PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

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August 2015, 9(3): 791-814. doi: 10.3934/ipi.2015.9.791

PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

Lili Chang ^1, , Wei Gong ^2, , Guiquan Sun ^1, and Ningning Yan ^3,

1.	Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shan'xi, China, China
2.	LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China
3.	LSEC, NCMIS, Institute of Systems Science, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China

Received November 2014 Revised March 2015 Published July 2015

Abstract
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This paper concerns the approximation of a Cauchy problem for the elliptic equation. The inverse problem is transformed into a PDE-constrained optimal control problem and these two problems are equivalent under some assumptions. Different from the existing literature which is also based on the optimal control theory, we consider the state equation in the sense of very weak solution defined by the transposition technique. In this way, it does not need to impose any regularity requirement on the given data. Moreover, this method can yield theoretical analysis simply and numerical computation conveniently. To deal with the ill-posedness of the control problem, Tikhonov regularization term is introduced. The regularized problem is well-posed and its solution converges to the non-regularized counterpart as the regularization parameter approaches zero. We establish the finite element approximation to the regularized control problem and the convergence of the discrete problem is also investigated. Then we discuss the first order optimality condition of the control problem further and obtain an efficient numerical scheme for the Cauchy problem via the adjoint state equation. The paper is ended with numerical experiments.

Keywords: finite element method., Tikhonov regularization, Elliptic equation, PDE-constrained optimal control, Cauchy problem.

Mathematics Subject Classification: Primary: 49J20, 65N21; Secondary: 65N3.

Citation: Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791

References:

