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- Beilina, Larisa, 1970, et al.
(author)
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Globally convergent and adaptive finite element methods in imaging of buried objects from experimental backscattering radar measurements
- 2015
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In: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427. ; 289, s. 371-391
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Journal article (peer-reviewed)abstract
- We consider a two-stage numerical procedure for imaging of objects buried in dry sand using time-dependent backscattering experimental radar measurements. These measurements are generated by a single point source of electric pulses and are collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. Our imaging problem is formulated as the inverse problem of the reconstruction of the spatially distributed dielectric constant, which is an unknown coefficient in Maxwell’s equations.
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| 3. |
- Beilina, Larisa, 1970, et al.
(author)
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Methods of quantitative reconstruction of shapes and refractive indices from experimental data
- 2015
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In: Inverse Problems and Applications; (papers from: 3rd Annual Workshop on Inverse Problems, 2013; Stockholm; Sweden; 2 - 6 May 2013); (Proceedings in Mathematics & Statistics). - Cham : Springer International Publishing. - 2194-1009 .- 2194-1017. - 9783319124988 ; 120, s. 13-41
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Conference paper (peer-reviewed)abstract
- In this chapter we summarize results of [5, 6, 14] and present new results of reconstruction of refractive indices and shapes of objects placed in the air from blind backscattered experimental data using two-stage numerical procedure of [4]. Data are collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. On the first stage the approximately globally convergent method of [4] is applied to get a good first approximation for the exact solution. Results of this stage are presented in [5, 14]. On the second stage the local adaptive finite element method of [1] is applied to refine the solution obtained on the first stage. In this chapter we briefly describe methods and present new results for both stages.
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| 5. |
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| 6. |
- Beilina, Larisa, 1970, et al.
(author)
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Reconstruction of shapes and refractive indices from backscattering experimental data using the adaptivity
- 2014
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In: Inverse Problems. - : IOP Publishing. - 1361-6420 .- 0266-5611. ; 30:10, s. Art. no. 105007-
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Journal article (peer-reviewed)abstract
- We consider the inverse problem of the reconstruction of the spatially distributed dielectric constant epsilon(r)(x), x is an element of R-3, which is an unknown coefficient in the Maxwell's equations, from time-dependent backscattering experimental radar data associated with a single source of electric pulses. The refractive index is n(x) = root epsilon(r)(x). The coefficient epsilon(r)(x) is reconstructed using a two-stage reconstruction procedure. In the first stage an approximately globally convergent method proposed is applied to get a good first approximation of the exact solution. In the second stage a locally convergent adaptive finite element method is applied, taking the solution of the first stage as the starting point of the minimization of the Tikhonov functional. This functional is minimized on a sequence of locally refined meshes. It is shown here that all three components of interest of targets can be simultaneously accurately imaged: refractive indices, shapes and locations.
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| 7. |
- Bondestam Malmberg, John, 1988
(author)
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A posteriori error estimate in the lagrangian setting for an inverse problem based on a new formulation of Maxwell’s system
- 2015
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In: Springer Proceedings in Mathematics and Statistics: 3rd Annual Workshop on Inverse Problems, 2013, Stockholm, Sweden, 2-6 May 2013. - Cham : Springer International Publishing. - 2194-1009 .- 2194-1017. - 9783319124988 ; 120, s. 43-53
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Conference paper (peer-reviewed)abstract
- In this paper we consider an inverse problem of determination of a dielectric permittivity function from a backscattered electromagnetic wave. The inverse problem is formulated as an optimal control problem for a certain partial differential equation derived from Maxwell’s system. We study a solution method based on finite element approximation and provide a posteriori error estimate for the use in an adaptive algorithm.
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| 8. |
- Bondestam Malmberg, John, 1988
(author)
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A Two-stage Numerical Procedure for an Inverse Scattering Problem
- 2015
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Licentiate thesis (other academic/artistic)abstract
- In this thesis we study a numerical procedure for the solution of the inverse problem of reconstructing location, shape and material properties (in particular refractive indices) of scatterers located in a known background medium. The data consist of time-resolved backscattered radar signals from a single source position. This relatively small amount of data and the ill-posed nature of the inversion are the main challenges of the problem. Mathematically, the problem is formulated as a coefficient inverse problem for a system of partial differential equations derived from Maxwell's equations. The numerical procedure is divided into two stages. In the first stage, a good initial approximation for the unknown coefficient is computed by an approximately globally convergent algorithm. This initial approximation is refined in the second stage, where an adaptive finite element method is employed to minimize a Tikhonov functional. An important tool for the second stage is a posteriori error estimates -- estimates in terms of known (computed) quantities -- for the difference between the computed coefficient and the true minimizing coefficient. This thesis includes four papers. In the first two, the a posteriori error analysis required for the adaptive finite element method in the second stage is extended from the previously existing indirect error estimators to direct ones. The last two papers concern verification of the two-stage numerical procedure on experimental data. We find that location and material properties of scatterers are obtained already in the first stage, while shapes are significantly improved in the second stage.
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| 10. |
- Bondestam Malmberg, John, 1988, et al.
(author)
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An Adaptive Finite Element Method in Quantitative Reconstruction of Small Inclusions from Limited Observations
- 2016
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In: ArXiv.
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Other publication (other academic/artistic)abstract
- We consider a coefficient inverse problem for the dielectric permittivity in Maxwell's equations, with data consisting of boundary measurements of one or two backscattered or transmitted waves. The problem is treated using a Lagrangian approach to the minimization of a Tikhonov functional, where an adaptive finite element method forms the basis of the computations. A new a posteriori error estimate for the coefficient is derived. The method is tested successfully in numerical experiments for the reconstruction of two, three, and four small inclusions with low contrast, as well as the reconstruction of a superposition of two Gaussian functions.
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