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Original Articles

A finite element model for the data completion problem: analysis and assessment

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Pages 1063-1086 | Received 24 Jul 2010, Accepted 06 Apr 2011, Published online: 27 Jun 2011

References

  • Andrieux, S, Baranger, T, and Ben Abda, A, 2006. Solving Cauchy problems by minimizing an energy-like functional, Inverse Probl. 22 (2006), pp. 115–134.
  • Eldén, L, and Simoncini, V, 2009. A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces, Inverse Probl. 25 (2009), Article ID 065002 (22pp).
  • Berntsson, F, and Eldén., L, 2001. Numerical solution of a Cauchy problem for the Laplace equation Inverse Probl. 17 (2001), pp. 839–853.
  • Leitão, A, 2000. An iterative method for solving elliptic Cauchy problem, Numer. Funct. Anal. Optim 21 (2000), pp. 715–742.
  • Tautenhahn, U, 1996. Optimal stable solution of Cauchy problems for elliptic equations, J. Anal. Appl. 15 (1996), pp. 961–984.
  • Jourhmane, M, and Nachaoui, A, 2002. Convergence of an alternating method to solve the Cauchy problem for Poisson's equation, Appl. Anal. 81 (2002), pp. 1065–1083.
  • Wei, T, Hon, YC, and Ling, L, 2007. Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Eng. Anal. Boundary Elements 31 (2007), pp. 373–385.
  • Cao, H, and Pereverzev, SV, 2007. The balancing principle for the regularization of elliptic Cauchy problems, Inverse Probl. 23 (2007), pp. 1943–1961.
  • Cimetière, A, Delvare, F, Jaoua, M, and Pons, F, 2001. Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Probl. 17 (2001), pp. 553–570.
  • Delvare, F, and Cimetière, A, 2008. A first order method for the Cauchy problem for the Laplace equation using BEM, Comput. Mech. 41 (2008), pp. 789–796.
  • Ben Abdallah, J, 2007. A conjugate gradient type method for the Steklov–Poincaré formulation of the Cauchy–Poisson problem, Int. J. Appl. Math. 3 (2007), pp. 27–40.
  • Johansson, T, 2004. An iterative procedure for solving a Cauchy problem for second order elliptic equations, Math. Nachr. 272 (2004), pp. 46–54.
  • Shigeta, T, and Young, DL, 2009. Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points, J. Comput. Phys. 228 (2009), pp. 1903–1915.
  • Andrieux, S, and Baranger, T, 2008. An optimization approach for the Cauchy problem in linear elasticity, Struct. Multidisc. Optim. 35 (2008), pp. 141–152.
  • Hào, DN, Duc, NV, and Lesnic, D, 2009. A non-local boundary value problem method for the Cauchy problem for elliptic equations, Inverse Probl. 25 (2009), p. 055002, (27pp).
  • Jin, B, and Zou, Y, 2008. A Bayesian inference approach to the ill-posed Cauchy problem of steady-state heat conduction, Int. J. Numer. Meth. Eng. 76 (2008), pp. 521–3544.
  • Ling, L, and Takeuchi, T, 2008. Boundary control for inverse Cauchy problems of the Laplace equations, Comput. Model. Eng. Sci. 29 (2008), pp. 45–54.
  • Kozlov, VA, Maz'ya, VG, and Fomin, AV, 1991. An iterative method for solving the Cauchy problem for elliptic equations Comput, Meth. Math.Phys. 31 (1991), pp. 45–52.
  • Engl, HW, and Leitão, A, 2001. A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim. 22 (2001), pp. 861–884.
  • Cheng, J, Hon, YC, Wei, T, and Yamamoto, M, 2001. Numerical computation of a Cauchy problem for Laplace's equation, ZAMM 81 (2001), pp. 665–674.
  • Qian, Z, Fu, C-L, and Xiong, X-T, 2006. Fourth-order modified method for the Cauchy problem for the Laplace equation, J. Comput. Appl. Math. 192 (2006), pp. 205–218.
  • Cao, H, Klibanov, MV, and Pereverzev, SV, 2009. A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation, Inverse Probl. 25 (2009), p. 035005.
  • Bourgeois, L, 2006. Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation, Inverse Probl. 22 (2006), pp. 413–430.
  • Ben Belgacem, F, and El Fekih, H, 2005. On Cauchy's problem I. A variational Steklov–Poincaré theory, Inverse Probl. 21 (2005), pp. 1915–1936.
