Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming

Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse prob...

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Published in:	Optimization letters Vol. 16; no. 5; pp. 1599 - 1609
Main Author:	Harrach, Bastian
Format:	Journal Article
Language:	English
Published:	Berlin, Heidelberg Springer 01.06.2022 Springer Berlin Heidelberg
Subjects:	Computational Intelligence Convexity Finitely many measurements Inverse Problem Loewner order Mathematics Mathematics and Statistics Monotonicity Numerical and Computational Physics Operations Research/Decision Theory Optimization Original Paper Simulation 35R30 Inverse Problem 90C22 Loewner order Convexity Finitely many measurements Monotonicity
ISSN:	1862-4480, 1862-4472, 1862-4480
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Abstract	Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.
AbstractList	Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.
Author	Harrach, Bastian
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References	Alessandrini, G., Vessella, S.: Lipschitz stability for the inverse conductivity problem. Adv. Appl. Math. 35(2), 207–241 (2005) Harrach, B.: On uniqueness in diffuse optical tomography. Inverse Problems 25(5), 055010 (2009) Uhlmann, G.: Electrical impedance tomography and Calderón’s problem. Inverse Problems 25(12), 123011 (2009) Calderón, A.P.: On an inverse boundary value problem. In: W.H. Meyer, M.A. Raupp (eds.) Seminar on Numerical Analysis and its Application to Continuum Physics, pp. 65–73. Brasil. Math. Soc., Rio de Janeiro (1980) Kenig, C., Salo, M.: Recent progress in the Calderón problem with partial data. Contemp. Math 615, 193–222 (2014) Alberti, G.S., Santacesaria, M.: Calderón’s inverse problem with a finite number of measurements. Forum Math. Sigma 7, e35 (2019) Harrach, B., Meftahi, H.: Global uniqueness and Lipschitz-stability for the inverse Robin transmission problem. SIAM J. Appl. Math. 79(2), 525–550 (2019) Rüland, A., Sincich, E.: Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Probl. Imaging 13(5), 1023–1044 (2019) Beretta, E., de Hoop, M.V., Francini, E., Vessella, S.: Stable determination of polyhedral interfaces from boundary data for the Helmholtz equation. Comm. Partial Differential Equations 40(7), 1365–1392 (2015) Harrach, B.: Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes. Inverse Problems 35(2), 024005 (2019) Harrach, B.: Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem. Numer. Math. 147, 29–70 (2021) Calderón, A.P.: On an inverse boundary value problem. Comput. Appl. Math. 25(2–3), 133–138 (2006) Klibanov, M.V., Li, J., Zhang, W.: Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data. Inverse Problems 35(3), 035005 (2019) Adler, A., Gaburro, R., Lionheart, W.: Electrical impedance tomography. In: O. Scherzer (ed.) Handbook of Mathematical Methods in Imaging, pp. 701–762. Springer (2015) Harrach, B.: An introduction to finite element methods for inverse coefficient problems in elliptic PDEs. Jahresber. Dtsch. Math. Ver. 123(3), 183–210 (2021) 1802_CR4 1802_CR3 1802_CR2 1802_CR1 1802_CR12 1802_CR13 1802_CR10 1802_CR11 1802_CR8 1802_CR7 1802_CR6 1802_CR14 1802_CR5 1802_CR15 1802_CR9
References_xml	– reference: Rüland, A., Sincich, E.: Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Probl. Imaging 13(5), 1023–1044 (2019) – reference: Alessandrini, G., Vessella, S.: Lipschitz stability for the inverse conductivity problem. Adv. Appl. Math. 35(2), 207–241 (2005) – reference: Beretta, E., de Hoop, M.V., Francini, E., Vessella, S.: Stable determination of polyhedral interfaces from boundary data for the Helmholtz equation. Comm. Partial Differential Equations 40(7), 1365–1392 (2015) – reference: Harrach, B.: Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes. Inverse Problems 35(2), 024005 (2019) – reference: Calderón, A.P.: On an inverse boundary value problem. Comput. Appl. Math. 25(2–3), 133–138 (2006) – reference: Kenig, C., Salo, M.: Recent progress in the Calderón problem with partial data. Contemp. Math 615, 193–222 (2014) – reference: Uhlmann, G.: Electrical impedance tomography and Calderón’s problem. Inverse Problems 25(12), 123011 (2009) – reference: Klibanov, M.V., Li, J., Zhang, W.: Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data. Inverse Problems 35(3), 035005 (2019) – reference: Calderón, A.P.: On an inverse boundary value problem. In: W.H. Meyer, M.A. Raupp (eds.) Seminar on Numerical Analysis and its Application to Continuum Physics, pp. 65–73. Brasil. Math. Soc., Rio de Janeiro (1980) – reference: Harrach, B., Meftahi, H.: Global uniqueness and Lipschitz-stability for the inverse Robin transmission problem. SIAM J. Appl. Math. 79(2), 525–550 (2019) – reference: Alberti, G.S., Santacesaria, M.: Calderón’s inverse problem with a finite number of measurements. Forum Math. Sigma 7, e35 (2019) – reference: Adler, A., Gaburro, R., Lionheart, W.: Electrical impedance tomography. In: O. Scherzer (ed.) Handbook of Mathematical Methods in Imaging, pp. 701–762. Springer (2015) – reference: Harrach, B.: On uniqueness in diffuse optical tomography. Inverse Problems 25(5), 055010 (2009) – reference: Harrach, B.: An introduction to finite element methods for inverse coefficient problems in elliptic PDEs. Jahresber. Dtsch. Math. Ver. 123(3), 183–210 (2021) – reference: Harrach, B.: Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem. Numer. Math. 147, 29–70 (2021) – ident: 1802_CR1 doi: 10.1007/978-1-4939-0790-8_14 – ident: 1802_CR14 doi: 10.3934/ipi.2019046 – ident: 1802_CR15 doi: 10.1088/0266-5611/25/12/123011 – ident: 1802_CR5 – ident: 1802_CR11 doi: 10.1137/18M1205388 – ident: 1802_CR7 doi: 10.1088/0266-5611/25/5/055010 – ident: 1802_CR4 doi: 10.1080/03605302.2015.1007379 – ident: 1802_CR13 doi: 10.1088/1361-6420/aafecd – ident: 1802_CR2 doi: 10.1017/fms.2019.31 – ident: 1802_CR12 doi: 10.1090/conm/615/12245 – ident: 1802_CR9 doi: 10.1365/s13291-021-00236-2 – ident: 1802_CR3 doi: 10.1016/j.aam.2004.12.002 – ident: 1802_CR6 doi: 10.1590/S0101-82052006000200002 – ident: 1802_CR8 doi: 10.1088/1361-6420/aaf6fc – ident: 1802_CR10 doi: 10.1007/s00211-020-01162-8
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SubjectTerms	Computational Intelligence Convexity Finitely many measurements Inverse Problem Loewner order Mathematics Mathematics and Statistics Monotonicity Numerical and Computational Physics Operations Research/Decision Theory Optimization Original Paper Simulation
Title	Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming
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