# zbMATH — the first resource for mathematics

Some novel numerical techniques for an inverse Cauchy problem. (English) Zbl 1446.65151
Summary: In this paper, we are interested in solving an elliptic inverse Cauchy problem. As it is well known this problem is one of highly ill posed problem in J. Hadamard’s sense [Lectures on Cauchy’s problem in linear partial differential equations. New York: Dover Publications (1952; Zbl 0049.34805)]. We first establish formally a relationship between the Cauchy problem and an interface problem illustrated in a rectangular structure divided into two domains. This relationship allows us to use classical methods of non-overlapping domain decomposition to develop some regularizing and stable algorithms for solving elliptic inverse Cauchy problem. Taking advantage of this relationship we reformulate this inverse problem into a fixed point one, based on Steklov-Poincaré operator. Thus, using the topological degree of Leray-Schauder we show an existence result. Finally, the efficiency and the accuracy of the developed algorithms are discussed.
##### MSC:
 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs 65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 47H11 Degree theory for nonlinear operators 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text:
##### References:
 [1] Aceto, L.; Ghelardoni, P.; Marletta, M., Numerical solution of forward and inverse Sturm-Liouville problems with an angular momentum singularity, Inverse Problems, 24, 1, 015001 (2007) · Zbl 1159.65078 [2] Bai, Z.-Z.; Buccini, A.; Hayami, K.; Reichel, L.; Yin, J.-F.; Zheng, N., Modulus-based iterative methods for constrained Tikhonov regularization, J. Comput. Appl. Math., 319, 1-13 (2017) · Zbl 1360.65112 [3] Gong, R.; Cheng, X.; Han, W., A fast solver for an inverse problem arising in bioluminescence tomography, J. Comput. Appl. Math., 267, 228-243 (2014) · Zbl 1293.92011 [4] Ramm, A. G.A. G., Inverse Problems : Mathematical and Analytical Techniques with Applications to Engineering (2004), Springer: Springer New York, Includes bibliographical references and index. URL http://site.ebrary.com/id/10139637 [5] Wen, Y.-W.; Ching, W.-K.; Ng, M., A semi-smooth Newton method for inverse problem with uniform noise, J. Sci. Comput., 1-20 (2017) [6] Chang, L.; Gong, W.; Sun, G.; Yan, N., PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem, Inverse Probl. Imaging, 9, 3, 791-814 (2015) · Zbl 1338.49006 [7] Fan, C.-M.; Li, P.-W.; Yeih, W., Generalized finite difference method for solving two-dimensional inverse Cauchy problems, Inverse Probl. Sci. Eng., 23, 5, 737-759 (2015) · Zbl 1329.65257 [8] Li, P.; Liu, K.; Zhong, W., Marching schemes for inverse scattering problems in waveguides with curved boundaries, J. Comput. Appl. Math., 328, 287-301 (2018) · Zbl 1378.78024 [9] Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations, iv+316 (1953), Dover Publications: Dover Publications New York [10] Bourgeois, L., A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation, Inverse Problems, 21, 3, 1087-1104 (2005) · Zbl 1071.35120 [11] Clason, C.; Klibanov, M. V., The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comput., 30, 1, 1-23 (2007) · Zbl 1159.65346 [12] Dardé, J.; Hannukainen, A.; Hyvönen, N., An $$H_{d i v}$$-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Numer. Anal., 51, 4, 2123-2148 (2013) · Zbl 1291.35434 [13] Klibanov, M. V.; Santosa, F., A computational quasi-reversibility method for Cauchy problems for Laplace’s equation, SIAM J. Appl. Math., 51, 6, 1653-1675 (1991) · Zbl 0769.35005 [14] Lattès, R.; Lions, J.-L., (Méthode de Quasi-Réversibilité et Applications. Méthode de Quasi-Réversibilité et Applications, Travaux et Recherches Mathématiques, No. 15 (1967), Dunod: Dunod Paris), 1-368 · Zbl 0159.20803 [15] Lattès, R.; Lions, J.l., The Method of Quasi-Reversibility. Applications to Partial Differential Equations, xx + 389 (1969), Elsevier: Elsevier New York, NY, Translated from the French edition and edited by Richard Bellman · Zbl 1220.65002 [16] Lattès, R.; Lions, J.-L., (The Method of Quasi-Reversibility. Applications to Partial Differential Equations. The Method of Quasi-Reversibility. Applications to Partial Differential Equations, Modern Analytic and Computational Methods in Science and Mathematics, No. 18 (1969), American Elsevier Publishing Co., Inc.: American Elsevier Publishing Co., Inc. New York), xx+389, Translated from the French edition and edited by Richard Bellman · Zbl 1220.65002 [17] Cimetière, A.; Delvare, F.; Jaoua, M.; Pons, F., Une méthode inverse à régularisation évanescente, C. R. Acad. Sci. IIB, 328, 9, 639-644 (2000) · Zbl 0991.74033 [18] Cimetière, A.; Delvare, F.; Jaoua, M.; Pons, F., Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17, 3, 553-570 (2001) · Zbl 0986.35128 [19] Jourhmane, M.; Nachaoui, A., An alternating method for an inverse Cauchy problem, Numer. Algorithms, 21, 1-4, 247-260 (1999), Numerical methods for partial differential equations (Marrakech, 1998) · Zbl 0944.65116 [20] Jourhmane, M.; Nachaoui, A., Convergence of an alternating method to solve the Cauchy problem for Poisson’s equation, Appl. Anal., 81, 5, 1065-1083 (2002) · Zbl 1060.35157 [21] Kozlov, V. A.; Mazya, V. G.; Fomin, A. V., An iterative method for solving the Cauchy problem for elliptic equations, Zh. Vychisl. Mat. Mat. Fiz., 31, 1, 64-74 (1991) · Zbl 0733.65056 [22] Bergam, A.; Chakib, A.; Nachaoui, A.; Nachaoui, M., Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem, Appl. Math. Comput., 346, 865-878 (2019) · Zbl 1429.65270 [23] Liu, J.-C.; Wei, T., A quasi-reversibility regularization method for an inverse heat conduction problem without initial data, Appl. Math. Comput., 219, 23, 10866-10881 (2013) · Zbl 1302.65216 [24] Andrieux, S.; Baranger, T. N.; Ben Abda, A., Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems, 22, 1, 115-133 (2006) · Zbl 1089.35084 [25] Chakib, A.; Nachaoui, A., Convergence analysis for finite element approximation to an inverse Cauchy problem, Inverse Problems, 22, 4, 1191-1206 (2006) · Zbl 1112.49027 [26] Quarteroni, A.; Valli, A., (Domain Decomposition Methods for Partial Differential Equations. Domain Decomposition Methods for Partial Differential Equations, Numerical Mathematics and Scientific Computation (1999), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York), xvi+360, Oxford Science Publications · Zbl 0931.65118 [27] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag, URL https://books.google.fr/books?id=RYo_AQAAIAAJ · Zbl 0559.47040 [28] Badeva, V.; Morozov, V., Problèmes Incorrectement Posées, Théorie et Applications (1997), Masson: Masson Paris
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.