An official website of the United States government
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 Tikhonov-like functional, which forms our convexification method. Theoretical results related to the convexification are also revisited in this work.
more »
« less
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 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
- Award ID(s):
- 1812693
- PAR ID:
- 10285012
- Date Published:
- Journal Name:
- Inverse Problems in Science and Engineering
- ISSN:
- 1741-5977
- Page Range / eLocation ID:
- 1 to 20
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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
-
We introduce a time-dimensional reduction method for the inverse source problem in linear elasticity, where the goal is to reconstruct the initial displacement and velocity fields from partial boundary measurements of elastic wave propagation. The key idea is to employ a novel spectral representation in time, using an orthonormal basis composed of Legendre polynomials weighted by exponential functions. This Legendre polynomial-exponential basis enables a stable and accurate decomposition in the time variable, effectively reducing the original space-time inverse problem to a sequence of coupled spatial elasticity systems that no longer depend on time. These resulting systems are solved using the quasi-reversibility method. On the theoretical side, we establish a convergence theorem ensuring the stability and consistency of the regularized solution obtained by the quasi-reversibility method as the noise level tends to zero. On the computational side, two-dimensional numerical experiments confirm the theory and demonstrate the method's ability to accurately reconstruct both the geometry and amplitude of the initial data, even under substantial measurement noise. The results highlight the effectiveness of the proposed framework as a robust and computationally efficient strategy for inverse elastic source problems.more » « less
-
We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the ``reduced dimensional method.'' Initially, we use the polynomial-exponential basis to approximate the inverse problem as a system of 1D nonlinear equations. We then employ a Picard iteration based on the quasi-reversibility method and a Carleman weight function. We will rigorously prove that the sequence derived from this iteration converges to the accurate solution for that 1D system without requesting a good initial guess of the true solution. The key tool for the proof is a Carleman estimate. We will also show some numerical examples.more » « less
-
In this work, we develop numerical methods to solve forward and inverse wave problems for a nonlinear Helmholtz equation defined in a spherical shell between two concentric spheres centred at the origin. A spectral method is developed to solve the forward problem while a combination of a finite difference approximation and the least squares method are derived for the inverse problem. Numerical examples are given to verify the method. ReferencesR. Askey. Orthogonal polynomials and special functions. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1975. doi: 10.1137/1.9781611970470G. Baruch, G. Fibich, and S. Tsynkov. High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension. Nonlinear Photonics. Optica Publishing Group, 2007. doi: 10.1364/np.2007.ntha6G. Fibich and S. Tsynkov. High-Order Two-Way Artificial Boundary Conditions for Nonlinear Wave Propagation with Backscattering. J. Comput. Phys. 171 (2001), pp. 632–677. doi: 10.1006/jcph.2001.6800G. Fibich and S. Tsynkov. Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions. J. Comput. Phys. 210 (2005), pp. 183–224. doi: 10.1016/j.jcp.2005.04.015P. M. Morse and K. U. Ingard. Theoretical Acoustics. International Series in Pure and Applied Physics. McGraw-Hill Book Company, 1968G. N. Watson. A treatise on the theory of Bessel functions. International Series in Pure and Applied Physics. Cambridge Mathematical Library, 1996. url: https://www.cambridge.org/au/universitypress/subjects/mathematics/real-and-complex-analysis/treatise-theory-bessel-functions-2nd-edition-1?format=PB&isbn=9780521483919more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1080/17415977.2021.1943384
