Carleman estimates for the regularization of ill-posed Cauchy problems |
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Institution: | 1. Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 31441, Saudi Arabia;2. Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, Iran;3. Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137–66731, Iran;4. Department of Mathematics, Kosar University of Bojnord, P.O. Box 94156–15458, Bojnord, Iran |
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Abstract: | This work is a survey of results for ill-posed Cauchy problems for PDEs of the author with co-authors starting from 1991. A universal method of the regularization of these problems is presented here. Even though the idea of this method was previously discussed for specific problems, a universal approach of this paper was not discussed, at least in detail. This approach consists in constructing of such Tikhonov functionals which are generated by unbounded linear operators of those PDEs. The approach is quite general one, since it is applicable to all PDE operators for which Carleman estimates are valid. Three main types of operators of the second order are among them: elliptic, parabolic and hyperbolic ones. The key idea is that convergence rates of minimizers are established using Carleman estimates. Generalizations to nonlinear inverse problems, such as problems of reconstructions of obstacles and coefficient inverse problems are also feasible. |
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Keywords: | Survey Carleman estimates Ill-posed Cauchy problems Convergence rates |
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