# American Institute of Mathematical Sciences

May  2016, 10(2): 379-407. doi: 10.3934/ipi.2016005

## Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems

 1 Institut de Mathématiques, Université de Toulouse, 118, Route de Narbonne, F-31062 Toulouse Cedex 9

Received  April 2015 Published  May 2016

We study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method proves to be efficient even with highly corrupted data.
Citation: Jérémi Dardé. Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems. Inverse Problems & Imaging, 2016, 10 (2) : 379-407. doi: 10.3934/ipi.2016005
##### References:
 [1] G. Alessandrini, L. Del 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 [2] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations,, Inverse problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123004. Google Scholar [3] K. A. Ames and L. E. Payne, Continuous dependence on modeling for some well-posed perturbations of the backward heat equation,, Journal of Inequalities and Applications, 3 (1999), 51. doi: 10.1155/S1025583499000041. Google Scholar [4] S. Andrieux, T. N. Baranger and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional,, Inverse problems, 22 (2006), 115. doi: 10.1088/0266-5611/22/1/007. Google Scholar [5] M. Azaïez, F. Ben Belgacem and H. El 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] A. Ben Abda, J. Blum, C. Boulbe and B. Faugeras, Minimization of an energy error functional to solve a Cauchy problem arising in plasma physics: The reconstruction of the magnetic flux in the vacuum surrounding the plasma in a Tokamak,, ARIMA 15 (2012), 15 (2012), 37. Google Scholar [7] L. Baratchart, L. Bourgeois and J. Leblond, Uniqueness results for inverse Robin problems with bounded coefficient,, J. Funct. Anal., 270 (2016), 2508. doi: 10.1016/j.jfa.2016.01.011. Google Scholar [8] F. Ben Belgacem, Why is the Cauchy problem severely ill-posed?,, Inverse problems, 23 (2007), 823. doi: 10.1088/0266-5611/23/2/020. Google Scholar [9] F. Ben Belgacem, D. T. Du and F. Jelassi, Extended-domain-Lavrentiev's regularization for the Cauchy problem,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/4/045005. Google Scholar [10] J. Blum, C. Boulbe and B. Faugeras, Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time,, Journal of Computational Physics, 231 (2012), 960. doi: 10.1016/j.jcp.2011.04.005. Google Scholar [11] Y. Boukari and H. Haddar, A convergent data completion algorithm using surface integral equations,, Inverse Problems, 31 (2015). doi: 10.1088/0266-5611/31/3/035011. Google Scholar [12] L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation,, Inverse Problems, 21 (2005), 1087. doi: 10.1088/0266-5611/21/3/018. Google Scholar [13] L. Bourgeois, Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation,, Inverse Problems, 22 (2006), 413. doi: 10.1088/0266-5611/22/2/002. Google Scholar [14] 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 [15] L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem,, Inverse Problems and Imaging, 4 (2010), 351. doi: 10.3934/ipi.2010.4.351. Google Scholar [16] L. Bourgeois and J. Dardé, The "exterior approach" to solve the inverse obstacle problem for the Stokes system,, Inverse Problems and Imaging, 8 (2014), 23. doi: 10.3934/ipi.2014.8.23. Google Scholar [17] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). doi: 10.1007/978-0-387-70914-7. Google Scholar [18] K. Bryan and L. F. Caudill, Jr., An inverse problem in thermal imaging,, SIAM J. Appl. Math., 