Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems
TLDR
A review paper of the role of Carleman estimates in the theory of multidimensional Coefficient Inverse Problems since the first inception of this idea in 1981 is given in this article.Abstract:
This is a review paper of the role of Carleman estimates in the theory of Multidimensional Coefficient Inverse Problems since the first inception of this idea in 1981.read more
Citations
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Journal ArticleDOI
Phaseless Inverse Scattering Problems in Three Dimensions
TL;DR: Three-dimensional inverse scattering problems in the frequency domain are considered in the case when only the modulus of the scattered field is given, while the phase is unknown and uniqueness theorems are proved.
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Carleman estimates for the regularization of ill-posed Cauchy problems
TL;DR: A survey of results for ill-posed Cauchy problems for PDEs of the author with co-authors starting from 1991 can be found in this article, where a universal method of the regularization of these problems is presented.
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Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs
TL;DR: In this paper, the existence of the regularized solution (i.e. the minimizer) is proved rather than assumed for ill-posed Cauchy problems, and numerical results are presented for the case of a 1-D quasilinear parabolic PDE with the lateral data given on one edge of the interval (0, 1).
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Convexification of restricted Dirichlet-to-Neumann map
TL;DR: In this paper, the restricted Dirichlet-to-Neumann map (DN) is defined and a new numerical concept for CIPs with restricted DN data for a broad class of PDEs of the second order, e.g. elliptic, parabolic and hyperbolic ones.
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Electrical Impedance Tomography with Restricted Dirichlet-to-Neumann Map Data
TL;DR: In this article, a new numerical method was proposed to reconstruct the isotropic electrical conductivity from measured restricted Dirichlet-to-Neumann map data in electrical impedance tomography (EIT) model.
References
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Book
Linear and Quasilinear Equations of Parabolic Type
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
Book
Regularization of Inverse Problems
TL;DR: Inverse problems have been studied in this article, where Tikhonov regularization of nonlinear problems has been applied to weighted polynomial minimization problems, and the Conjugate Gradient Method has been used for numerical realization.