# American Institute of Mathematical Sciences

August  2021, 15(4): 599-618. doi: 10.3934/ipi.2021006

## A regularization operator for source identification for elliptic PDEs

 Faculty of Science and Technology, Norwegian University of Life Sciences, P.O. Box 5003, NO-1432 Ås, Norway

* Corresponding author: B. F. Nielsen

Received  August 2020 Revised  October 2020 Published  August 2021 Early access  December 2020

Fund Project: This work was supported by The Research Council of Norway, project number 239070

We study a source identification problem for a prototypical elliptic PDE from Dirichlet boundary data. This problem is ill-posed, and the involved forward operator has a significant nullspace. Standard Tikhonov regularization yields solutions which approach the minimum $L^2$-norm least-squares solution as the regularization parameter tends to zero. We show that this approach 'always' suggests that the unknown local source is very close to the boundary of the domain of the PDE, regardless of the position of the true local source.

We propose an alternative regularization procedure, realized in terms of a novel regularization operator, which is better suited for identifying local sources positioned anywhere in the domain of the PDE. Our approach is motivated by the classical theory for Tikhonov regularization and yields a standard quadratic optimization problem. Since the new methodology is derived for an abstract operator equation, it can be applied to many other source identification problems. This paper contains several numerical experiments and an analysis of the new methodology.

Citation: Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems and Imaging, 2021, 15 (4) : 599-618. doi: 10.3934/ipi.2021006
##### References:
 [1] B. Abdelaziz, A. El Badia and A. El Hajj, Direct algorithms for solving some inverse source problems in 2D elliptic equations, Inverse Problems, 31 (2015), 105002, 26pp. doi: 10.1088/0266-5611/31/10/105002. [2] C. J. S. Alves, J. B. Abdallah and M. Jaoua, Recovery of cracks using a point-source reciprocity gap function, Inverse Problems in Science and Engineering, 12 (2004), 519-534.  doi: 10.1080/1068276042000219912. [3] S. Baillet, J. C. Mosher and R. M. Leahy, Electromagnetic brain mapping, IEEE Signal Processing Magazine, 18 (2001), 14-30.  doi: 10.1109/79.962275. [4] A. Ben Abda, F. Ben Hassen, J. Leblond and M. Mahjoub, Sources recovery from boundary data: A model related to electroencephalography, Mathematical and Computer Modelling, 49 (2009), 2213-2223.  doi: 10.1016/j.mcm.2008.07.016. [5] X. Cheng, R. Gong and W. Han, A new Kohn-Vogelius type formulation for inverse source problems, Inverse Problems and Imaging, 9 (2015), 1051-1067.  doi: 10.3934/ipi.2015.9.1051. [6] A. El Badia and T. Ha-Duong, Some remarks on the problem of source identification from boundary measurements, Inverse Problems, 14 (1998), 883-891.  doi: 10.1088/0266-5611/14/4/008. [7] A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663.  doi: 10.1088/0266-5611/16/3/308. [8] R. Elul, The genesis of the EEG, International Review of Neurobiology, 15 (1972), 227-272.  doi: 10.1016/S0074-7742(08)60333-5. [9] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, 1996. [10] M. Hanke and W. Rundell, On rational approximation methods for inverse source problems, Inverse Problems and Imaging, 5 (2011), 185-202.  doi: 10.3934/ipi.2011.5.185. [11] F. Hettlich and W. Rundell, Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems, 12 (1996), 251-266.  doi: 10.1088/0266-5611/12/3/006. [12] M. Hinze, B. Hofmann and T. N. T. Quyen, A regularization approach for an inverse source problem in elliptic systems from single Cauchy data, Numerical Functional Analysis and Optimization, 40 (2019), 1080-1112.  doi: 10.1080/01630563.2019.1596953. [13] V. Isakov, Inverse Problems for Partial Differential Equations, Second edition. Applied Mathematical Sciences, 127. Springer, New York, 2006. [14] K. Kunisch and X. Pan, Estimation of interfaces from boundary measurements, SIAM J. Control Optim., 32 (1994), 1643-1674.  doi: 10.1137/S0363012992226338. [15] B. F. Nielsen, M. Lysaker and P. Grøttum, Computing ischemic regions in the heart with the bidomain model; first steps towards validation, IEEE Transactions on Medical Imaging, 32 (2013), 1085-1096.  doi: 10.1109/TMI.2013.2254123. [16] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, 1993. [17] W. Ring, Identification of a core from boundary data, SIAM Journal on Applied Mathematics, 55 (1995), 677-706.  doi: 10.1137/S0036139993256308. [18] S. J. Song and J. G. Huang, Solving an inverse problem from bioluminescence tomography by minimizing an energy-like functional, J. Comput. Anal. Appl., 14 (2012), 544-558. [19] D. Wang, R. M. Kirby, R. S. MacLeod and C. R. Johnson, Inverse electrocardiographic source localization of ischemia: An optimization framework and finite element solution, Journal of Computational Physics, 250 (2013), 403-424.  doi: 10.1016/j.jcp.2013.05.027. [20] X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems, 33 (2017), 035001, 18pp. doi: 10.1088/1361-6420/aa573c. [21] D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001, 21pp. doi: 10.1088/1361-6420/aaccda. [22] D. Zhang, Y. Guo, J. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Communications in Computational Physics, 25 (2019), 1328-1356.  doi: 10.4208/cicp.oa-2018-0020.

