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Articles

Numerical reconstruction of unknown boundary data in the Cauchy problem for Laplace’s equation

Pages 1255-1267 | Received 24 Feb 2012, Accepted 24 Oct 2012, Published online: 23 Nov 2012

References

  • O.M.Alifanov, Inverse Heat Transfer Problems, Springer, Berlin, 1994.
  • E.J.Kassab, E.Divo, and J.S.Kapat, Multi-dimensional heat flux reconstruction using narrow-band thermochromic liquid crystal thermography, Inverse Probl. Sci. Eng. 9(7) (2001), pp. 537–559.
  • L.Bourgeois and J.Darde, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data. Inverse Prob. 26 (2010), p. 095016.
  • HuiCao, 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 Prob. 25 (2009), p. 035005.
  • S.Kabanikhin, M.A.Bektemesov, A.T.Ayapbergenova, and D.V.Nechaev, Optimization methods of solving continuation problems. Vichisli. Tekhnol. Comp. Tech. SB RAS 9 (2004), pp. 50–60.
  • V.A.Kozlov, A.F.Fomin, and V.G.Maz’ja, An iterative method for solving the Cauchy problem for elliptic equations, USSR, Comput. Math. Math. Phys. 31 (1991), pp. 45–52.
  • M.M.Lavrent’ev, On the Cauchy problem for the Laplace equation, Izvest. Akad. Nauk. SSSR, Ser. Mat.20 (1956), pp. 819–842.
  • L.E.Payne, Bounds in the Cauchy problem for the Laplace equation, Arch. Ration. Mech. Anal.5 (1960), pp. 35–45.
  • V.K.Ivanov, On ill-posed problems, Mat. Sbornik61(2) (1963), pp. 211–223 (in Russian).
  • N.Zabaras and J.Liu, An analysis of two-dimensional linear inverse heat transfer problems using an integral method, Num. Heat Transf. 13 (1988), pp. 527–33.
  • F.P.Vasil’ev, Methods for Solving Extremal Problems, Nauka, Moscow, 1981.
  • A.Hasanov, P.Duchateau, and B.Pektas, An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation, J. Inverse Ill-posed Prob. 14(5) (2006), pp. 435–463.
  • L.Beilina, M.V.Klibanov, and M. Yu.Kokurin, Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem, J. Math. Sci.167(3) (2009), pp. 279–325.
  • L.Beilina and M.V.Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012.
  • A.I.Egorov, On optimal control of processes in distributed objects, J. Appl. Math. Mech.27(4) (1963), pp. 1045–1058.
  • A.I.Egorov, Optimal control of thermal and diffusion processes, Nauka, Moskow, 1978 (In Russian).
  • L.V.Petuhov and V.F.Troitskiy, Variational optimization problems, Appl. Math. Mech. 34(2) (1972), pp. 578–588 (In Russian).
  • V.I.Plotnikov,Necessary and sufficient optimality conditions and uniqueness conditions for optimizing functions for control systems of general type, Math. USSR Izv. 6(3) (1972), pp. 649–676 (In Russian).
  • L.S.Pontryagin, Selected works, The Mathematical Theory of Optimal Processes, Vol. 4, CRC Press, New York, 1987.
  • O.G.Provorova, On a question of control process described by a quasi linear parabolic equation, Upavliaemye sistremy, Nauka, Siberian Branch, Novosibirsk, 1973 (In Russian).
  • B.D.Tajibaev, On optimality conditions in one control problem, Upavliaemye sistremy, Nauka, Siberian Branch, Novosibirsk, 1988 (In Russian).
  • V.S.Belonosov, Interior estimates for solutions to quasiparabolic systems, Siber. Math. J.37(1) (1996), pp. 17–32.
  • O.A.Ladyzenskaja, V.A.Solonnikov, and N.N.Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type. Translations of Mathematical Monographs, Vol. 23, AMS, Providence, RI, 1968.
  • A.Tikhonov and V.Arsenin, Solution of Ill-Posed Problems, John Wiley, New York, 1977.

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