulations we apply domain decomposition finite element-finite difference method for minimization of the Lagrangian. Our computational study shows efficiency of the proposed method in the reconstruction of the conductivity function in three dimensions.