Convexification for the Inversion of a Time Dependent Wave Front in a Heterogeneous Medium

Abstract

An inverse scattering problem for the 3D acoustic equation in time domain is considered. The unknown spatially distributed speed of sound is the subject of the solution of this problem. A single location of the point source is used. Using a Carleman Weight Function, a globally strictly convex cost functional is constructed. A new Carleman estimate is proven. Global convergence of the gradient projection method is proven. Numerical experiments are conducted.

Convexification of a 3-D coefficient inverse scattering problem

“…After [2], a number of works on the convexification was published by the first author with coauthors, in which the theory is combined with numerical results, see, e.g. [16,26,27,28,29]. We also refer to [3] where a different version of the convexification is developed for a CIP for the hyperbolic equation u tt = ∆u + q (x) u and numerical results are presented.…”

“…This means that each new CIP requires it own version of the convexification, and these versions differ from each other quite significantly. Currently the convexification is developed analytically and tested numerically for CIPs for the Helmholtz equation [16,26,29], two hyperbolic equations [3,5,28] and Electrical Impedance Tomography [27]. The goal of this paper is to develop analytically and implement computationally the convexification method for a CIP for a parabolic PDE.…”

On the travel time tomography problem in 3D

“…Remark 6.1. Although Theorem 6.1, so as other theorems of this section, is valid only for sufficiently large values of the parameter λ, our past computational experience with the convexification demonstrates that usually once can select quite reasonable values of λ ∈ [1,3] in the numerical practice [1,19,20,22,23,21].…”

“…The suitable values of numbers N and K should be chosen numerically, see, e.g. [7,10,11,20,22,23,21,24] for such choices for a variety of inverse problems.…”

“…Since smallness conditions are not imposed on R, then this is the global convergence by our above definition. After the theory was cleared in [1], a number of works were published in which numerical studies of the convexification were conducted [19,20,22,23,21]. We also refer here to the publication [2] for a different version of the convexification for a hyperbolic coefficient inverse problem with a non vanishing initial condition.…”