| 1. |
- Beilina, Larisa, 1970, et al.
(författare)
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A new approximate mathematical model for global convergence for a coefficient inverse problem with backscattering data
- 2012
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Ingår i: Journal of Inverse and Ill-Posed Problems. - : Walter de Gruyter GmbH. - 0928-0219 .- 1569-3945. ; 20:4, s. 513-565
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Tidskriftsartikel (refereegranskat)abstract
- An approximately globally convergent numerical method for a 3d coefficient inverse problem for a hyperbolic equation with backscattering data is presented. A new approximate mathematical model is presented as well. An approximation is used only on the first iteration and amounts to the truncation of a certain asymptotic series. A significantly new element of the convergence analysis is that the so-called "tail functions" are estimated. Numerical results in 2d and 3d cases are discussed, including the one for a quite heterogeneous medium.
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| 2. |
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| 3. |
- Beilina, Larisa, 1970, et al.
(författare)
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Adaptive FEM with relaxation for a hyperbolic coefficient inverse problem
- 2013
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Ingår i: Applied Inverse Problems, Springer Proceedings in Mathematics & Statistics (Select Contributions from the First Annual Workshop on Inverse Problems, Gothenburg, Sweden, 2-3 June 2011). - New York, NY : Springer New York. - 2194-1009 .- 2194-1017. - 9781461478164 ; 48, s. 129-153
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Konferensbidrag (refereegranskat)abstract
- Recent research of publications (Beilina and Johnson, Numerical Mathematics and Advanced Applications: ENUMATH 2001, Springer, Berlin, 2001; Beilina, Applied and Computational Mathematics 1, 158-174, 2002; Beilina and Johnson, Mathematical Models and Methods in Applied Sciences 15, 23-37, 2005; Beilina and Clason, SIAM Journal on Scientific Computing 28, 382-402, 2006; Beilina, Applicable Analysis 90, 1461-1479, 2011; Beilina and Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012; Beilina and Klibanov, Journal of Inverse and Ill-posed Problems 18, 85-132, 2010; Beilina and Klibanov, Inverse Problems 26, 045012, 2010; Beilina and Klibanov, Inverse Problems 26, 125009, 2010; Beilina et al., Journal of Mathematical Sciences 167, 279-325, 2010) have shown that adaptive finite element method presents a useful tool for solution of hyperbolic coefficient inverse problems. In the above publications improvement in the image reconstruction is achieved by local mesh refinements using a posteriori error estimate in the Tikhonov functional and in the reconstructed coefficient. In this paper we apply results of the above publications and present the relaxation property for the mesh refinements and a posteriori error estimate for the reconstructed coefficient for a hyperbolic CIP, formulate an adaptive algorithm, and apply it to the reconstruction of the coefficient in hyperbolic PDE. Our numerical examples present performance of the two-step numerical procedure on the computationally simulated data where on the first step we obtain good approximation of the exact coefficient using approximate globally convergent method of Beilina and Klibanov (Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012), and on the second step we take this solution for further improvement via adaptive mesh refinements.
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| 4. |
- Beilina, Larisa, 1970, et al.
(författare)
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Approximate global convergence in imaging of land mines from backscattered data
- 2013
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Ingår i: Applied Inverse Problems, Springer Proceedings in Mathematics & Statistics (Select Contributions from the First Annual Workshop on Inverse Problems, Gothenburg, Sweden, 2-3 June 2011). - New York, NY : Springer New York. - 2194-1009 .- 2194-1017. - 9781461478164 ; 48, s. 15-36
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Konferensbidrag (refereegranskat)abstract
- We present new model of an approximate globally convergent method in the most challenging case of the backscattered data. In this case data for the coefficient inverse problem are given only at the backscattered side of the medium which should be reconstructed. We demonstrate efficiency and robustness of the proposed technique on the numerical solution of the coefficient inverse problem in two dimensions with the time-dependent backscattered data. Goal of our tests is to reconstruct dielectrics in land mines which is the special case of interest in military applications. Our tests show that refractive indices and locations of dielectric abnormalities are accurately imaged.
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| 5. |
- Beilina, Larisa, 1970, et al.
