Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data

Abstract

A version of the convexification numerical method for a Coefficient Inverse Problem for a 1D hyperbolic PDE is presented. The data for this problem are generated by a single measurement event. This method converges globally. The most important element of the construction is the presence of the Carleman Weight Function in a weighted Tikhonovlike functional. This functional is strictly convex on a certain bounded set in a Hilbert space, and the diameter of this set is an arbitrary positive number. The global convergence of the gradient projection method is established. Computational results demonstrate a good performance of the numerical method for noisy data.

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

“…Applications of this technique are in detection and identification of explosives, see Figure 1 and Section 7. The current paper continues a series of works of this research group that establish a variety of versions of the convexification principle to numerically solve coefficient inverse problems for many partial differential equations [11,12,21,14,23,24,26,27,29,30,50]. Furthermore, the convexification also works for numerical solutions of ill-posed Cauchy problems for quasilinear PDEs [20].…”

“…The inverse problem in this paper, Problem 1.1, is identical with the inverse problem in [50,51]. Although the convexification method of [50,51] is effective, still there are some rooms to improve. The method of [50,51] has two stages.…”

“…To overcome these limitations, Klibanov and Ioussoupova introduced the convexification method [14]. Since then, it has been intensively applied to solve coefficient inverse problems [1,11,12,15,16,20,23,24,27,30,50,51]. The reconstructions due to the convexification are successful even for the challenging case of experimental data [11,24,51].…”

“…Although the convexification method of [50,51] is effective, still there are some rooms to improve. The method of [50,51] has two stages. On stage 1, the authors used the well-known change of variables of [48,Chapter 2] to reduce the original inverse problem to the inverse problem of computing the potential of a 1D hyperbolic equation.…”

“…On stage 1, the authors used the well-known change of variables of [48,Chapter 2] to reduce the original inverse problem to the inverse problem of computing the potential of a 1D hyperbolic equation. This resulting inverse problem amounts to the computation of the potential, and it can be solved using one of two versions of the convexificaton method of either [50] or [51]. On stage 2, the authors computed the dielectric constant from the knowledge of the reconstructed potential of stage 1.…”