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.

• Preface
• Acknowledgments
• Contents
• 1 Topics of this book
• 2 Carleman estimates and Hölder stability for ill-posed Cauchy problems
• 3 Global uniqueness for coefficient inverse problems and Lipschitz stability for a hyperbolic CIP
• 4 The quasi-reversibility numerical method for ill-posed Cauchy problems for linear PDEs
• 5 Convexification for ill-posed Cauchy problems for quasi-linear PDEs
• 6 A special orthonormal basis in L₂(a, b) for the convexification for CIPs without the initial conditions—restricted Dirichlet-to-Neumann map
• 7 Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data
• 8 Convexification for a coefficient inverse problem for a hyperbolic equation with a single location of the point source
• 9 Convexification for an inverse parabolic problem
• 10 Experimental data and convexification for the recovery of the dielectric constants of buried targets using the Helmholtz equation
• 11 Travel time tomography with formally determined incomplete data in 3D
• 12 Numerical solution of the linearized travel time tomography problem with incomplete data
• Bibliography
• Index