| 1. |
- 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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| 2. |
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| 3. |
- Beilina, Larisa, 1970, et al.
(author)
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Adaptive Hybrid Finite Element/Difference method for Maxwell's equations
- 2010
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In: TWMS Journal of Pure and Applied Mathematics. - 1683-3511. ; 1:2, s. 176-197
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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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| 4. |
- Beilina, Larisa, 1970
(author)
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Adaptive Hybrid Finite Element/Difference method for Maxwell's equations: an a priory error estimate and efficiency
- 2010
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In: Applied and Computational Mathematics. - 1683-3511. ; 9:2, s. 176-197
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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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| 5. |
- Beilina, Larisa, 1970, et al.
(author)
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Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem
- 2010
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In: Journal of Mathematical Sciences, JMS, Springer. - : Springer Science and Business Media LLC. - 1072-3374 .- 1573-8795. ; 167:3, s. 279-325
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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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| 6. |
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| 7. |
- Beilina, Larisa, 1970, et al.
(author)
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Synthesis of global convergence and adaptivity for a hyperbolic coefficient inverse problem in 3D
- 2010
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In: Journal of Inverse and Ill-Posed Problems. - 0928-0219 .- 1569-3945. ; 18:1, s. 85-132
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Journal article (peer-reviewed)abstract
- A globally convergent numerical method for a 3-Dimensional Coefficient Inverse Problem for a hyperbolic equation is presented. A new globally convergent theorem is proven. It is shown that this technique provides a good first guess for the Finite Element Adaptive method (adaptivity) method. This leads to a synthesis of both approaches. Numerical results are presented.
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| 8. |
- Klibanov, M. V., et al.
(author)
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Picosecond scale experimental verification of a globally convergent algorithm for a coefficient inverse problem
- 2010
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In: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 26:4
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Journal article (peer-reviewed)abstract
- A globally convergent algorithm by the first and third authors for a 3D hyperbolic coefficient inverse problem is verified on experimental data measured in the picosecond scale regime. Quantifiable images of dielectric abnormalities are obtained. The total measurement timing of a 100 ps pulse for one detector location was 1.2 ns with 20 ps (= 0.02 ns) time step between two consecutive readings. Blind tests have consistently demonstrated an accurate imaging of refractive indexes of dielectric abnormalities. At the same time, it is shown that a modified gradient method is inapplicable to this kind of experimental data. This inverse algorithm is also applicable to other types of imaging modalities, e. g. acoustics. Potential applications are in airport security, imaging of land mines, imaging of defects in non-distractive testing, etc.
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| 9. |
- Xin, J., et al.
(author)
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Globally convergent numerical methods for coefficient inverse problems for imaging inhomogeneities
- 2010
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In: Computing in Science and Engineering, (CISE). - 1521-9615. ; 12:5, s. 64-77
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Journal article (peer-reviewed)abstract
- How can we differentiate between an underground stone and a land mine? We discuss a class of methods for solving such problems. This class of methods concerns globally convergent numerical methods for Coefficient Inverse Problems, unlike conventional locally convergent algorithms. Numerical results are presented modeling imaging of the spatially distributed dielectric permittivity function in an environment where antipersonnel land mines are embedded along with stones. While these results are concerned with the first generation of globally convergent algorithms, images obtained by the most recent second generation are also presented for a generic case of imaging of the dielectric permittivity function. The mathematical apparatus is sketched only very briefly with references to corresponding publications.
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