More Like this

1. Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data

   https://doi.org/10.1080/17415977.2021.1943384  

   Truong, Trung; Nguyen, Dinh-Liem; Klibanov, Michael V. (January 2021, Inverse Problems in Science and Engineering)

   null (Ed.)

   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 ill-posed 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 quasi-reversibility approach. The cost functional for the weighted quasi-reversibility method is constructed as a Tikhonov-like 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 Tikhonov-like functional without any advanced a priori knowledge about it. 

   more » « less
2. Inverse scattering without phase: Carleman convexification and phase retrieval via the Wentzel--Kramers--Brillouin approximation

   Le, T T; Nguyen, P M; Nguyen, L H (June 2025, Preprint arXiv:2506.21699)

   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. 

   more » « less
3. Analysis of the monotonicity method for an anisotropic scatterer with a conductive boundary

   https://doi.org/10.1088/1361-6420/ad7053  

   Harris, Isaac; Hughes, Victor; Lee, Heejin (August 2024, Inverse Problems)

   Abstract In this paper, we consider the inverse scattering problem associated with an anisotropic medium with a conductive boundary. We will assume that the corresponding far–field pattern is known/measured and we consider two inverse problems. First, we show that the far–field data uniquely determines the boundary coefficient. Next, since it is known that anisotropic coefficients are not uniquely determined by this data we will develop a qualitative method to recover the scatterer. To this end, we study the so–called monotonicity method applied to this inverse shape problem. This method has recently been applied to some inverse scattering problems but this is the first time it has been applied to an anisotropic scatterer. This method allows one to recover the scatterer by considering the eigenvalues of an operator associated with the far–field operator. We present some simple numerical reconstructions to illustrate our theory in two dimensions. For our reconstructions, we need to compute the adjoint of the Herglotz wave function as an operator mapping intoH¹of a small ball. 

   more » « less
4. Numerical reconstruction for 3D nonlinear SAR imaging via a version of the convexification method

   Vo Anh Khoa, Michael Victor (April 2023, Contemporary mathematics)

   This work extends the applicability of our recent convexification- based algorithm for constructing images of the dielectric constant of buried or occluded target. We are orientated towards the detection of explosive-like targets such as antipersonnel land mines and improvised explosive devices in the non-invasive inspections of buildings. In our previous work, the method is posed in the perspective that we use multiple source locations running along a line of source to get a 2D image of the dielectric function. Mathematically, we solve a 1D coefficient inverse problem for a hyperbolic equation for each source location. Different from any conventional Born approximation-based technique for synthetic-aperture radar, this method does not need any linearization. In this paper, we attempt to verify the method using several 3D numerical tests with simulated data. We revisit the global convergence of the gradient descent method of our computational approach. 

   more » « less
5. Reduced inverse Born series: a computational study

   https://doi.org/10.1364/JOSAA.473683  

   Markel, Vadim_A; Schotland, John_C (November 2022, Journal of the Optical Society of America A)

   We investigate the inverse scattering problem for scalar waves. We report conditions under which the terms in the inverse Born series cancel in pairs, leaving only one term at each order. We refer to the resulting expansion as the reduced inverse Born series. The reduced series can also be derived from a nonperturbative inversion formula. Our results are illustrated by numerical simulations that compare the performance of the reduced series to the full inverse Born series and the Newton–Kantorovich method. 

   more » « less