| 1. |
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| 2. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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A Discontinuous Galerkin Approach for Stabilized Maxwell’s Equations in Pseudo-Frequency Domain
- 2023
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In: Springer Proceedings in Mathematics and Statistics. - 2194-1009 .- 2194-1017. - 9783031358708
-
Conference paper (peer-reviewed)abstract
- This paper concerns the study of a stabilized discontinuous Galerkin finite element method for the Maxwell’s equations in pseudo-frequency domain obtained through Laplace transformation in time. The model problem is considered in the special case assuming constant dielectric permittivity function in a boundary neighborhood. The discontinuous Galerkin finite element method (DGFEM) is formulated and the convergence is addressed in a priori setting where we derive optimal order error bound of the scheme in a L2 -based triple norm. Finally, our numerical examples confirm predicted convergence of the proposed scheme.
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| 3. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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A posteriori error estimates in a globally convergent FEM for a hyperbolic coefficient inverse problem
- 2010
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In: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 26:11
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Journal article (peer-reviewed)abstract
- This study concerns a posteriori error estimates in a globally convergent numerical method for a hyperbolic coefficient inverse problem. Using the Laplace transform the model problem is reduced to a nonlinear elliptic equation with a gradient dependent nonlinearity. We investigate the behavior of the nonlinear term in both a priori and a posteriori settings and derive optimal a posteriori error estimates for a finite-element approximation of this problem. Numerical experiments justify the efficiency of a posteriori estimates in the globally convergent approach.
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| 4. |
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| 5. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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A Priori Error Estimates and Computational Studies for a Fermi Pencil-Beam Equation
- 2018
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In: Journal of Computational and Theoretical Transport. - : Informa UK Limited. - 2332-4325 .- 2332-4309. ; 47:1-3, s. 125-151
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Journal article (peer-reviewed)abstract
- We derive a priori error estimates for the standard Galerkin and streamline diffusion finite element methods for the Fermi pencil-beam equation obtained from a fully three-dimensional Fokker-Planck equation in space and velocity variables. For a constant transport cross-section, there is a closed form analytic solution available for the Fermi equation with a data as product of Dirac functions. Our objective is to study the case of nonconstant, nonincreasing transport cross-section. Therefore we start with a theoretical, that is, a priori, error analysis for a Fermi model with modified initial data in L-2. Then we construct semi-streamline-diffusion and characteristic streamline-diffusion schemes and consider an adaptive algorithm for local mesh refinements. To derive the stability estimates, for simplicity, we rely on the assumption of nonincreasing transport cross-section. Different numerical examples, in two space dimensions are justifying the theoretical results. Implementations show significant reduction of the computational error by using such adaptive procedure.
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| 6. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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A stabilized P1 domain decomposition finite element method for time harmonic Maxwell's equations
- 2023
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In: Mathematics and Computers in Simulation. - : Elsevier BV. - 0378-4754. ; 204, s. 556-574
-
Journal article (peer-reviewed)abstract
- One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell's equations is to render their hyperbolic character to elliptic form. This paper is devoted to the study of a stabilized linear, domain decomposition, finite element method for the time harmonic Maxwell's equations, in a dual form, obtained through the Laplace transformation in time. The model problem is for the particular case of the dielectric permittivity function which is assumed to be constant in a boundary neighborhood. The discrete problem is coercive in a symmetrized norm, equivalent to the discrete norm of the model problem. This yields discrete stability, which together with continuity guarantees the well-posedness of the discrete problem, cf Arnold et al. (2002) [3], Di Pietro and Ern (2012) [45]. The convergence is addressed both in a priori and a posteriori settings. In the a priori error estimates we confirm the theoretical convergence of the scheme in a L2-based, gradient dependent, triple norm. The order of convergence is O(h) in weighted Sobolev space Hw2(ohm), and hence optimal. Here, the weight w := w(epsilon, s) where epsilon is the dielectric permittivity function and s is the Laplace transformation variable. We also derive, similar, optimal a posteriori error estimates controlled by a certain, weighted, norm of the residuals of the computed solution over the domain and at the boundary (involving the relevant jump terns) and hence independent of the unknown exact solution. The a posteriori approach is used, e.g. in constructing adaptive algorithms for the computational purposes, which is the subject of a forthcoming paper. Finally, through implementing several numerical examples, we validate the robustness of the proposed scheme. (c) 2022 The Author(s). Published by Elsevier B.V. on behalf of International Association for Mathematics and Computers in Simulation (IMACS). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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| 7. |
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| 8. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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Adaptive approximate globally convergent algorithm with backscattered data.
