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Maxwell's system; coefficient inverse problem; Tikhonov functional; Lagrangian approach; a posteriori error estimate
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.
[1] Assous, F., Degond, P., Heintze, E., Raviart, P. A., Segre, J.: On a finite-element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 109 (1993), 222-237. DOI 10.1006/jcph.1993.1214 | MR 1253460 | Zbl 0795.65087
[2] Bakushinsky, A. B., Kokurin, M. Y., Smirnova, A.: Iterative Methods for Ill-Posed Problems. An Introduction. Inverse and Ill-Posed Problems Series 54 Walter de Gruyter, Berlin (2011). MR 2757493 | Zbl 1215.47013
[3] Beilina, L.: Adaptive hybrid FEM/FDM methods for inverse scattering problems. Inverse Problems and Information Technologies, V.1, N.3 73-116 (2002).
[4] Beilina, L.: Adaptive finite element method for a coefficient inverse problem for Maxwell's system. Appl. Anal. 90 (2011), 1461-1479. DOI 10.1080/00036811.2010.502116 | MR 2832218 | Zbl 1223.78010
[5] Beilina, L.: Energy estimates and numerical verification of the stabilized domain decomposition finite element/finite difference approach for time-dependent Maxwell's system. Cent. European J. Math. 11 (2013), 702-733, DOI 10.2478/s11533-013-0202-3. DOI 10.2478/s11533-013-0202-3 | MR 3015394 | Zbl 1267.78044
[6] Beilina, L., Cristofol, M., Niinimäki, K.: Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations. Inverse Probl. Imaging 9 1-25 (2015). MR 3305884 | Zbl 1308.35296
[7] Beilina, L., Hosseinzadegan, S.: An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations. ArXiv:1510.07525 (2015). MR 3502111
[8] Beilina, L., Klibanov, M. V.: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. Springer, New York (2012). Zbl 1255.65168
[9] Beilina, L., Klibanov, M. V., Kokurin, M. Yu.: Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem. J. Math. Sci., New York 167 279-325 (2010), translation from Probl. Mat. Anal. 46 3-44 (2010), Russian original. DOI 10.1007/s10958-010-9921-1 | MR 2839023 | Zbl 1286.65147
[10] Beilina, L., Thành, N. T., Klibanov, M. V., Malmberg, J. B.: Reconstruction of shapes and refractive indices from backscattering experimental data using the adaptivity. Inverse Probl. 30 28 pages, Article ID 105007 (2014). MR 3274607 | Zbl 1327.35429
[11] Bellassoued, M., Cristofol, M., Soccorsi, E.: Inverse boundary value problem for the dynamical heterogeneous Maxwell's system. Inverse Probl. 28 18 pages, Article ID 095009 (2012). MR 2966179 | Zbl 1250.35181
[12] Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15 Springer, Berlin (2002). DOI 10.1007/978-1-4757-3658-8_7 | MR 1894376 | Zbl 1012.65115
[13] P. Ciarlet, Jr., H. Wu, J. Zou: Edge element methods for Maxwell's equations with strong convergence for Gauss' laws. SIAM J. Numer. Anal. 52 (2014), 779-807. DOI 10.1137/120899856 | MR 3188392
[14] Cohen, G. C.: Higher Order Numerical Methods for Transient Wave Equations. Scientific Computation Springer, Berlin (2002). MR 1870851 | Zbl 0985.65096
[15] Courant, R., Friedrichs, K., Lewy, H.: On the partial difference equations of mathematical physics. IBM J. Res. Dev. 11 (1967), 215-234. DOI 10.1147/rd.112.0215 | MR 0213764 | Zbl 0145.40402
[16] Engl, H. W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and Its Applications 375 Kluwer Academic Publishers, Dordrecht (1996). MR 1408680 | Zbl 0859.65054
[17] Eriksson, K., Estep, D., Johnson, C.: Applied Mathematics: Body and Soul. Vol. 3. Calculus in Several Dimensions. Springer, Berlin (2004). MR 2020205
[18] Hosseinzadegan, S.: Iteratively regularized adaptive finite element method for reconstruction of coefficients in Maxwell's system. Master's Thesis in Applied Mathematics Department of Mathematical Sciences, Chalmers University of Technology and Gothenburg University (2015).
[19] Ito, K., Jin, B., Takeuchi, T.: Multi-parameter Tikhonov regularization. Methods Appl. Anal. 18 (2011), 31-46. MR 2804535 | Zbl 1285.65032
[20] Johnson, C., Szepessy, A.: Adaptive finite element methods for conservation laws based on a posteriori error estimates. Commun. Pure Appl. Math. 48 199-234 (1995). DOI 10.1002/cpa.3160480302 | MR 1322810 | Zbl 0826.65088
[21] Klibanov, M. V., Bakushinsky, A. B., Beilina, L.: Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess. J. Inverse Ill-Posed Probl. 19 83-105 (2011). DOI 10.1515/jiip.2011.024 | MR 2794397 | Zbl 1279.35113
[22] Křížek, M., Neittaanmäki, P.: Finite Element Approximation of Variational Problems and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics 50 Longman Scientific & Technical, Harlow; John Wiley & Sons, New York (1990). MR 1066462
[23] Ladyzhenskaya, O. A.: The Boundary Value Problems of Mathematical Physics. Applied Mathematical Sciences 49 Springer, New York (1985). DOI 10.1007/978-1-4757-4317-3 | MR 0793735 | Zbl 0588.35003
[24] Malmberg, J. B.: A posteriori error estimation in a finite element method for reconstruction of dielectric permittivity. ArXiv:1502.07658 (2015). MR 3279172
[25] Malmberg, J. B.: A posteriori error estimate in the Lagrangian setting for an inverse problem based on a new formulation of Maxwell's system. L. Beilina Inverse Problems and Applications Selected Papers Based on the Presentations at the Third Annual Workshop on Inverse Problems, Stockholm, 2013 Springer Proceedings in Mathematics and Statistics 120 (2015), 43-53. MR 3343199 | Zbl 1319.78014
[26] Munz, C. D., Omnes, P., Schneider, R., Sonnendrücker, E., Voß, U.: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161 (2000), 484-511. DOI 10.1006/jcph.2000.6507 | MR 1764247 | Zbl 0970.78010
[27] PETSc Team: PETSc, Portable, Extensible Toolkit for Scientific Computation.
[28] Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer Series in Computational Physics Springer, New York (1984). MR 0725856 | Zbl 0534.49001
[29] Scott, L. R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990), 483-493. DOI 10.1090/S0025-5718-1990-1011446-7 | MR 1011446 | Zbl 0696.65007
[30] Smith, D. R., Schultz, S., Markoš, P., Soukoulis, C. M.: Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients. Phys. Rev. B 65 (2002), DOI:10.1103/PhysRevB.65.195104. DOI 10.1103/PhysRevB.65.195104
[31] Tikhonov, A. N., Goncharskiy, A. V., Stepanov, V. V., Kochikov, I. V.: Ill-posed problems of the image processing. DAN USSR 294 832-837 (1987). MR 0898748
[32] Tikhonov, A. N., Goncharsky, A. V., Stepanov, V. V., Yagola, A. G.: Numerical Methods for the Solution of Ill-Posed Problems. Rev. Mathematics and Its Applications 328 Kluwer Academic Publishers, Dordrecht (1995). MR 1350538 | Zbl 0831.65059
[33] WavES: The software package.
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