×

zbMATH — the first resource for mathematics

Adaptive finite element method for a coefficient inverse problem for Maxwell’s system. (English) Zbl 1223.78010
Summary: 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.

MSC:
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Software:
L-BFGS
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Engl HW, Regularization of Inverse Problems (2000)
[2] Tikhonov AN, Numerical Methods for the Solution of Ill-Posed Problems (1995)
[3] DOI: 10.1142/S0218202505003885 · Zbl 1160.76363
[4] Beilina L, Appl. Comput. Math. 2 pp 119– (2003)
[5] Eriksson K, Computational Differential Equations (1996)
[6] DOI: 10.1017/S0962492900002531
[7] DOI: 10.1088/0266-5611/26/4/045012 · Zbl 1193.65165
[8] DOI: 10.1007/s10958-010-9921-1 · Zbl 1286.65147
[9] Ainsworth M, A Posteriori Error Estimation in Finite Element Analysis (2000)
[10] Eriksson K, Applied Mathematics: Body and Soul. Calculus in Several Dimensions (2004)
[11] DOI: 10.1515/9783110203042 · Zbl 1162.65001
[12] Beilina L, Numerical Mathematics and Advanced Applications – ENUMATH 2001 (2001)
[13] DOI: 10.1137/050631252 · Zbl 1104.65313
[14] DOI: 10.1007/BF01389668 · Zbl 0625.65107
[15] DOI: 10.1016/0021-9991(90)90181-Y · Zbl 0703.65082
[16] DOI: 10.1109/22.75280
[17] DOI: 10.1093/acprof:oso/9780198508885.001.0001 · Zbl 1024.78009
[18] Elmkies A, Numeri. Anal., C. R. Acad. Sci. Paris 324 pp 1287– (1997) · Zbl 0877.65081
[19] Joly P, Lecture Notes in Computational Science and Engineering (2003)
[20] Jin J, The Finite Element Methods in Electromagnetics (1993)
[21] Brenner SC, The Mathematical Theory of Finite Element Methods (1994)
[22] Cohen GC, Highre Order Numerical Methods for Transient Wave Equations (2002)
[23] Hughes TJR, The Finite Element Method (1987)
[24] Becker R, Adaptive finite elements for optimal control problems (2001)
[25] DOI: 10.1090/S0025-5718-1980-0572855-7
[26] Pironneau O, Optimal Shape Design for Elliptic Systems (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.