- 2013
-
In: Inverse Problems and Large-Scale Computations. Springer Proceedings in Mathematics and Statistics. Larisa Beilina, Yury V. Shestopalov (Eds.). - Cham : Springer International Publishing. - 2194-1009 .- 2194-1017. - 9783319006598 ; 52, s. 1-20
-
Conference paper (peer-reviewed)abstract
- We construct, analyze and implement an approximately globally convergent finite element scheme for a hyperbolic coefficient inverse problem in the case of backscattering data. This extends the computational aspects introduced in Asadzadeh and Beilina (Inv. Probl. 26, 115007, 2010), where using Laplace transformation, the continuous problem is reduced to a nonlinear elliptic equation with a gradient dependent nonlinearity. We investigate the behavior of the nonlinear term and discuss the stability issues as well as optimal a posteriori error bounds, based on an adaptive procedure and due to the maximal available regularity of the exact solution. Numerical implementations justify the efficiency of adaptive a posteriori approach in the globally convergent setting.
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| 9. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
-
Finite element schemes for Fermi equation
- 2017
-
In: International Conference of Numerical Analysis and Applied Mathematics 2016, ICNAAM 2016, Rhodes, Greece, 19-25 September 2016. AIP Conference Proceedings. Vol. 1863, nr. 1, 370007. - : Author(s). - 0094-243X .- 1551-7616. - 9780735415386
-
Conference paper (peer-reviewed)abstract
- A priori error estimates are derived for the streamline diffusion (SD) finite element methods for the Fermi pencil-beam equation. Two-dimensional numerical examples confirm our theoretical investigations.
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| 10. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
-
Stability and Convergence Analysis of a Domain Decomposition FE/FD Method for Maxwell's Equations in the Time Domain
- 2022
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In: Algorithms. - : MDPI AG. - 1999-4893. ; 15:10
-
Journal article (peer-reviewed)abstract
- Stability and convergence analyses for the domain decomposition finite element/finite difference (FE/FD) method are presented. The analyses are developed for a semi-discrete finite element scheme for time-dependent Maxwell's equations. The explicit finite element schemes in different settings of the spatial domain are constructed and a domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies.
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| 11. |
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| 12. |
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| 13. |
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| 14. |
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| 15. |
- Beilina, Larisa, 1970, et al.
(author)
-
A new approximate mathematical model for global convergence for a coefficient inverse problem with backscattering data
- 2012
-
In: Journal of Inverse and Ill-Posed Problems. - : Walter de Gruyter GmbH. - 0928-0219 .- 1569-3945. ; 20:4, s. 513-565
-
Journal article (peer-reviewed)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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| 16. |
- Beilina, Larisa, 1970, et al.
(author)
-
A Posteriori Error Estimates and Adaptive Error Control for Permittivity Reconstruction in Conductive Media
- 2023
-
In: Springer Proceedings in Mathematics and Statistics. - 2194-1009 .- 2194-1017. - 9783031358708
-
Conference paper (peer-reviewed)abstract
- An inverse problem of reconstruction of the spatially distributed dielectric permittivity function in the Maxwell’s system is considered. The reconstruction method is based on the optimization approach to find stationary point of the Tikhonov functional. A posteriori estimates for the corresponding Tikhonov functional and for the reconstructed dielectric permittivity function are derived. Based on these estimates two adaptive conjugate gradient algorithms are formulated. Our numerical tests show feasibility of application of an adaptive optimization algorithm for reconstruction of dielectric permittivity function using anatomically realistic breast phantom of MRI database produced in University of Wisconsin [53].
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| 17. |
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| 18. |
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| 19. |
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| 20. |
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| 21. |
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| 22. |
- Beilina, Larisa, 1970, et al.
(author)
-
Adaptive FEM with relaxation for a hyperbolic coefficient inverse problem
- 2013
-
In: 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
-
Conference paper (peer-reviewed)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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| 23. |
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| 24. |
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| 25. |
- Beilina, Larisa, 1970
(author)
-
Adaptive finite element/difference methods for time-dependent inverse scattering problems
- 2003
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Doctoral thesis (other academic/artistic)abstract
- In this thesis we develop adaptive hybrid finite element/difference methods for inverse time-domain acoustic and elastic scattering, where we seek to find the location and form of a (small) unknown object inside a large homogeneous body from measured wave-reflection data. We formulate the inverse problem as an optimal control problem, where we seek to reconstruct unknown material coefficients with best least squares wave fit to data. We solve the equations of optimality expressing stationarity of an associated Lagrangian by a quasi-Newton method, where in each step we compute the gradient by solving a forward and an adjoint wave equation. In the first part of the thesis we develop an explicit hybrid finite element/difference method for time-dependent acoustic wave propagation in two and three space dimensions, combining the flexibility of finite elements with the efficiency of finite differences. In the second part we then apply this method to inverse acoustic scattering in two and three space dimensions. We prove an a posteriori error estimate for the error in the Lagrangian and we formulate a corresponding adaptive method, where the finite element mesh covering the object is refined from residual feed-back. The forward and inverse solvers are implemented in object-oriented form in C++. We demonstrate the performance of the adaptive hybrid method for inverse scattering in several examples. In the third and fourth parts we extend to elastic scattering.