[1]	R. A. Adams, Sobolev Spaces,, Academic Press, (1975).
[2]	G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement,, Inverse Problems, 19 (2003), 973. doi: 10.1088/0266-5611/19/4/312.
[3]	G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion,, J. Comp. Appl. Math., 198 (2007), 307. doi: 10.1016/j.cam.2005.06.048.
[4]	S. Andrieux and A. A. Ben, Identification of planar cracks by complete overdetermined data: inversion formulae,, Inverse Problems, 12 (1996), 553. doi: 10.1088/0266-5611/12/5/002.
[5]	M. Azaïez, F. B. Belgacem and H. E. Fekih, On Cauchy's problem: II. Completion, regularization and approximation,, Inverse Problems, 22 (2006), 1307. doi: 10.1088/0266-5611/22/4/012.
[6]	F. B. Belgacem and H. E. Fekih, On Cauchy's problem: I. A variational Steklov-Poincaré theory,, Inverse Problems, 21 (2005), 1915. doi: 10.1088/0266-5611/21/6/008.
[7]	M. Berggren, Approximation of very weak solutions to boundary value problems,, SIAM J. Numer. Anal., 42 (2004), 860. doi: 10.1137/S0036142903382048.
[8]	L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/9/095016.
[9]	S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods,, $3^{th}$ edition, (2008). doi: 10.1007/978-0-387-75934-0.
[10]	M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements,, M2AN Math. Model. Numer. Anal., 35 (2001), 595. doi: 10.1051/m2an:2001128.
[11]	E. Casas and J. P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations,, SIAM J. Control Optim., 45 (2006), 1586. doi: 10.1137/050626600.
[12]	A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem,, Inverse Problems, 22 (2006), 1191. doi: 10.1088/0266-5611/22/4/005.
[13]	J. Cheng, Y. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation,, Z. Angew. Math. Mech., 81 (2001), 665. doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V.
[14]	P. G. Ciarlet, The Finite Element Methods for Elliptic Problems,, North-Holland, (1978). doi: 10.1137/1.9780898719208.
[15]	A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization,, Inverse Problems, 17 (2001), 553. doi: 10.1088/0266-5611/17/3/313.
[16]	F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology,, Calcolo, 16 (1979), 459. doi: 10.1007/BF02576643.
[17]	K. Deckelnick, A. Günther and M. Hinze, Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains,, SIAM J. Control Optim., 48 (2009), 2798. doi: 10.1137/080735369.
[18]	H. W. Engl and A. Leitão, A Mann iterative regularization method for elliptic Cauchy problems,, Numer. Funct. Anal. Optim., 22 (2001), 861. doi: 10.1081/NFA-100108313.
[19]	D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method,, Numer. Funct. Anal. Optim., 12 (1991), 299. doi: 10.1080/01630569108816430.
[20]	A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements,, J. Ind. Univ. Math., 38 (1989), 527. doi: 10.1512/iumj.1989.38.38025.
[21]	A. V. Fursikov, The Cauchy problem for a second-order elliptic equation in a conditionally well-posed formulation,, Tr. Mosk. Mat. Obs., 52 (1991), 139.
[22]	H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations,, J. Inverse Ill-Posed Problems, 5 (1997), 437. doi: 10.1515/jiip.1997.5.5.437.
[23]	W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography,, Inverse Problems, 22 (2006), 1659. doi: 10.1088/0266-5611/22/5/008.
[24]	W. Han, J. Huang, K. Kazmi and Y. Chen, A numerical method for a Cauchy problem for elliptic partial differential equations,, Inverse Problems, 23 (2007), 2401. doi: 10.1088/0266-5611/23/6/008.
[25]	M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comput. Optim. Appl., 30 (2005), 45. doi: 10.1007/s10589-005-4559-5.
[26]	M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer, (2009).
[27]	G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977. doi: 10.1088/0266-5611/13/4/006.
[28]	V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (2006).
[29]	M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation,, Appl. Anal., 81 (2002), 1065. doi: 10.1080/0003681021000029819.
[30]	S. I. Kabanikhin and A. L. Karchevsky, Optimizational method for solving the Cauchy problem for an elliptic equation,, J. Inverse Ill-Posed problems, 3 (1995), 21. doi: 10.1515/jiip.1995.3.1.21.
[31]	M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation,, SIAM J. Appl. Math., 51 (1991), 1653. doi: 10.1137/0151085.
[32]	R. V. Kohn and M. S. Vogelius, Determining conductivity by boundary measurements: II. Interior results,, Comm. Pure Appl. Math., 38 (1985), 643. doi: 10.1002/cpa.3160380513.
[33]	V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations,, U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45.
[34]	R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications],, Travaux et Recherches Mathématiques, (1967).
[35]	R. Li and W. Liu, The C++ software library AFEPack., Available from: , ().
[36]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971).
[37]	J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Vol. II, (1972).
[38]	M. Maher, K. Moez and T. Anis, A Cauchy problem for an inverse problem in image inpainting,, in Industrial Engineering and Systems Management (IESM), (2013).
[39]	S. Pereverzev and E. Schock, Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces,, Numer. Funct. Anal. Optim., 21 (2000), 901. doi: 10.1080/01630560008816993.
[40]	N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Akademie Verlag, (1995).
[41]	A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems,, Winston and Sons, (1977).
[42]	W. Weikl, H. Andra and E. Schnack, An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body,, Inverse Problems, 17 (2001), 1957. doi: 10.1088/0266-5611/17/6/325.
[43]	K. Yosida, Functional Analysis,, $5^{th}$ edition, (1978).

show all references

References:

