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An Iterative Method for the Inverse Eddy Current Problem with Total Variation Regularization

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Abstract

Conductivity reconstruction in an inverse eddy current problem is considered in the present paper. With the electric field measurement on part of domain boundary, we formulate the reconstruction problem as a constrained optimization problem with total variation regularization. The existence and stability are proved for the solution to the optimization problem. The finite element method is employed to discretize the optimization problem. The gradient Lipschitz property of the objective functional is established for the discrete optimization problem. We propose a novel modification to the traditional alternating direction method of multipliers, and prove the convergence of the modified algorithm. Finally, we show some numerical experiments to illustrate the efficiency of the proposed method.

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References

  1. Ammari, H., Bao, G., Fleming, J.L.: An inverse source problem for Maxwell’s equations in magnetoencephalography. SIAM J. Appl. Math. 62, 1369–1382 (2002)

    Article  MathSciNet  Google Scholar 

  2. Ammari, H., Chen, J., Chen, Z., Garnier, J., Volkov, D.: Target detection and characterization from electromagnetic induction data. J. Math. Pures Appl. 101, 54–75 (2014)

    Article  MathSciNet  Google Scholar 

  3. Ammari, H., Chen, J., Chen, Z., Volkov, D., Wang, H.: Detection and classification from electromagnetic induction data. J. Comput. Phys. 301, 201–217 (2015)

    Article  MathSciNet  Google Scholar 

  4. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learni 3(1), 1–22 (2011)

    Google Scholar 

  5. Bringmans, L.: Discrete inverse Sobolev inequalities with applications to the edge and face lemma for the finite element method, Master Thesis, ETH Zurich, Department of Mathematics, February 2020, https://doi.org/10.3929/ethz-b-000401981

  6. Buffa, A., Ammari, H., Nédélec, J.C.: A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60, 1805–1823 (2000)

    Article  MathSciNet  Google Scholar 

  7. Cai, J.F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8, 337–369 (2009)

    Article  MathSciNet  Google Scholar 

  8. Chan, T., Tai, X.: Identification of discontinuous coefficients in elliptic problems using total variation regularization. SIAM J. Sci. Comput. 25, 881–904 (2003)

    Article  MathSciNet  Google Scholar 

  9. Chen, J., Chen, Z., Cui, T., Zhang, L.-B.: An adaptive finite method for the eddy current model with circuit/field couplings. SIAM J. Sci. Comput. 32, 1020–1042 (2010)

    Article  MathSciNet  Google Scholar 

  10. Chen, J., Liang, Y., Zou, J.: Mathematical and numerical study of a three-dimensional inverse eddy current problem. SIAM J. Appl. Math. 80, 1467–1492 (2020)

    Article  MathSciNet  Google Scholar 

  11. Chen, Z., Zou, J.: An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems. SIAM J. Control Optim. 37, 892–910 (1999)

    Article  MathSciNet  Google Scholar 

  12. Ciarlet, P.: The finite element method for elliptic problems. North-Holland Publishing Company, Amsterdam (1978)

    Google Scholar 

  13. Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems, ESAIM. Math Model. Numer. Anal. 3, 627–649 (1999)

    Article  Google Scholar 

  14. Gehre, M., Jin, B., Lu, X.: An analysis of finite element approximation in electrical impedance tomography. Inverse Problems 30, 958–964 (2013)

    MathSciNet  Google Scholar 

  15. Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009)

    Article  MathSciNet  Google Scholar 

  16. Haber, E., Horesh, L., Tenorio, L.: Numerical methods for experimental design of large-scale linear ill-posed inverse problems. Inverse Probl. 24, 055012 (2008)

    Article  MathSciNet  Google Scholar 

  17. Haber, E.: Computational Methods in Geophysical Electromagnetics. Society for Industrial and Applied Mathematics, (2014)

  18. He, B., Yuan, X.: On the \(O(1/n)\) convergence rate of the douglas-rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)

    Article  MathSciNet  Google Scholar 

  19. Hong, M., Luo, Z., Razaviyayn, M.: Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. SIAM J. Optim. 26, 337–364 (2016)

    Article  MathSciNet  Google Scholar 

  20. Parallel Hierarchical Grid : http://lsec.cc.ac.cn/phg/

  21. Hiptmair, R., Xu, J.: Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45, 2483–2509 (2007)

    Article  MathSciNet  Google Scholar 

  22. Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4, 460–489 (2005)

    Article  MathSciNet  Google Scholar 

  23. Rodríguez, A., Camano, J., Valli, A.: Inverse source problems for eddy current equations. Inverse Problems 28, 015006 (2012)

    Article  MathSciNet  Google Scholar 

  24. Rudinleonid, I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MathSciNet  Google Scholar 

  25. Tamburrino, A., Rubinacci, G.: A new non-iterative inversion method for electrical resistance tomography. Inverse Problems 18, 1809–1829 (2002)

    Article  MathSciNet  Google Scholar 

  26. Zhdanov, M.S.: Foundations of Geophysical Electromagnetic Theory and Methods: Elsevier, (2018)

  27. Yang, J., Zhang, Y., Yin, W.: An efficient TV-L1 algorithm for debluring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput. 31, 2842–2865 (2009)

    Article  MathSciNet  Google Scholar 

  28. Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for \({\ell }_1\)-minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1, 143–168 (2008)

    Article  MathSciNet  Google Scholar 

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Funding

The work was supported by National Key R &D Program of China 2019YFA0709600, 2019YFA0709602 and by NSFC under the grant 11871300.

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  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Junqing Chen & Zehao Long

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  1. Junqing Chen
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  2. Zehao Long
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Correspondence to Junqing Chen.

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Chen, J., Long, Z. An Iterative Method for the Inverse Eddy Current Problem with Total Variation Regularization. J Sci Comput 99, 38 (2024). https://doi.org/10.1007/s10915-024-02501-9

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