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| 26. |
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| 27. |
- Beilina, Larisa, 1970
(author)
-
Adaptive Finite Element Method for a coefficient inverse problem for the Maxwell's system
- 2011
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In: Applicable Analysis. - : Informa UK Limited. - 0003-6811 .- 1563-504X. ; 90:10, s. 1461-1479
-
Journal article (peer-reviewed)abstract
- We consider a coefficient inverse problem for Maxwell's system in 3-D. The coefficient of interest is the dielectric permittivity function. Only backscattering single measurement data are used. The problem is formulated as an optimization problem. The key idea is to use the adaptive finite element method for the solution. Both analytical and numerical results are presented. Similar ideas for inverse problems for the complete time dependent Maxwell's system were not considered in the past.
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| 28. |
- Beilina, Larisa, 1970
(author)
-
Adaptive Finite Element Method for an Electromagnetic Coefficient Inverse Problem
- 2010
-
In: 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. 1052-1055
-
Conference paper (peer-reviewed)abstract
- We present an adaptive finite element method for solution of an electromagnetic coefficient inverse problem to reconstruct the dielectric permittivity and magnetic permeability functions. The inverse problem is formulated as an optimal control problem, where we solve the equations of optimality expressing stationarity of an associated Lagrangian by a quasi-Newton method: in each step we compute the gradient by solving a forward and an adjoint equation. We formulate an adaptive algorithm which can be used to efficiently solve electromagnetic coefficient inverse problem.
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| 29. |
- Beilina, Larisa, 1970, et al.
(author)
-
Adaptive finite element method in nanophotonic simulations
- 2017
-
In: International Conference of Numerical Analysis and Applied Mathematics 2016, ICNAAM 2016, Rhodes, Greece, 19-25 September 2016. Vol. 1863, nr. 1, 370004. - : Author(s). - 0094-243X .- 1551-7616. - 9780735415386
-
Conference paper (other academic/artistic)abstract
- The problem of constructing nanophotonic structures of arbitrary geometry with prescribed properties was studied using an adaptive optimization algorithm. Stability estimates for the forward and adjoint problems involved in this algorithm are presented. A numerical example illustrates the construction of nanostructure in two dimensions.
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| 30. |
- Beilina, Larisa, 1970, et al.
(author)
-
Adaptive finite element method in reconstruction of dielectrics from backscattered data
- 2013
-
In: Applied Inverse Problems, Springer Proceedings in Mathematics & Statistics. Larisa Beilina (Ed.). - New York, NY : Springer New York. - 2194-1009 .- 2194-1017. - 9781461478157 ; 48, s. 51-73
-
Conference paper (peer-reviewed)abstract
- The validity of the adaptive finite element method for reconstruction of dielectrics in a symmetric structure is verified on time-resolved data in two dimensions. This problem has practical interest in the reconstruction of the structure of photonic crystals or in the imaging of land mines. Dielectric permittivity, locations, and shapes/sizes of dielectric abnormalities are accurately imaged using adaptive finite element algorithm.
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| 31. |
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| 32. |
- Beilina, Larisa, 1970, et al.
(author)
-
Adaptive Hybrid Finite Element/Difference method for Maxwell's equations
- 2010
-
In: TWMS Journal of Pure and Applied Mathematics. - 1683-3511. ; 1:2, s. 176-197
-
Journal article (peer-reviewed)abstract
- An explicit, adaptive, hybrid finite element/finite difference method is proposed for the numerical solution of Maxwell’s equations in the time domain. The method is hybrid in the sense that different numerical methods, finite elements and finite differences, are used in different parts of the computational domain. Thus, we combine the flexibility of finite elements with the efficiency of finite differences. Furthermore, an a posteriori error estimate is derived for local adaptivity and error control inside the subregion, where finite elements are used. Numerical experiments illustrate the usefulness of computational adaptive error control of proposed new method.
