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This paper addresses the challenging and interesting inverse problem of reconstructing the spatially varying dielectric constant of a medium from phaseless backscattering measurements generated by single-point illumination. The underlying mathematical model is governed by the three-dimensional Helmholtz equation, and the available data consist solely of the magnitude of the scattered wave field. To address the nonlinearity and servere ill-posedness of this phaseless inverse scattering problem, we introduce a robust, globally convergent numerical framework combining several key regularization strategies. Our method first employs a phase retrieval step based on the Wentzel--Kramers--Brillouin (WKB) ansatz, where the lost phase information is reconstructed by solving a nonlinear optimization problem. Subsequently, we implement a Fourier-based dimension reduction technique, transforming the original problem into a more stable system of elliptic equations with Cauchy boundary conditions. To solve this resulting system reliably, we apply the Carleman convexification approach, constructing a strictly convex weighted cost functional whose global minimizer provides an accurate approximation of the true solution. Numerical simulations using synthetic data with high noise levels demonstrate the effectiveness and robustness of the proposed method, confirming its capability to accurately recover both the geometric location and contrast of hidden scatterers.
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Solving the inverse scattering problem via Carleman-based contraction mapping
This paper addresses the inverse scattering problem in a domain $$\Omega$$. The input data, measured outside $$\Omega$$, involve the waves generated by the interaction of plane waves with various directions and unknown scatterers that are fully occluded inside $$\Omega$$. The output of this problem is the spatial dielectric constant of these scatterers. Our approach to solving this problem consists of two primary stages. Initially, we eliminate the unknown dielectric constant from the governing equation, resulting in a system of partial differential equations. Subsequently, we develop the Carleman contraction mapping method to effectively tackle this system. It is noteworthy to highlight the robustness of this method. It does not require a precise initial guess of the true solution, and its computational cost is relatively inexpensive. Some numerical examples are presented.
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- Award ID(s):
- 2208159
- PAR ID:
- 10612298
- Publisher / Repository:
- preprint arXiv:2404.04145
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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