56 (1996), 715. doi: 10.1137/S0036139994277828. Google Scholar [19] E. Burman, A stabilized nonconforming finite element method for the elliptic Cauchy problem,, Mathematics of Computation, (2016). doi: 10.1090/mcom/3092. Google Scholar [20] H. Cao, M. V. Klibanov and S. V. Pereverzev, A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/3/035005. Google Scholar [21] S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements,, Inverse problems, 15 (1999), 1425. doi: 10.1088/0266-5611/15/6/303. Google Scholar [22] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, Classics in Applied Mathematics, (2002). doi: 10.1137/1.9780898719208. Google Scholar [23] 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 [24] G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems,, Electronic Journal of Differential Equations, 8 (1994), 1. Google Scholar [25] C. Clason and M. V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium,, SIAM Journal on Scientific Computing, 30 (2007), 1. doi: 10.1137/06066970X. Google Scholar [26] J. Dardé, The 'exterior approach': A new framework to solve inverse obstacle problems,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/1/015008. Google Scholar [27] J. Dardé, A. Hannukainen and N. Hyvönen, An $H_text{div}$-Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems,, SIAM J. Numer. Anal., 51 (2013), 2123. doi: 10.1137/120895123. Google Scholar [28] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Mathematics and its Applications, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar [29] D. Fasino and G. Inglese, An inverse Robin problem for Laplace's equation: Theoretical results and numerical methods,, Inverse problems, 15 (1999), 41. doi: 10.1088/0266-5611/15/1/008. Google Scholar [30] P. Fernandes and G. Gilardi, Magnetostatic and Electrostatic Problems in Inhomogeneous Anisotropic Media with Irregular Boundary and Mixed Boundary Conditions,, Math. Models Methods Appl. Sci., 7 (1997), 957. doi: 10.1142/S0218202597000487. Google Scholar [31] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Classic in Applied Mathematics, (2011). doi: 10.1137/1.9781611972030. Google Scholar [32] H. Harbrecht and J. Tausch, On the numerical solution of a shape optimization problem for the heat equation,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/110855703. Google Scholar [33] M. Ikehata and M. Kawashita, The enclosure method for the heat equation,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/7/075005. Google Scholar [34] M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for cauchy problems for Laplace's equation,, SIAM Journal on Applied Mathematics, 51 (1991), 1653. doi: 10.1137/0151085. Google Scholar [35] R. Lattès and J. L. Lions, The Method of Quasi-reversibility: Applications to Partial Differential Equations,, American Elsevier Publishing Company, (1969). Google Scholar [36] J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations,, ESAIM Control Optim. Calc. Var., 18 (2012), 712. doi: 10.1051/cocv/2011168. Google Scholar [37] R. E. Puzyrev and A. A. Shlapunov, On an Ill-posed problem for the heat equation,, Journal of Siberian Federal University, 5 (2012), 337. Google Scholar [38] E. Sincich, Lipschitz stability for the inverse Robin problem,, Inverse problems, 23 (2007), 1311. doi: 10.1088/0266-5611/23/3/027. Google Scholar [39] A. N. Tykhonov, Solution of incorrectly formulated problems and the regularization method,, Soviet Math. Dokl., 4 (1063). Google Scholar

