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doi: 10.3934/ipi.2021006

A regularization operator for source identification for elliptic PDEs

Ole Løseth Elvetun and  Bjørn Fredrik Nielsen ,

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  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.

Keywords: Inverse source problems, PDE-constrained optimization, Tikhonov regularization, nullspace, Helmholtz equation.
Mathematics Subject Classification: Primary: 35R30, 47A52; Secondary: 65F22.
Citation: Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. Elul, The genesis of the EEG, International Review of Neurobiology, 15 (1972), 227-272.  doi: 10.1016/S0074-7742(08)60333-5.  Google Scholar

[9]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

V. Isakov, Inverse Problems for Partial Differential Equations, Second edition. Applied Mathematical Sciences, 127. Springer, New York, 2006.  Google Scholar

[14]

K. Kunisch and X. Pan, Estimation of interfaces from boundary measurements, SIAM J. Control Optim., 32 (1994), 1643-1674.  doi: 10.1137/S0363012992226338.  Google Scholar

[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.  Google Scholar

[16]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, 1993.  Google Scholar

[17]

W. Ring, Identification of a core from boundary data, SIAM Journal on Applied Mathematics, 55 (1995), 677-706.  doi: 10.1137/S0036139993256308.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. Elul, The genesis of the EEG, International Review of Neurobiology, 15 (1972), 227-272.  doi: 10.1016/S0074-7742(08)60333-5.  Google Scholar

[9]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

V. Isakov, Inverse Problems for Partial Differential Equations, Second edition. Applied Mathematical Sciences, 127. Springer, New York, 2006.  Google Scholar

[14]

K. Kunisch and X. Pan, Estimation of interfaces from boundary measurements, SIAM J. Control Optim., 32 (1994), 1643-1674.  doi: 10.1137/S0363012992226338.  Google Scholar

[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.  Google Scholar

[16]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, 1993.  Google Scholar

[17]

W. Ring, Identification of a core from boundary data, SIAM Journal on Applied Mathematics, 55 (1995), 677-706.  doi: 10.1137/S0036139993256308.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Comparison of the true source and the inverse solution using standard Tikhonov regularization with $ \alpha = 10^{-3} $
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Figure 2.  Recovered source, Example 1, with the regularization parameter $ \alpha = 10^{-3} $. The true source is depicted in panel (a) in Figure 1
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Figure 3.  L-shaped domain, Example 2. Comparison of the true source and the inverse solutions, using the regularization parameter $ \alpha = 10^{-3} $
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Figure 4.  Source at the boundary, Example 3. Comparison of the true source and the inverse solutions, using the regularization parameter $ \alpha = 10^{-4} $
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Figure 5.  Source at the boundary, Example 3. Inverse solution computed with standard Tikhonov regularization, $ \alpha = 10^{-4} $
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Figure 6.  Vector field of $ \sigma $
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Figure 7.  State equation with a tensor, Example 4. Comparison of the true source and the inverse solutions, using the regularization parameter $ \alpha = 10^{-4} $
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Figure 8.  Two disjoint sources, Example 5. Comparison of the true sources and the inverse solutions, using the regularization parameter $ \alpha = 10^{-3} $
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Figure 9.  Three disjoint sources, Example 5. Comparison of the true sources and the inverse solutions, using the regularization parameter $ \alpha = 10^{-3} $
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Figure 10.  Example 6, $ 5 \% $ and $ 20\% $ noise. The true source is shown in panel (a) in Figure 8
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Figure 11.  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
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Figure 12.  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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