Regularization Methods for an Ill-Posed Elliptic Problem
Keywords:
Regularization methods, Ill-Posed Elliptic ProblemAbstract
We study an abstract elliptic Cauchy problem associated with an unbounded self-adjoint positive operator, which has a continuous spectrum. It is well-known that such a problem is severely ill-posed; that is, the solution does not depend continuously on the Cauchy data.
References
. H. Brezis and A. Pazy. Convergence and approximation of Semigroups of Nonlinear Operators in Banach Spaces. J. Func. Anal., 9:63-74, 1972.
. A.Tikhonov, V.Arsénine: Methods of regularization of Ill-posed problems.Ed 1976.
. D. D. Ang, R. Gorenflo, V. K .Le, and D. D. Trong, Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction, Vol. 1792 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2002.
. M.. Denisov, E. V. Zakharov, A. V. Kalinin and V. V. Kalinin, ‘’Numerical solution of the inverse electrocardiography Problem With the use of the tikhonov regularization method,’’ Computational Mathematics and Cybernetics, vol, 32, no. 2, pp. 61-68, 2008.
. M. M. Lavrentev, V. G. Romanov, and G. P. Shishatskii, III-Posed Problems in Mathematical Physics and Analysis, American Mathematical Society, RI, USA, 1986.
. L. E. Payne, Improperly Posed Problems in Partial Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1975.
. G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella, ‘’The Stability for the Cauchy problem for elliptic equations,’’Inverse Problems, vol. 25, no. 12, Article ID 123 004, 47 pages, 2009.
. U. Tautenhahn, ‘’Optimal stable solution of Cauchy problems For elliptic equations,’’ Zeitschrift für Analysis und ihre Anwendungen, vol. 15, no. 4, pp. 961-984, 1996.
. Z. Qian, C-L. Fu and Z-P. Li, ‘‘two regularization methods For a Cauchy problem for the Laplace equation,’’ Journal of Mathematical Analysis and Application, vol. 338, no. 1, pp.479 489, 2008
. D. N. Hào, N. V. Duc, and D. Lesnic,’’A non-local boundary Value problem method for the Cauchy problem for elliptic Equation,’’Inverse problems, vol. 25, no. 5, Article ID 055002, 27 pages, 2009.
. X-L. Feng, L. Eldén, and C.-L. Fu,’’A quasi-boundary- value Method for the Cauchy problem for elliptic equations with Nonhomogeneous Neumann data,’’ Journal of Inverse and III-Posed Problems, vol. 18, no. 6, pp. 617-645, 2010.
. V. Mel’ nikova,’’Regularization of ill- posed differential problems,’’ Siberian Mathematical Journal, vol. 33, no. 2, pp. 289-298, 1992.
. P. N. Vabishchevich and A. Yu. Denisenko, ‘’ Regularization Of nonstationary problems for elliptic equations,’’ Journal of Engineering Physics and Thermophysics, vol. 65, no. 6, pp. 1195-1199, 1993.
. V. A. Kozlov and V.G. Maz’ya,’’On iterative procedure for solv-ing ill-posed boundary value problems that preserve differential Equations,’’ Leningrad Mathematical Journal, vol. 1, no. 5, pp. 1207-1228, 1990.
. D. N. Hào,’’A mollification method for ill-posed problems,’’ Numerische Mathematik, vol. 68, no. 4, pp. 469-506, 1994.
. N. Dunford and J. Schwartz, Linear Opertors-Part Il, John Wiley & Sons, New York, NY? USA? 1967
. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
. S. G. Krein and Ju. I. Petunin,’’Scales of Banach spaces,’’ Uspekhi Matematicheskikh Nauk, vol. 21, no. 2(128), pp. 89-168, 1966.
. H. O. Fattorini, the Cauchy Problem, vol. 18 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley Publishing, Reading, Mass, USA? 1983.
. K. Schmüdgen, Unbounded Self- Adjoint Operators on Hilbert Space, vol. 265 of Graduate Texts in Mathematics, Springer, Dordrecht The Netherlands, 2012.
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