Carleman estimates for coefficient inverse problems and numerical applications



Carleman estimates for coefficient inverse problems and numerical applications

M.V. Klibanov and A. Timonov

(Inverse and ill-posed problems series)

VSP, 2004

大学図書館所蔵 件 / 5



Includes bibliographical references (p. [273]-282)



This is the first book dedicated to applying the Carleman estimates to coefficient inverse problems. Written in a readable and concise manner, the book introduces the reader to the essence of the method of Carleman estimates, which is one of the most powerful tools for the mathematical treatment of coefficient inverse problems. The core of the book is two most recent advances of the authors. These are the global uniqueness of a multidimensional coefficient inverse problem for a nonlinear parabolic equation and the so-called convexification framework for constructing globally convergent algorithms for a broad class of inverse problems. Several applications of the convexification to magnetotelluric frequency sounding, electrical impedance tomography, infra-red optical sensing of biotissies, and time reversal are considered.


Chapter 1Introduction 1.1. Some historical remarks 1.2. Mathematical background 1.3. The concept of overdetermination 1.4. Uniqueness results in one dimension 1.5. Uniqueness results in high dimensions 1.6. A brief overview of uniqueness results Chapter 2. Carleman estimates and ill-posed Cauchy problems 2.1. A second order partial differential operator 2.2. Examples of Carleman estimates 2.3. Uniqueness and the Holder stability 2.4. The Lipschitz stability for a hyperbolic Cauchy problem 2.5. Error estimates in the method of quasireversibility Chapter 3. Global uniqueness results in high dimensions 3.1. Estimating a Volterra-like operator 3.2. An inverse problem for a hyperbolic equation 3.3. Some inverse problems for a parabolic equation 3.4. Inverse problems for a elliptic equation 3.5. The global uniqueness in a 2D inverse conductivity problem Chapter 4. The global uniqueness of a nonlinear parabolic problem 4.1. Problem formulation 4.2. Statement of the main result 4.3. An estimate of an integral 4.4. The integro-differential inequality 4.5. Domains 4.6. Notations 4.7. A Carleman estimate 4.8. Proof of the main result Chapter 5. On the numerical solution of coefficient inverse problems 5.1. Some traditional methods 5.2. Using the Dirichlet-to-Neumann map 5.3. Convexification 5.4. Strict convexity of J_{\lambda, \kappa}(q) Chapter 6. Some globally convergent convexification algorithms 6.1. Model problems in one dimension 6.2. The recurrence minimization method 6.3. Two numerical methods for nonlinear convex programming 6.4. Error estimates 6.5. Unifying framework Chapter 7. Some applied problems 7.1. Magnetotelluric sounding of layered media 7.2. Magnetotelluric sounding of 2D inhomogeneous media 7.3. Space electric sounding of 2D inhomogeneous media 7.4. Near-infrared optical sensing of layered biotissues 7.5. Computational time reversal Bibliography

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