Identifying the source term in the potential equation with weighted sparsity regularization
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- by Ole Løseth Elvetun and Bjørn Fredrik Nielsen;
- Math. Comp. 93 (2024), 2811-2836
- DOI: https://doi.org/10.1090/mcom/3941
- Published electronically: February 1, 2024
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Abstract:
We explore the possibility for using boundary measurements to recover a sparse source term $f(x)$ in the potential equation. Employing weighted sparsity regularization and standard results for subgradients, we derive simple-to-check criteria which assure that a number of sinks ($f(x)<0$) and sources ($f(x)>0$) can be identified. Furthermore, we present two cases for which these criteria always are fulfilled: (a) well-separated sources and sinks, and (b) many sources or sinks located at the boundary plus one interior source/sink. Our approach is such that the linearity of the associated forward operator is preserved in the discrete formulation. The theory is therefore conveniently developed in terms of Euclidean spaces, and it can be applied to a wide range of problems. In particular, it can be applied to both isotropic and anisotropic cases. We present a series of numerical experiments. This work is motivated by the observation that standard methods typically suggest that internal sinks and sources are located close to the boundary.References
- Abdellatif El Badia, Inverse source problem in an anisotropic medium by boundary measurements, Inverse Problems 21 (2005), no. 5, 1487–1506. MR 2173407, DOI 10.1088/0266-5611/21/5/001
- Amel Ben Abda, Fahmi Ben Hassen, Juliette Leblond, and Moncef Mahjoub, Sources recovery from boundary data: a model related to electroencephalography, Math. Comput. Modelling 49 (2009), no. 11-12, 2213–2223. MR 2527938, DOI 10.1016/j.mcm.2008.07.016
- Martin Burger, Hendrik Dirks, and Jahn Müller, Inverse problems in imaging, Large scale inverse problems, Radon Ser. Comput. Appl. Math., vol. 13, De Gruyter, Berlin, 2013, pp. 135–179. MR 3185329
- Emmanuel J. Candes and Terence Tao, Decoding by linear programming, IEEE Trans. Inform. Theory 51 (2005), no. 12, 4203–4215. MR 2243152, DOI 10.1109/TIT.2005.858979
- Eduardo Casas, Christian Clason, and Karl Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM J. Control Optim. 50 (2012), no. 4, 1735–1752. MR 2974716, DOI 10.1137/110843216
- Caroline Chaux, Patrick L. Combettes, Jean-Christophe Pesquet, and Valérie R. Wajs, A variational formulation for frame-based inverse problems, Inverse Problems 23 (2007), no. 4, 1495–1518. MR 2348078, DOI 10.1088/0266-5611/23/4/008
- Yun-Sung Chung and Soon-Yeong Chung, Identification of the combination of monopolar and dipolar sources for elliptic equations, Inverse Problems 25 (2009), no. 8, 085006, 16. MR 2529196, DOI 10.1088/0266-5611/25/8/085006
- David L. Donoho and Michael Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via $l^1$ minimization, Proc. Natl. Acad. Sci. USA 100 (2003), no. 5, 2197–2202. MR 1963681, DOI 10.1073/pnas.0437847100
- Vincent Duval and Gabriel Peyré, Exact support recovery for sparse spikes deconvolution, Found. Comput. Math. 15 (2015), no. 5, 1315–1355. MR 3394712, DOI 10.1007/s10208-014-9228-6
- Vincent Duval and Gabriel Peyré, Sparse regularization on thin grids I: the Lasso, Inverse Problems 33 (2017), no. 5, 055008, 29. MR 3628904, DOI 10.1088/1361-6420/aa5e12
- A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis, Inverse Problems 16 (2000), no. 3, 651–663. MR 1766228, DOI 10.1088/0266-5611/16/3/308
- Ole Løseth Elvetun and Bjørn Fredrik Nielsen, A regularization operator for source identification for elliptic PDEs, Inverse Probl. Imaging 15 (2021), no. 4, 599–618. MR 4259669, DOI 10.3934/ipi.2021006
- Ole Løseth Elvetun and Bjørn Fredrik Nielsen, Modified Tikhonov regularization for identifying several sources, Int. J. Numer. Anal. Model. 18 (2021), no. 6, 740–757. MR 4400821
- O. L. Elvetun and B. F. Nielsen, Box constraints and weighted sparsity regularization for identifying sources in elliptic PDEs, arXiv, 2022.
