Klibanov M.V., Li J. Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data

De Gruyter, 2021. — 344 p. — (Inverse and Ill-Posed Problems 63). — ISBN 978-3-11-074541-2.Обратные задачи и оценки Карлемана: глобальная единственность, глобальная сходимость и экспериментальные данные (обратные и некорректные задачи)This book summarizes the main analytical and numerical results of Carleman estimates. In the analytical part, Carleman estimates for three main types of Partial Differential Equations (PDEs) are derived. In the numerical part, first numerical methods are proposed to solve ill-posed Cauchy problems for both linear and quasilinear PDEs. Next, various versions of the convexification method are developed for a number of Coefficient Inverse Problems.
Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data (Inverse and Ill-Posed Problems)The following four topics are discussed in this book:
Topic 1: Derivation of Carleman estimates and conditional stability estimates for
some ill-posed Cauchy problems, Chapter 2.
Topic 2: Global uniqueness for multidimensional CIPs on the basis of the BK
method, Chapter 3.
Topic 3: Carleman estimates for numerical methods for ill-posed Cauchy problems
for PDEs, Chapters 4 and 5.
Topic 4: The convexification globally convergent numerical concept for CIPs: a far
reaching consequence of the BK method, Chapters 6–12.Carleman estimates and Hölder stability for ill-posed Cauchy problems
Global uniqueness for coefficient inverse problems and Lipschitz stability for a hyperbolic CIP
The quasi-reversibility numerical method for ill-posed Cauchy problems for linear PDEs
Convexification for ill-posed Cauchy problems for quasi-linear PDEs
A special orthonormal basis in L2(a, b) for the convexification for CIPs without the initial conditions—restricted Dirichlet-to-Neumann map
Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data
Convexification for a coefficient inverse problem for a hyperbolic equation with a single location of the point source
Convexification for an inverse parabolic problem
Experimental data and convexification for the recovery of the dielectric constants of buried targets using the Helmholtz equation
Travel time tomography with formally determined incomplete data in 3D
Numerical solution of the linearized travel time tomography problem with incomplete data