The reconstruction of physical properties of a medium from boundary measurements, known as inverse scattering problems, presents significant challenges. The present study aims to validate a newly developed convexification method for a 3D coefficient inverse problem in the case of buried unknown objects in a sandbox, using experimental data collected by a microwave scattering facility at The University of North Carolina at Charlotte. Our study considers the formulation of a coupled quasilinear elliptic system based on multiple frequencies. The system can be solved by minimizing a weighted Tikhonovlike functional, which forms our convexification method. Theoretical results related to the convexification are also revisited in this work.
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Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data
This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and illposed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation using a change of variables. Next, using a truncated expansion with respect to a special Fourier basis, we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a weighted quasireversibility approach. The cost functional for the weighted quasireversibility method is constructed as a Tikhonovlike functional that involves a Carleman Weight Function. Our numerical study shows that, using a version of the gradient descent method, one can find the minimizer of this Tikhonovlike functional without any advanced a priori knowledge about it.
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 Award ID(s):
 1812693
 NSFPAR ID:
 10285012
 Date Published:
 Journal Name:
 Inverse Problems in Science and Engineering
 ISSN:
 17415977
 Page Range / eLocation ID:
 1 to 20
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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