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##### References:
 [1] B. Abdelaziz, A. El Badia and A. El Hajj, Direct algorithms for solving some inverse source problems in 2D elliptic equations, Inverse Problems, 31 (2015), 105002, 26pp. doi: 10.1088/0266-5611/31/10/105002. [2] C. J. S. Alves, J. B. Abdallah and M. Jaoua, Recovery of cracks using a point-source reciprocity gap function, Inverse Problems in Science and Engineering, 12 (2004), 519-534.  doi: 10.1080/1068276042000219912. [3] S. Baillet, J. C. Mosher and R. M. Leahy, Electromagnetic brain mapping, IEEE Signal Processing Magazine, 18 (2001), 14-30.  doi: 10.1109/79.962275. [4] A. Ben Abda, F. Ben Hassen, J. Leblond and M. Mahjoub, Sources recovery from boundary data: A model related to electroencephalography, Mathematical and Computer Modelling, 49 (2009), 2213-2223.  doi: 10.1016/j.mcm.2008.07.016. [5] X. Cheng, R. Gong and W. Han, A new Kohn-Vogelius type formulation for inverse source problems, Inverse Problems and Imaging, 9 (2015), 1051-1067.  doi: 10.3934/ipi.2015.9.1051. [6] A. El Badia and T. Ha-Duong, Some remarks on the problem of source identification from boundary measurements, Inverse Problems, 14 (1998), 883-891.  doi: 10.1088/0266-5611/14/4/008. [7] A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663.  doi: 10.1088/0266-5611/16/3/308. [8] R. Elul, The genesis of the EEG, International Review of Neurobiology, 15 (1972), 227-272.  doi: 10.1016/S0074-7742(08)60333-5. [9] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, 1996. [10] M. Hanke and W. Rundell, On rational approximation methods for inverse source problems, Inverse Problems and Imaging, 5 (2011), 185-202.  doi: 10.3934/ipi.2011.5.185. [11] F. Hettlich and W. Rundell, Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems, 12 (1996), 251-266.  doi: 10.1088/0266-5611/12/3/006. [12] M. Hinze, B. Hofmann and T. N. T. Quyen, A regularization approach for an inverse source problem in elliptic systems from single Cauchy data, Numerical Functional Analysis and Optimization, 40 (2019), 1080-1112.  doi: 10.1080/01630563.2019.1596953. [13] V. Isakov, Inverse Problems for Partial Differential Equations, Second edition. Applied Mathematical Sciences, 127. Springer, New York, 2006. [14] K. Kunisch and X. Pan, Estimation of interfaces from boundary measurements, SIAM J. Control Optim., 32 (1994), 1643-1674.  doi: 10.1137/S0363012992226338. [15] B. F. Nielsen, M. Lysaker and P. Grøttum, Computing ischemic regions in the heart with the bidomain model; first steps towards validation, IEEE Transactions on Medical Imaging, 32 (2013), 1085-1096.  doi: 10.1109/TMI.2013.2254123. [16] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, 1993. [17] W. Ring, Identification of a core from boundary data, SIAM Journal on Applied Mathematics, 55 (1995), 677-706.  doi: 10.1137/S0036139993256308. [18] S. J. Song and J. G. Huang, Solving an inverse problem from bioluminescence tomography by minimizing an energy-like functional, J. Comput. Anal. Appl., 14 (2012), 544-558. [19] D. Wang, R. M. Kirby, R. S. MacLeod and C. R. Johnson, Inverse electrocardiographic source localization of ischemia: An optimization framework and finite element solution, Journal of Computational Physics, 250 (2013), 403-424.  doi: 10.1016/j.jcp.2013.05.027. [20] X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems, 33 (2017), 035001, 18pp. doi: 10.1088/1361-6420/aa573c. [21] D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001, 21pp. doi: 10.1088/1361-6420/aaccda. [22] D. Zhang, Y. Guo, J. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Communications in Computational Physics, 25 (2019), 1328-1356.  doi: 10.4208/cicp.oa-2018-0020.
Comparison of the true source and the inverse solution using standard Tikhonov regularization with $\alpha = 10^{-3}$
Recovered source, Example 1, with the regularization parameter $\alpha = 10^{-3}$. The true source is depicted in panel (a) in Figure 1
L-shaped domain, Example 2. Comparison of the true source and the inverse solutions, using the regularization parameter $\alpha = 10^{-3}$
Source at the boundary, Example 3. Comparison of the true source and the inverse solutions, using the regularization parameter $\alpha = 10^{-4}$
Source at the boundary, Example 3. Inverse solution computed with standard Tikhonov regularization, $\alpha = 10^{-4}$
Vector field of $\sigma$
State equation with a tensor, Example 4. Comparison of the true source and the inverse solutions, using the regularization parameter $\alpha = 10^{-4}$
Two disjoint sources, Example 5. Comparison of the true sources and the inverse solutions, using the regularization parameter $\alpha = 10^{-3}$
Three disjoint sources, Example 5. Comparison of the true sources and the inverse solutions, using the regularization parameter $\alpha = 10^{-3}$
Example 6, $5 \%$ and $20\%$ noise. The true source is shown in panel (a) in Figure 8
Inhomogeneous Helmholtz equation with $\epsilon = -1$. Comparison of the inverse solutions, using the regularization parameter $\alpha = 10^{-3}$. The true source is displayed in Figure 1a
Inhomogeneous Helmholtz equation with $\epsilon = -100$. Comparison of the inverse solutions, using the regularization parameter $\alpha = 10^{-3}$. The true source is displayed in Figure 1a
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