A primal dual mixed finite element method for inverse identification of the diffusion coefficient and its relation to the Kohn-Vogelius penalty method

Authors

  • Erik Burman Department of Mathematics, University College London, United Kingdom

DOI:

https://doi.org/10.17268/sel.mat.2023.01.02

Keywords:

Unique continuation, Mixed finite element method, Stability

Abstract

We revisit the celebrated Kohn-Vogelius penalty method and discuss how to use it for the unique continuation problem where data is given in the bulk of the domain. We then show that the primal-dual mixed finite element methods for the elliptic Cauchy problem introduced in [1] (E. Burman, M. Larson, L. Oksanen, Primal-dual mixed finite element methods for the elliptic Cauchy problem, SIAM J. Num. Anal., 56 (6), 2018) can be interpreted as a Kohn-Vogelius penalty method and modify it to allow for unique continuation using data in the bulk. We prove that the resulting linear system is invertible for all data. Then we show that by introducing a singularly perturbed Robin condition on the discrete level sufficient regularization is obtained so that error estimates can be shown using conditional stability. Finally we show how the method can be used for the identification of the diffusivity coefficient in a second order elliptic operator with partial data. Some numerical examples are presented showing the performance of the method for unique continuation and for impedance computed tomography with partial data.

References

Burman E, Larson MG, Oksanen L. Primal-dual mixed finite element methods for the elliptic Cauchy problem. SIAM J Numer Anal. 2018;56(6):3480-509. Available from: https://doi.org/10.1137/17M1163335.

Calderón AP. On an inverse boundary value problem. In: Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980). Soc. Brasil. Mat., Rio de Janeiro; 1980. p. 65-73.

Uhlmann G. Electrical impedance tomography and Calderón’s problem. Inverse Problems. 2009;25(12):123011, 39. Available from: https://doi.org/10.1088/0266-5611/25/12/123011.

Kohn RV, McKenney A. Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems. 1990;6(3):389-414. Available from: http://stacks.iop.org/0266-5611/6/389.

Knowles I. A variational algorithm for electrical impedance tomography. Inverse Problems. 1998;14(6):1513-25. Available from: https://doi.org/10.1088/0266-5611/14/6/010.

Gehre M, Jin B, Lu X. An analysis of finite element approximation in electrical impedance tomography. Inverse Problems. 2014;30(4):045013, 24. Available from: https://doi.org/10.1088/0266-5611/30/4/045013.

Rondi L. Discrete approximation and regularisation for the inverse conductivity problem. Rend Istit Mat Univ Trieste. 2016;48:315-52. Available from: https://doi.org/10.13137/2464-8728/13162.

Hinze M, Kaltenbacher B, Quyen TNT. Identifying conductivity in electrical impedance tomography with total variation regularization. Numer Math. 2018;138(3):723-65. Available from: https://doi.org/10.1007/s00211-017-0920-8.

Harrach B. An introduction to finite element methods for inverse coefficient problems in elliptic PDES. Jahresber Dtsch Math-Ver. 2021;123(3):183-210. Available from: https://doi.org/10.1365/s13291-021-00236-2.

Choulli M. Comments on the determination of the conductivity by boundary measurements. J Math Anal Appl. 2023;517(2):Paper No. 126638, 31. Available from: https://doi.org/10.1016/j.jmaa.2022.126638.

Burman E. Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J Sci Comput. 2013;35(6):A2752-80. Available from: https://doi.org/10.1137/130916862.

Burman E. Error estimates for stabilized finite element methods applied to ill-posed problems. C R Math Acad Sci Paris. 2014;352(7-8):655-9. Available from: https://doi.org/10.1016/j.crma.2014.06.008.

Burman E. Stabilised finite element methods for ill-posed problems with conditional stability. In: Building bridges: connections and challenges in modern approaches to numerical partial differential equations. vol. 114 of Lect. Notes Comput. Sci. Eng. Springer, [Cham]; 2016. p. 93-127.

Burman E. The elliptic Cauchy problem revisited: control of boundary data in natural norms. C R Math Acad Sci Paris. 2017;355(4):479-84. Available from: https://doi.org/10.1016/j.crma.2017.02.014.

Burman E, Hansbo P, Larson MG. Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems. 2018;34(3):035004, 36. Available from: https://doi.org/10.1088/1361-6420/aaa32b.

Burman E, Nechita M, Oksanen L. Unique continuation for the Helmholtz equation using stabilized finite element methods. J Math Pures Appl (9). 2019;129:1-22. Available from: https://doi.org/10.1016/j.matpur.2018.10.003.

Burman E, Nechita M, Oksanen L. A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime. Numer Math. 2020;144(3):451-77. Available from: https://doi.org/10.1007/s00211-019-01087-x.

Wexler A, Fry B, Neuman M. Impedance-computed tomography algorithm and system. Applied optics. 1985;24(23):3985-92.

Kohn RV, Vogelius M. Relaxation of a variational method for impedance computed tomography. Comm Pure Appl Math. 1987;40(6):745-77. Available from: https://doi.org/10.1002/cpa.3160400605.

Ben Belgacem F, Girault V, Jelassi F. Full discretization of Cauchy's problem by Lavrentiev-finite element method. SIAM J Numer Anal. 2022;60(2):558-84. Available from: https://doi.org/10.1137/21M1401310.

Ben Belgacem F, Jelassi F, Girault V. Uniqueness failure for the finite element Cauchy-Poisson's problem. Comput Math Appl. 2023;135:77-92. Available from: https://doi.org/10.1016/j.camwa.2023.01.010.

Raviart PA, Thomas JM. A mixed finite element method for 2nd order elliptic problems. In: Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975). Lecture Notes in Math., Vol. 606. Springer, Berlin; 1977.p.292-315.

Alessandrini G, Rondi L, Rosset E, Vessella S. The stability for the Cauchy problem for elliptic equations. Inverse Problems. 2009;25(12):123004, 47. Available from: https://doi.org/10.1088/0266-5611/25/12/123004.

Robbiano L. Théoreme d'unicité et controle pour les équations hyperboliques. In: Nonlinear partial differential equations and their applications. College de France Seminar, Vol. XIII (Paris, 1994/1996). vol. 391 of Pitman Res. Notes Math. Ser. Longman, Harlow; 1998. p. 294-302.

Downloads

Published

2023-06-14

How to Cite

Burman, E. (2023). A primal dual mixed finite element method for inverse identification of the diffusion coefficient and its relation to the Kohn-Vogelius penalty method. Selecciones Matemáticas, 10(01), 16 - 28. https://doi.org/10.17268/sel.mat.2023.01.02

Issue

Vol. 10 No. 01 (2023): Special Issue

Section

Articles

License

Copyright (c) 2023 Selecciones Matemáticas

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.