show all references

##### References:
 [1] G. Alessandrini, L. Del 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 [2] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations,, Inverse problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123004. Google Scholar [3] K. A. Ames and L. E. Payne, Continuous dependence on modeling for some well-posed perturbations of the backward heat equation,, Journal of Inequalities and Applications, 3 (1999), 51. doi: 10.1155/S1025583499000041. Google Scholar [4] S. Andrieux, T. N. Baranger and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional,, Inverse problems, 22 (2006), 115. doi: 10.1088/0266-5611/22/1/007. Google Scholar [5] M. Azaïez, F. Ben Belgacem and H. El 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] A. Ben Abda, J. Blum, C. Boulbe and B. Faugeras, Minimization of an energy error functional to solve a Cauchy problem arising in plasma physics: The reconstruction of the magnetic flux in the vacuum surrounding the plasma in a Tokamak,, ARIMA 15 (2012), 15 (2012), 37. Google Scholar [7] L. Baratchart, L. Bourgeois and J. Leblond, Uniqueness results for inverse Robin problems with bounded coefficient,, J. Funct. Anal., 270 (2016), 2508. doi: 10.1016/j.jfa.2016.01.011. Google Scholar [8] F. Ben Belgacem, Why is the Cauchy problem severely ill-posed?,, Inverse problems, 23 (2007), 823. doi: 10.1088/0266-5611/23/2/020. Google Scholar [9] F. Ben Belgacem, D. T. Du and F. Jelassi, Extended-domain-Lavrentiev's regularization for the Cauchy problem,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/4/045005. Google Scholar [10] J. Blum, C. Boulbe and B. Faugeras, Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time,, Journal of Computational Physics, 231 (2012), 960. doi: 10.1016/j.jcp.2011.04.005. Google Scholar [11] Y. Boukari and H. Haddar, A convergent data completion algorithm using surface integral equations,, Inverse Problems, 31 (2015). doi: 10.1088/0266-5611/31/3/035011. Google Scholar [12] L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation,, Inverse Problems, 21 (2005), 1087. doi: 10.1088/0266-5611/21/3/018. Google Scholar [13] L. Bourgeois, Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation,, Inverse Problems, 22 (2006), 413. doi: 10.1088/0266-5611/22/2/002. Google Scholar [14] 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 [15] L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem,, Inverse Problems and Imaging, 4 (2010), 351. doi: 10.3934/ipi.2010.4.351. Google Scholar [16] L. Bourgeois and J. Dardé, The "exterior approach" to solve the inverse obstacle problem for the Stokes system,, Inverse Problems and Imaging, 8 (2014), 23. doi: 10.3934/ipi.2014.8.23. Google Scholar [17] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). doi: 10.1007/978-0-387-70914-7. Google Scholar [18] K. Bryan and L. F. Caudill, Jr., An inverse problem in thermal imaging,, SIAM J. Appl. Math., 56 (1996), 715. doi: 10.1137/S0036139994277828. Google Scholar [19] E. Burman, A stabilized nonconforming finite element method for the elliptic Cauchy problem,, Mathematics of Computation, (2016). doi: 10.1090/mcom/3092. Google Scholar [20] H. Cao, M. V. Klibanov and S. V. Pereverzev, A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/3/035005. Google Scholar [21] S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements,, Inverse problems, 15 (1999), 1425. doi: 10.1088/0266-5611/15/6/303. Google Scholar [22] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, Classics in Applied Mathematics, (2002). doi: 10.1137/1.9780898719208. Google Scholar [23] 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 [24] G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems,, Electronic Journal of Differential Equations, 8 (1994), 1. Google Scholar [25] C. Clason and M. V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium,, SIAM Journal on Scientific Computing, 30 (2007), 1. doi: 10.1137/06066970X. Google Scholar [26] J. Dardé, The 'exterior approach': A new framework to solve inverse obstacle problems,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/1/015008. Google Scholar [27] J. Dardé, A. Hannukainen and N. Hyvönen, An $H_text{div}$-Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems,, SIAM J. Numer. Anal., 51 (2013), 2123. doi: 10.1137/120895123. Google Scholar [28] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Mathematics and its Applications, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar [29] D. Fasino and G. Inglese, An inverse Robin problem for Laplace's equation: Theoretical results and numerical methods,, Inverse problems, 15 (1999), 41. doi: 10.1088/0266-5611/15/1/008. Google Scholar [30] P. Fernandes and G. Gilardi, Magnetostatic and Electrostatic Problems in Inhomogeneous Anisotropic Media with Irregular Boundary and Mixed Boundary Conditions,, Math. Models Methods Appl. Sci., 7 (1997), 957. doi: 10.1142/S0218202597000487. Google Scholar [31] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Classic in Applied Mathematics, (2011). doi: 10.1137/1.9781611972030. Google Scholar [32] H. Harbrecht and J. Tausch, On the numerical solution of a shape optimization problem for the heat equation,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/110855703. Google Scholar [33] M. Ikehata and M. Kawashita, The enclosure method for the heat equation,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/7/075005. Google Scholar [34] M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for cauchy problems for Laplace's equation,, SIAM Journal on Applied Mathematics, 51 (1991), 1653. doi: 10.1137/0151085. Google Scholar [35] R. Lattès and J. L. Lions, The Method of Quasi-reversibility: Applications to Partial Differential Equations,, American Elsevier Publishing Company, (1969). Google Scholar [36] J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations,, ESAIM Control Optim. Calc. Var., 18 (2012), 712. doi: 10.1051/cocv/2011168. Google Scholar [37] R. E. Puzyrev and A. A. Shlapunov, On an Ill-posed problem for the heat equation,, Journal of Siberian Federal University, 5 (2012), 337. Google Scholar [38] E. Sincich, Lipschitz stability for the inverse Robin problem,, Inverse problems, 23 (2007), 1311. doi: 10.1088/0266-5611/23/3/027. Google Scholar [39] A. N. Tykhonov, Solution of incorrectly formulated problems and the regularization method,, Soviet Math. Dokl., 4 (1063). Google Scholar
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