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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

 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

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.
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:
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A variational Steklov-Poincaré theory,, Inverse Problems, 21 (2005), 1915.  doi: 10.1088/0266-5611/21/6/008.  Google Scholar [7] M. Berggren, Approximation of very weak solutions to boundary value problems,, SIAM J. Numer. Anal., 42 (2004), 860.  doi: 10.1137/S0036142903382048.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] P. G. Ciarlet, The Finite Element Methods for Elliptic Problems,, North-Holland, (1978).  doi: 10.1137/1.9780898719208.  Google Scholar [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.  Google Scholar [16] F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology,, Calcolo, 16 (1979), 459.  doi: 10.1007/BF02576643.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer, (2009).   Google Scholar [27] G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977.  doi: 10.1088/0266-5611/13/4/006.  Google Scholar [28] V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (2006).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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).   Google Scholar [35] R. Li and W. Liu, The C++ software library AFEPack., Available from: , ().   Google Scholar [36] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971).   Google Scholar [37] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Vol. II, (1972).   Google Scholar [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).   Google Scholar [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.  Google Scholar [40] N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Akademie Verlag, (1995).   Google Scholar [41] A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems,, Winston and Sons, (1977).   Google Scholar [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.  Google Scholar [43] K. Yosida, Functional Analysis,, $5^{th}$ edition, (1978).   Google Scholar

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] M. Berggren, Approximation of very weak solutions to boundary value problems,, SIAM J. Numer. Anal., 42 (2004), 860.  doi: 10.1137/S0036142903382048.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] P. G. Ciarlet, The Finite Element Methods for Elliptic Problems,, North-Holland, (1978).  doi: 10.1137/1.9780898719208.  Google Scholar [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.  Google Scholar [16] F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology,, Calcolo, 16 (1979), 459.  doi: 10.1007/BF02576643.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer, (2009).   Google Scholar [27] G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977.  doi: 10.1088/0266-5611/13/4/006.  Google Scholar [28] V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (2006).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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).   Google Scholar [35] R. Li and W. Liu, The C++ software library AFEPack., Available from: , ().   Google Scholar [36] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971).   Google Scholar [37] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Vol. II, (1972).   Google Scholar [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).   Google Scholar [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.  Google Scholar [40] N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Akademie Verlag, (1995).   Google Scholar [41] A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems,, Winston and Sons, (1977).   Google Scholar [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.  Google Scholar [43] K. Yosida, Functional Analysis,, $5^{th}$ edition, (1978).   Google Scholar
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