Abstract
We consider an inverse problem of reconstructing the conductivity function in a hyperbolic equation using single space-time domain noisy observations of the solution on the backscattering boundary of the computational domain. We formulate our inverse problem as an optimization problem and use Lagrangian approach to minimize the corresponding Tikhonov functional. We present a theorem of a local strong convexity of our functional and derive error estimates between computed and regularized as well as exact solutions of this functional, correspondingly. In numerical simulations we apply domain decomposition finite element-finite difference method for minimization of the Lagrangian. Our computational study shows efficiency of the proposed method in the reconstruction of the conductivity function in three dimensions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A.B. Bakushinsky, M.Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems (Springer, 2004)
A. Bakushinsky, M.Y. Kokurin, A. Smirnova, Iterative Methods for Ill-posed Problems, vol. 54, Inverse and Ill-Posed Problems Series (De Gruyter, 2011)
L. Beilina, Domain Decomposition finite element/finite difference method for the conductivity reconstruction in a hyperbolic equation. Commun Nonlinear Sci Numer Simul. Elsevier (2016). https://doi.org/10.1016/j.cnsns.2016.01.016
L. Beilina, K. Samuelsson, K. Åhlander, Efficiency of a hybrid method for the wave equation, in Proceedings of the International Conference on Finite Element Methods: Three dimensional problems. GAKUTO International Series, Mathematical Sciences and Applications, 15 (2001)
L. Beilina, C. Johnson, A posteriori error estimation in computational inverse scattering. Math Models Appl Sci 1, 23–35 (2005)
L. Beilina, M.V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems (Springer, New-York, 2012)
L. Beilina, M.V. Klibanov, MYu. Kokurin, Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem. J Math Sci 167, 279–325 (2010)
L. Beilina, N.T. Thà nh, M.V. Klibanov, J. Bondestam-Malmberg, Reconstruction of shapes and refractive indices from blind backscattering experimental data using the adaptivity. Inverse Probl. 30, 105007 (2014)
L. Beilina, M. Cristofol, K. Niinimäki, Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations. Inverse Probl Imaging 9(1), 1–25 (2015)
C. Bellis, M. Bonnet, B.B. Guzina, Apposition of the topological sensitivity and linear sampling approaches to inverse scattering. Wave Motion 50, 891–908 (2013)
S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, Berlin, 1994)
H.D. Bui, A. Constantinescu, H. Maigre, Numerical identification of planar cracks in elastodynamics using the instantaneous reciprocity gap. Inverse Probl 20, 993–1001 (2004)
H.D. Bui, A. Constantinescu, H. Maigre, An exact inversion formula for determining a planar fault from boundary measurements. J Inverse Ill Posed Probl 13, 553–565 (2005)
Y.T. Chow, J. Zou, A numerical method for reconstructing the coefficient in a wave equation. Numer Methods Partial Differ Equ 31, 289–307 (2015)
G.C. Cohen, Higher Order Numerical Methods for Transient Wave Equations (Springer, 2002)
R. Courant, K. Friedrichs, H. Lewy, On the partial differential equations of mathematical physics. IBM J Res Dev 11(2), 215–234 (1967)
M. Cristofol, S. Li, E. Soccorsi, Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary. Math Control Relat Fields 6(3), 407–427 (2016)
H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic Publishers, Boston, 2000)
B. Engquist, A. Majda, Absorbing boundary conditions for the numerical simulation of waves. Math. Comp. 31, 629–651 (1977)
S.N. Fata, B.B. Guzina, A linear sampling method for near-field inverse problems in elastodynamics. Inverse Probl 20, 713–736 (2004)
M.V. Klibanov, A.B. Bakushinsky, L. Beilina, Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess. J Inverse Ill Posed Probl 19(1), 83–105 (2011)
O.A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics (Springer, Berlin, 1985)
PETSc, Portable, Extensible Toolkit for Scientific Computation. http://www.mcs.anl.gov/petsc/
O. Pironneau, Optimal Shape Design for Elliptic Systems (Springer, Berlin, 1984)
A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov, A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer, London, 1995)
WavES, the software package. http://www.waves24.com
Acknowledgements
The research of L. B. is partially supported by the sabbatical programme at the Faculty of Science, University of Gothenburg, Sweden, and the research of K. N. was supported by the Swedish Foundation for Strategic Research. The computations were performed on resources at Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Beilina, L., Niinimäki, K. (2018). Numerical Studies of the Lagrangian Approach for Reconstruction of the Conductivity in a Waveguide. In: Beilina, L., Smirnov, Y. (eds) Nonlinear and Inverse Problems in Electromagnetics. PIERS PIERS 2017 2017. Springer Proceedings in Mathematics & Statistics, vol 243. Springer, Cham. https://doi.org/10.1007/978-3-319-94060-1_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-94060-1_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94059-5
Online ISBN: 978-3-319-94060-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)