(författare)
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Global convergence for Inverse Problems
- 2010
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Ingår i: AIP Conference Proceedings; International Conference on Numerical Analysis and Applied Mathematics Rhodes, GREECE, SEP 19-25, 2010. - : AIP. - 0094-243X .- 1551-7616. - 9780735408340 ; 1281, s. 1056-1058
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Konferensbidrag (refereegranskat)abstract
- A globally convergent numerical method for a multidimensional Coefficient Inverse Problem for a hyperbolic equation is presented. It is shown that this technique provides a good starting point for the finite element adaptive method (adaptivity). This leads to a natural two-stage numerical procedure, which synthesizes both these methods.
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| 6. |
- Beilina, Larisa, 1970, et al.
(författare)
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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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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427. ; 289, s. 371-391
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Tidskriftsartikel (refereegranskat)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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| 7. |
- Beilina, Larisa, 1970, et al.
(författare)
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Globally strongly convex cost functional for a coefficient inverse problem
- 2015
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Ingår i: Nonlinear Analysis. - : Elsevier BV. - 1468-1218. ; 22, s. 272-288
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Tidskriftsartikel (refereegranskat)abstract
- A Carleman Weight Function (CWF) is used to construct a new cost functional for a Coefficient Inverse Problems for a hyperbolic PDE. Given a bounded set of an arbitrary size in a certain Sobolev space, one can choose the parameter of the CWF in such a way that the constructed cost functional will be strongly convex on that set. Next, convergence of the gradient method, which starts from an arbitrary point of that set, is established. Since restrictions on the size of that set are not imposed, then this is the global convergence.
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| 8. |
- Beilina, Larisa, 1970, et al.
(författare)
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New a posteriori error estimates for adaptivity technique and global convergence for a hyperbolic coefficient inverse problem
- 2011
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Ingår i: Journal of Mathematical Sciences. - : Springer Science and Business Media LLC. - 1573-8795 .- 1072-3374. ; 172:4, s. 449-476
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Tidskriftsartikel (refereegranskat)abstract
- The coefficient inverse problem for a hyperbolic equation is studied by using the two-stage numerical procedure consisting of the global convergence method and the adaptive finite element method. We obtain new a posteriori error estimates for the Lagrangian and for the unknown coefficient, which is important at the second stage of procedure from the computational point of view. The results are illustrated by numerical experiments. Bibliography: 23 titles. Illustrations: figures. © 2011 Springer Science+Business Media, Inc.
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| 9. |
- Beilina, Larisa, 1970, et al.
(författare)
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Reconstruction from blind experimental data for an inverse problem for a hyperbolic equation
- 2014
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 30:2
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of the reconstruction of dielectrics from blind backscattered experimental data. The reconstruction is done from time domain data, as opposed to a more conventional case of frequency domain data. Experimental data were collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. This system sends electromagnetic pulses into the medium and collects the time-resolved backscattered data on a part of a plane. The spatially distributed dielectric constant epsilon(r)(x), x is an element of R-3 is the unknown coefficient of a wave-like PDE. This coefficient is reconstructed from those data in blind cases. To do this, a globally convergent numerical method is used.
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| 10. |
- Beilina, Larisa, 1970, et al.
(författare)
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Relaxation property for the adaptivity for ill-posed problems
- 2014
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Ingår i: Applicable Analysis. - : Informa UK Limited. - 0003-6811 .- 1563-504X. ; 93:2, s. 223-253
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Tidskriftsartikel (refereegranskat)abstract
- Adaptive finite element method (adaptivity) is known to be an effective numerical tool for some ill-posed problems. The key advantage of the adaptivity is the image improvement with local mesh refinements. A rigorous proof of this property is the central part of this paper. In terms of coefficient inverse problems with single measurement data, the authors consider the adaptivity as the second stage of a two-stage numerical procedure. The first stage delivers a good approximation of the exact coefficient without an advanced knowledge of a small neighborhood of that coefficient. This is a necessary element for the adaptivity to start iterations from. Numerical results for the two-stage procedure are presented for both computationally simulated and experimental data.
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