- Ole Løseth Elvetun and Bjørn Fredrik Nielsen, Weighted sparsity regularization for source identification for elliptic PDEs, J. Inverse Ill-Posed Probl. 31 (2023), no. 5, 687–709. MR 4649207, DOI 10.1515/jiip-2021-0057
- Jean-Jacques Fuchs, On sparse representations in arbitrary redundant bases, IEEE Trans. Inform. Theory 50 (2004), no. 6, 1341–1344. MR 2094894, DOI 10.1109/TIT.2004.828141
- Markus Grasmair, Markus Haltmeier, and Otmar Scherzer, Necessary and sufficient conditions for linear convergence of $\ell ^1$-regularization, Comm. Pure Appl. Math. 64 (2011), no. 2, 161–182. MR 2766525, DOI 10.1002/cpa.20350
- Martin Hanke and William Rundell, On rational approximation methods for inverse source problems, Inverse Probl. Imaging 5 (2011), no. 1, 185–202. MR 2773431, DOI 10.3934/ipi.2011.5.185
- Frank Hettlich and William Rundell, Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems 12 (1996), no. 3, 251–266. MR 1391538, DOI 10.1088/0266-5611/12/3/006
- Victor Isakov, Inverse problems for partial differential equations, 2nd ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006. MR 2193218
- Ji-Chuan Liu, An inverse source problem of the Poisson equation with Cauchy data, Electron. J. Differential Equations (2017), Paper No. 119, 19. MR 3651916
- F. Lucka, S. Pursiainen, M. Burger, and C. H. Wolters, Hierarchical Bayesian inference for the EEG inverse problem using realistic FE head models: depth localization and source separation for focal primary currents, Neuro image 61 (2012), no. 4, 1364–1382.
- R. D. Pascual-Marqui, Standardized low-resolution brain electromagnetic tomography (sLORETA): technical details, Methods Findings Exp. Clinical Pharmacology, 24 (2002), Suppl. D, 5–12.
- Joel A. Tropp, Greed is good: algorithmic results for sparse approximation, IEEE Trans. Inform. Theory 50 (2004), no. 10, 2231–2242. MR 2097044, DOI 10.1109/TIT.2004.834793
- Hui Zhang, Wotao Yin, and Lizhi Cheng, Necessary and sufficient conditions of solution uniqueness in 1-norm minimization, J. Optim. Theory Appl. 164 (2015), no. 1, 109–122. MR 3296287, DOI 10.1007/s10957-014-0581-z
Bibliographic Information
- Ole Løseth Elvetun
- Affiliation: Faculty of Science and Technology, Norwegian University of Life Sciences, P.O. Box 5003, NO-1432 Ås, Norway
- MR Author ID: 1003304
- Email: ole.elvetun@nmbu.no
- Bjørn Fredrik Nielsen
- Affiliation: Faculty of Science and Technology, Norwegian University of Life Sciences, P.O. Box 5003, NO-1432 Ås, Norway
- Email: bjorn.f.nielsen@nmbu.no
- Received by editor(s): January 13, 2023
- Received by editor(s) in revised form: November 3, 2023
- Published electronically: February 1, 2024
- © Copyright 2024 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 2811-2836
- MSC (2020): Primary 35R30, 47A52, 65F22
- DOI: https://doi.org/10.1090/mcom/3941
- MathSciNet review: 4780346