  • Azaïez, M, Ben Belgacem, F, and El Fekih, H, 2006. On Cauchy's problem. II. Completion, regularization and approximation, Inverse Probl. 22 (2006), pp. 1307–1336.
  • Groetsch, GW, and Guacaneme, J, 1987. Arcangeli's Method for Fredholm equations of the first kind, Proc. Am. Math. Soc. 99 (1987), pp. 256–260.
  • Bakushinsky, A, and Goncharsky, A, 1994. Ill-posed Problems: Theory and Applications. Dordrecht, The Netherlands: Kluwer Academic; 1994.
  • Adams, DA, 1975. Sobolev Spaces. New York: Academic Press; 1975.
  • Hanke, M, 1995. "Conjugate Gradient type Methods for Ill-posed Problems". In: Pitman Research Notes in Mathematics Series. Vol. 327. Harlow: Longman Scientific and Technical; 1995.
  • Ben Belgacem, F, 2007. Why is the Cauchy's problem severely ill-posed?, Inverse Probl. 23 (2007), pp. 823–836.
  • Ciarlet, P-G, 1978. The Finite Element Method for Elliptic Problems. Amsterdam: North Holland; 1978.
  • Lucht, W, A finite element method for an ill-posed problem, Seventh conference on the numerical treatment of differential equations (Halle, 1994), Appl. Numer. Math. 18 (1995), pp. 253–266.
  • Quarteroni, A, and Valli, A, 1999. Domain Decomposition Methods for Partial Differential Equations. Oxford: Oxford University Press; 1999.
  • Ben Belgacem, F, El Fekih, H, and Jelassi, F, 2008. The Lavrentiev regularization of the data completion problem, Inverse Probl. 24 (2008), p. 045009, (14pp).
  • Du, DT, and Jelassi, F, A preconditioned Richardson's regularization for the data completion problem and the Kozlov–Maz'ya–Fomin method ARIMA, (accepted).
  • Ben Belgacem, F, and Kaber, S-M, 2008. Quadratic optimization in ill-posed problems, Inverse Probl. 24 (2008), Article ID 055002 (15pp).
  • Hager, WW, 2000. Iterative methods for nearly singular linear systems, SIAM J. Sci. Comput. 22 (2000), pp. 747–766.
  • Lavrentiev, MM, 1967. Some Improperly Posed Problems of Mathematical Physics.. New York: Springer-Verlag; 1967.
  • Engl, HW, Hanke, M, and Neubauer, A, 1996. "Regularization of Inverse Probl.". In: Mathematics and its Applications. Vol. 375. Dordrecht, The Netherlands: Kluwer Academic; 1996.
  • Tikhonov, AN, and Arsenin, VY, 1977. Solution to Ill-posed Problems. New York: Winston-Wiley; 1977.
  • Phillips, DL, 1962. A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9 (1962), pp. 84–97.
  • Morozov, VA, 1966. On the solution of functional equations by the method of regularization, Sov. Math. Dokl. 7 (1966), pp. 414–417.
  • Nair, MT, and Tautenhahn, U, 2004. Lavrentiev regularization for linear ill-posed problems under general source conditions, J. Anal. Appl. 23 (2004), pp. 167–185.
  • Plato, R, 1998. The method of conjugate residuals for solving the Galerkin equations associated with symmetric positive semidefinite ill-posed problems, SIAM J. Numer. Anal. 35 (1998), pp. 1621–1645.
  • El Badia, A, and Farah, M, 2007. Identification of dipole sources in an elliptic equation from boundary measurements: application to the inverse EEG problem, J. Inverse Ill-posed Probl. 14 (2007), pp. 331–353.
  • Nara, T, 2008. An algebraic method for identification of dipoles and quadrupoles, Inverse Probl. 24 (2008), Article ID 025010 (19pp).
  • Isakov, V, 1998. "Inverse Probl. for Partial Differential Equations". In: Applied Mathematical Sciences. Vol. 127. New York: Springer-Verlag; 1998.
  • Saltel, E, and Hecht, F, 1995. EMC2, Un logiciel d'édition de maillage et de contours bidimensionnels. Technical Report No. 118. Rocquencourt, FRANCE: INRIA; 1995.
  • Lucquin, B, and Pironeau, O, Introduction to Scientific Computing, (English) Translated from the French by Michel Kern, John Wiley & Sons, Chichester, 1998.
  • Frayssé, V, and Giraud, L, 2000. A set of conjugate gradient routines for real and complex arithmetics. CERFACS Technical Report TR/PA/00/47, METEOPOLE Campus, Toulouse; 2000.

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