Abstract

This work concerns the use of the iterative algorithm (KMF algorithm) proposed by Kozlov, Mazya and Fomin to solve the Cauchy problem for Laplace’s equation. This problem consists to recovering the lacking data on some part of the boundary using the over specified conditions on the other part of the boundary. We describe an alternating formulation of the KMF algorithm and its relationship with a classical formulation. The implementation of this algorithm for a regular domain is performed by the finite element method using the software Freefem. The numerical tests developed show the effectiveness of the proposed algorithm since it allows to have more accurate results as well as reducing the number of iterations needed for convergence.

References

• Inglese, I. An inverse problem in corrosion detection, inverse problem 13 (1997), 977-994.
• Lesnic, D., Elliot, L., and Ingham, D. B., An iterative boundary element method for the Laplace equation, Engineering Analysis with Boundary Elements 20 (1997), 123-133.
• Zhdanov, M. S. 2002 Geophysical inverse theory and regularization problems, Elsevier science B.V
• Yamashita, Y. Theoretical studies on the inverse problem in electrocardiography and the uniqueness of the solution, IEEE trans. Biom. Engrg (1982), 719-725.
• Hadamard, J. 1923 Lectures on Cauchy’s in Linear Partial Differential Equations, London Oxford University Press. .
• Isakov, V. 1998 Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences Springer-Verlag (127), New York.
• Dautray, R., and Lions,J. L. 1988 Mathematical analysis and numerical methods for science and technology Functional and Variational Methods (2) Springer, Berlein.
• Hohage, J., 2002 Lectures Notes in Inverse problems, University of Gottingen.
• Kohn, R., V., and McKenney, A.Numerical implementation of a variational method for electrical impedance tomography, Inverse Problems 6 (2002), 389-414.
• Lattes, R. and Lions, J. L. 1967 Mèthode de Quasi-reversibilité et Applications Dunod, Paris .
• Klibanov, M.V., and Santosa, F. A computational Quasi-reversibility method for Cauchy problems for Laplace’s equation, SIAM J.Appl. Math 51 (1991) 1653-1675.
• Bourgeois, L. A mixed formulation of Quasi-reversibility method to solve the Cauchy problem for Laplace’s equation, Inverse problem 21 (2005), 1087-1104
• Qin, H., H., and Wei, T., Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation, Mathematics and Computers in Simulation 80 (2009),352–366
• Fasino, D., and Inglese; G. An inverse Robin problem for Laplace’s equation: theoretical results and numerical methods, Inverse Problems 15 (1999) ,41–48
• Cimetière, A., Delvare, F., Jaoua, M. and Pons, F. , Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems 17 (2001), 553-570
• Pasquetti, R. and Petit, D. Inverse diffusion by boundary elements, Engrg. Anal. Boundary Elements 15(1995) ,197–205
• Falk, R.S. and Monk, P.B. Logarithmic convexity of dicrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation, Math. Comput. 47(1986), 135-149,
• Hon, Y., C., and Wei, T. Backus Gilbert algorithm for the Cauchy problem of the Laplace equation, Inverse Problems 17 (2001) 261-271.
• Andrieux, S., Abda, A.B., and Baranger, T.N. Data completion via an energy error functional. C.R.Mecanique 333 (2005) 171-177.
• Kozlov,V.A., Maz’ya, V.G., and Fomin, D.V. An iterative method for solving the Cauchy problem for elliptic equation, Comput. Math. Phys. 31(1991) 45-52.
• Lesnic, D., Elliott, L. and Ingham, D. B. An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation, Engineering Analysis with Boundary Elements (20) 1997, 123-133
• jourhmane, M and Nachaoui, A. An alternating method for an inverse Cauchy problem Numerical Algorithms (21) (1999) , 247–260
• Engl, H.W., and Leitão, A. A mann iterative regularization method for elliptic Cauchy problems,Numerical Functional Analysis and Optimization, (22) 2001, 861 — 884
• Jourhmane, M., Lesnic, D., and Mera, N.S. Relaxation procedures for an iterative algorithm for solving the Cauchy problem for the Laplace equation Engineering Analysis with Boundary Elements (28) 2004 , 655–665
• Essaouini, M., Nachaoui, A., and El Hajji, S. Numerical method for solving a class of nonlinear elliptic inverse problems, Journal of computational and Applied Mathematics (162) 2004, 165-181
• Tajani, C., Abouchabaka, J., and Abdoun, A 2011 Numerical simulation of an inverse problem: Testing the influence data”, in.proc.of the 4th International Conference on Approximation Methods and Numerical Modeling in Environment and Natural , 29-32
• Baumeister, J., and Leitao, A. Iterative methods for ill-posed problems modeled by partial differential equations, Journal of Inverse and ill-posed problem (9) 2001, 1-17
• Jourhmane, M., and Nachaoui, A. Convergence of an Alternating Method to Solve the Cauchy Problem for Poisson’s Equation Applicable Analysis (81) 2002, 1065–1083
• Marin, L., Elliot, L., Ingham, D., and Lesnic, D. Boundary element method for the Cauchy problem in linear elasticity Engineering Anllysis with boundary Elements (25) 2001, 783-793.