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| 33. |
- Beilina, Larisa, 1970
(author)
-
Adaptive Hybrid Finite Element/Difference method for Maxwell's equations: an a priory error estimate and efficiency
- 2010
-
In: Applied and Computational Mathematics. - 1683-3511. ; 9:2, s. 176-197
-
Journal article (peer-reviewed)abstract
- In this work we extend our previous study where anexplicit adaptive hybrid finite element/finite difference method wasproposed for the numerical solution of Maxwell's equations in the timedomain. Here we derive a priori error estimate in finite elementmethod and present numerical examples where we indicate the rate ofconvergence of the hybrid method. We compare also hybrid finiteelement/finite difference method with pure finite element method andshow that we devise an optimized method. In our three dimensionalcomputations the hybrid approach is about 3 times faster than acorresponding highly optimized finite element method. We conclude thatthe hybrid approach may be an important tool to reduce the executiontime and memory requirements for large scale computations.
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| 34. |
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| 35. |
- Beilina, Larisa, 1970, et al.
(author)
-
Adaptive optimization algorithm for the computational design of nanophotonic structures
- 2016
-
In: International Conference on Electromagnetics in Advanced Applications (ICEAA), 2016. - : IEEE. - 9781467398114
-
Conference paper (peer-reviewed)abstract
- We consider the problem of the construction of the nanophotonic structures of arbitrary geometry with prescribed desired properties. We illustrate the efficiency of our adaptive optimization algorithm on the construction of nanophotonic structure in two dimensions.
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| 36. |
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| 37. |
- Beilina, Larisa, 1970, et al.
(author)
-
Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem
- 2010
-
In: Journal of Mathematical Sciences, JMS, Springer. - : Springer Science and Business Media LLC. - 1072-3374 .- 1573-8795. ; 167:3, s. 279-325
-
Journal article (peer-reviewed)abstract
- A new framework of the functional analysis is developed for the finite element adaptive method (adaptivity) for the Tikhonov regularization functional for some ill-posed problems. As a result, the relaxation property for adaptive mesh refinements is established. An application to a multidimensional coefficient inverse problem for a hyperbolic equation is discussed. This problem arises in the inverse scattering of acoustic and electromagnetic waves. First, a globally convergent numerical method provides a good approximation for the correct solution of this problem. Next, this approximation is enhanced via the subsequent application of the adaptivity. Analytical results are verified computationally. Bibliography: 30 titles. Illustration: 2 figures.
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| 38. |
- Beilina, Larisa, 1970, et al.
(author)
-
An Adaptive Finite Element/Finite Difference Domain Decomposition Method for Applications in Microwave Imaging
- 2022
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In: Electronics. - : MDPI AG. - 2079-9292. ; 11:9
-
Journal article (peer-reviewed)abstract
- A new domain decomposition method for Maxwell's equations in conductive media is presented. Using this method, reconstruction algorithms are developed for the determination of the dielectric permittivity function using time-dependent scattered data of an electric field. All reconstruction algorithms are based on an optimization approach to find the stationary point of the Lagrangian. Adaptive reconstruction algorithms and space-mesh refinement indicators are also presented. Our computational tests show the qualitative reconstruction of the dielectric permittivity function using an anatomically realistic breast phantom.
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| 39. |
- Beilina, Larisa, 1970, et al.
(author)
-
An adaptive finite element method for solving 3D electromagnetic volume integral equation with applications in microwave thermometry
- 2022
-
In: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 459
-
Journal article (peer-reviewed)abstract
- An adaptive finite element method (AFEM) for the numerical solution of an electromagnetic volume integral equation (VIE) is presented. To solve the model VIE, the problem is formulated as an optimal control problem for minimization of Tikhonov's regularization functional. A posteriori error estimates in the obtained finite element reconstruction and in the underlying Tikhonov's functional are derived. Based on these estimates, adaptive finite element algorithms are formulated and numerically tested on the problem of microwave hyperthermia in cancer treatment. In this problem, the temperature change of a target in the computational domain results in the change of its dielectric properties. Numerical examples of monitoring this change show robust and qualitative three dimensional reconstructions of the target using the proposed adaptive algorithms. (C) 2022 The Author(s). Published by Elsevier Inc.
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| 40. |
- Beilina, Larisa, 1970, et al.
(author)
-
An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations
- 2016
-
In: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 61:3, s. 253-286
-
Journal article (peer-reviewed)abstract
- We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell's system using limited boundary observations of the electric field in 3D. We derive a posteriori error estimates in the Tikhonov functional to be minimized and in the regularized solution of this functional, as well as formulate the corresponding adaptive algorithm. Our numerical experiments justify the efficiency of our a posteriori estimates and show significant improvement of the reconstructions obtained on locally adaptively refined meshes.
