Carleman estimates for coefficient inverse problems and numerical applications.
type: Item
instance of: scholarly article
title: Carleman estimates for coefficient inverse problems and numerical applications.
zbMATH Open document ID: 1069.65106
author: Michael V. Klibanov
author: A. A. Timonov
published in: Inverse and Ill-Posed Problems Series
publication date: 2004-12-09T00:00:00Z
review text: This book is devoted to the mathematical treatment and numerical solution of non-overdetermined coefficient inverse problems with the lateral data via Carleman estimates. The problems of determining coefficients of partial differential equations from the boundary data available from observations are frequently arising in many real-world situations. These problems are very difficult, since they are nonlinear and ill-posed in the sense of Hadamard. The last means that either there is no solution, or if there is a solution it may not be unique or does not continuously depend on the data (in some natural metrics). Due to their vital importance in practice, during the last 40 years there has been a considerable effort on establishing the uniqueness and stability results as well as developing stable numerical methods for such problems. Among various approaches to obtaining stability results the method of Carleman estimates has been recognized very powerful. This method allows to prove the global uniqueness and stability estimates for a broad class of coefficient inverse problems. Discovered in 1939 by the Swedish mathematician Torcio Carleman, a priori estimates of a weighted norm of the solution and its derivatives were originally used for establishing the uniqueness result for an elliptic Cauchy problem. In the early eighties of the last century this technique was applied to coefficient inverse problems by Bukhgeim and Klibanov for the first time and since then it became one of the most powerful techniques in the theory of inverse problems. However, most of the results are obtained for linear differential equations and not much attention has been paid to applicability of this technique to the numerical methods. In this book the authors present the Carleman estimate technique for standard coefficient inverse problems for linear partial differential equations and also apply it to proving the global uniqueness of coefficient inverse problems for nonlinear partial different…
Mathematics Subject Classification ID: 65M32
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 2121577
zbMATH Keywords: coefficient inverse problems
zbMATH Keywords: ill-posed problems
zbMATH Keywords: Carleman estimates
zbMATH Keywords: convexification
zbMATH Keywords: textbook
zbMATH Keywords: stability
zbMATH Keywords: elliptic Cauchy problem
zbMATH Keywords: convergence
zbMATH Keywords: method of quasireversibility
zbMATH Keywords: nonlinear parabolic equation
zbMATH Keywords: method of quasisolution
reviewed by: Dinh Nho Hào
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2121577