[1]	R. A. Adams, Sobolev Spaces,, Academic Press, (1975).
[2]	G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement,, Inverse Problems, 19 (2003), 973. doi: 10.1088/0266-5611/19/4/312.
[3]	G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion,, J. Comp. Appl. Math., 198 (2007), 307. doi: 10.1016/j.cam.2005.06.048.
[4]	S. Andrieux and A. A. Ben, Identification of planar cracks by complete overdetermined data: inversion formulae,, Inverse Problems, 12 (1996), 553. doi: 10.1088/0266-5611/12/5/002.
[5]	M. Azaïez, F. B. Belgacem and H. E. Fekih, On Cauchy's problem: II. Completion, regularization and approximation,, Inverse Problems, 22 (2006), 1307. doi: 10.1088/0266-5611/22/4/012.
[6]	F. B. Belgacem and H. E. Fekih, On Cauchy's problem: I. A variational Steklov-Poincaré theory,, Inverse Problems, 21 (2005), 1915. doi: 10.1088/0266-5611/21/6/008.
[7]	M. Berggren, Approximation of very weak solutions to boundary value problems,, SIAM J. Numer. Anal., 42 (2004), 860. doi: 10.1137/S0036142903382048.
[8]	L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/9/095016.
[9]	S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods,, $3^{th}$ edition, (2008). doi: 10.1007/978-0-387-75934-0.
[10]	M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements,, M2AN Math. Model. Numer. Anal., 35 (2001), 595. doi: 10.1051/m2an:2001128.
[11]	E. Casas and J. P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations,, SIAM J. Control Optim., 45 (2006), 1586. doi: 10.1137/050626600.
[12]	A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem,, Inverse Problems, 22 (2006), 1191. doi: 10.1088/0266-5611/22/4/005.
[13]	J. Cheng, Y. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation,, Z. Angew. Math. Mech., 81 (2001), 665. doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V.
[14]	P. G. Ciarlet, The Finite Element Methods for Elliptic Problems,, North-Holland, (1978). doi: 10.1137/1.9780898719208.
[15]	A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization,, Inverse Problems, 17 (2001), 553. doi: 10.1088/0266-5611/17/3/313.
[16]	F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology,, Calcolo, 16 (1979), 459. doi: 10.1007/BF02576643.
[17]	K. Deckelnick, A. Günther and M. Hinze, Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains,, SIAM J. Control Optim., 48 (2009), 2798. doi: 10.1137/080735369.
[18]	H. W. Engl and A. Leitão, A Mann iterative regularization method for elliptic Cauchy problems,, Numer. Funct. Anal. Optim., 22 (2001), 861. doi: 10.1081/NFA-100108313.
[19]	D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method,, Numer. Funct. Anal. Optim., 12 (1991), 299. doi: 10.1080/01630569108816430.
[20]	A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements,, J. Ind. Univ. Math., 38 (1989), 527. doi: 10.1512/iumj.1989.38.38025.
[21]	A. V. Fursikov, The Cauchy problem for a second-order elliptic equation in a conditionally well-posed formulation,, Tr. Mosk. Mat. Obs., 52 (1991), 139.
[22]	H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations,, J. Inverse Ill-Posed Problems, 5 (1997), 437. doi: 10.1515/jiip.1997.5.5.437.
[23]	W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography,, Inverse Problems, 22 (2006), 1659. doi: 10.1088/0266-5611/22/5/008.
[24]	W. Han, J. Huang, K. Kazmi and Y. Chen, A numerical method for a Cauchy problem for elliptic partial differential equations,, Inverse Problems, 23 (2007), 2401. doi: 10.1088/0266-5611/23/6/008.
[25]	M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comput. Optim. Appl., 30 (2005), 45. doi: 10.1007/s10589-005-4559-5.
[26]	M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer, (2009).
[27]	G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977. doi: 10.1088/0266-5611/13/4/006.
[28]	V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (2006).
[29]	M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation,, Appl. Anal., 81 (2002), 1065. doi: 10.1080/0003681021000029819.
[30]	S. I. Kabanikhin and A. L. Karchevsky, Optimizational method for solving the Cauchy problem for an elliptic equation,, J. Inverse Ill-Posed problems, 3 (1995), 21. doi: 10.1515/jiip.1995.3.1.21.
[31]	M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation,, SIAM J. Appl. Math., 51 (1991), 1653. doi: 10.1137/0151085.
[32]	R. V. Kohn and M. S. Vogelius, Determining conductivity by boundary measurements: II. Interior results,, Comm. Pure Appl. Math., 38 (1985), 643. doi: 10.1002/cpa.3160380513.
[33]	V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations,, U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45.
[34]	R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications],, Travaux et Recherches Mathématiques, (1967).
[35]	R. Li and W. Liu, The C++ software library AFEPack., Available from: , ().
[36]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971).
[37]	J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Vol. II, (1972).
[38]	M. Maher, K. Moez and T. Anis, A Cauchy problem for an inverse problem in image inpainting,, in Industrial Engineering and Systems Management (IESM), (2013).
[39]	S. Pereverzev and E. Schock, Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces,, Numer. Funct. Anal. Optim., 21 (2000), 901. doi: 10.1080/01630560008816993.
[40]	N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Akademie Verlag, (1995).
[41]	A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems,, Winston and Sons, (1977).
[42]	W. Weikl, H. Andra and E. Schnack, An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body,, Inverse Problems, 17 (2001), 1957. doi: 10.1088/0266-5611/17/6/325.
[43]	K. Yosida, Functional Analysis,, $5^{th}$ edition, (1978).

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