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| 41. |
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| 42. |
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| 43. |
- Beilina, Larisa, 1970
(author)
-
Application of the finite element method in a quantitative imaging technique
- 2016
-
In: Journal of Computational Methods in Sciences and Engineering. - 1472-7978. ; 16:4, s. 755-771
-
Journal article (peer-reviewed)abstract
- We present the Finite Element Method (FEM) for the numerical solution of the multidimensional coefficient inverse problem (MCIP) in two dimensions. This method is used for explicit reconstruction of the coefficient in the hyperbolic equation using data resulted from a single measurement. To solve our MCIP we use approximate globally convergent method and then apply FEM for the resulted equation. Our numerical examples show quantitative reconstruction of the sound speed in small tumor-like inclusions.
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| 44. |
- Beilina, Larisa, 1970, et al.
(author)
-
Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems
- 2012
-
Book (other academic/artistic)abstract
- Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems is the first book in which two new concepts of numerical solutions of multidimensional Coefficient Inverse Problems (CIPs) for a hyperbolic Partial Differential Equation (PDE) are presented: Approximate Global Convergence and the Adaptive Finite Element Method (adaptivity for brevity). Two central questions for CIPs are addressed: How to obtain a good approximation for the exact solution without any knowledge of a small neighborhood of this solution, and how to refine it given the approximation. The book also combines analytical convergence results with recipes for various numerical implementations of developed algorithms. The developed technique is applied to two types of blind experimental data, which are collected both in a laboratory and in the field. The result for the blind backscattering experimental data collected in the field addresses a real-world problem of imaging of shallow explosives.
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| 45. |
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| 46. |
- Beilina, Larisa, 1970, et al.
(author)
-
Approximate global convergence in imaging of land mines from backscattered data
- 2013
-
In: 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
-
Conference paper (peer-reviewed)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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| 47. |
- Beilina, Larisa, 1970, et al.
(author)
-
Computational design of acoustic materials using an adaptive optimization algorithm
- 2018
-
In: Applied Mathematics and Information Sciences. - : Natural Sciences Publishing. - 1935-0090 .- 2325-0399. ; 12:1, s. 33-43
-
Journal article (peer-reviewed)abstract
- © 2018 NSP Natural Sciences Publishing Cor. We consider the problem of design of the acoustic structure of arbitrary geometry with prescribed desired properties. We use optimization approach for the solution of this problem and minimize the Tikhonov functional on adaptively refined meshes. These meshes are refined locally only in places where the acoustic structure should be designed. Our special symmetric mesh refinement strategy together with interpolation procedure allows the construction of the symmetric acoustic material with prescribed properties. Efficiency of the presented adaptive optimization algorithm is illustrated on the construction of the symmetric acoustic material in two dimensions.
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| 48. |
- Beilina, Larisa, 1970, et al.
(author)
-
Determining the conductivity for a nonautonomous hyperbolic operator in a cylindrical domain
- 2018
-
In: Mathematical Methods in the Applied Sciences. - : Wiley. - 0170-4214 .- 1099-1476. ; 41:5, s. 2012-2030
-
Journal article (peer-reviewed)abstract
- This paper is devoted to the reconstruction of the conductivity coefficient for a nonautonomous hyperbolic operator an infinite cylindrical domain. Applying a local Carleman estimate, we prove the uniqueness and a Holder stability in the determination of the conductivity using a single measurement data on the lateral boundary. Our numerical examples show good reconstruction of the location and contrast of the conductivity function in 3 dimensions.
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| 49. |
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| 50. |
- Beilina, Larisa, 1970
(author)
-
Domain decomposition finite element/finite difference method for the conductivity reconstruction in a hyperbolic equation
- 2016
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In: Communications in Nonlinear Science and Numerical Simulation. - : Elsevier BV. - 1007-5704. ; 37, s. 222-237
-
Journal article (peer-reviewed)abstract
- We present domain decomposition finite element/finite difference method for the solution of hyperbolic equation. The domain decomposition is performed such that finite elements and finite differences are used in different subdomains of the computational domain: finite difference method is used on the structured part of the computational domain and finite elements on the unstructured part of the domain. Explicit discretizations for both methods are constructed such that the finite element and the finite difference schemes coincide on the common structured overlapping layer between computational subdomains. Then the resulting approach can be considered as a pure finite element scheme which avoids instabilities at the interfaces. We derive an energy estimate for the underlying hyperbolic equation with absorbing boundary conditions and illustrate efficiency of the domain decomposition method on the reconstruction of the conductivity function in three dimensions.
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