##
**Carleman estimates for coefficient inverse problems and numerical applications.**
*(English)*
Zbl 1069.65106

Inverse and Ill-Posed Problems Series. Utrecht: VSP (ISBN 90-6764-405-6/hbk). iv, 282 p. (2004).

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 differential equations. The second part of the book is devoted to the construction of globally convergent numerical methods for coefficient inverse problems and concrete applied problems in geophysics of exploration, medical optical imaging and computational time reversal.

The book consists of 7 chapters. In Chapter 1 some historical remarks, mathematical background and the concept of overdetermination are given. Further, a brief overview of uniqueness results is presented. In Chapter 2 the authors apply the Carleman estimates to establishing the uniqueness and HĂ¶lder stability results for ill-posed Cauchy problems for general partial differential equations and verify these for some second order parabolic, elliptic and hyperbolic equations. In this section the Lipschitz stability estimates also established. Besides, the authors use the method of Carleman estimates for establishing the stability estimates and rates of convergence of the method of quasireversibility. Section 3 is devoted to the global uniqueness in high dimensions by using the Carleman estimates and Section 4 to that of an inverse problem for a nonlinear parabolic equation in high dimensions.

In Chapter 5 the authors review some methods for the numerical solution of coefficient inverse problems and introduce their convexification approach. The core of this concept is the approximation of the original inverse problems by a family of Cauchy-like problems for a second order operator differential equation, whose operator does not contain explicitly an unknown coefficient. Once such an approximating problem is derived, the method of quasisolutions is applied to this problem to derive inverse formulas. To ensure the strict convexity of the resulting objective functional, the Carleman weighted functions are used. The authors claim (p. 171) that a minimizer of the introduced objective functionals is a quasisolution. The main feature of convexification is that it allows for constructing the strictly convex functionals on a correctness set. This ensures the uniqueness and stability of both quasisolution of the auxiliary problem and the approximate solution of the original coefficient inverse problem. Both the problem of local minima and ill-posedness of an original coefficient inverse problem are, as the authors claim, simultaneously treated within the framework of convexification.

In Chapter 6 the authors present the recurrence minimization method for constructing the convexification algorithms for a broad class of coefficient inverse problems. This method reduces computing the quasisolution of the auxiliary problem to solving more simple nonlinear convex minimization problems and provides the global convergence. The last chapter is the application of the convexification approach to high resolution imaging of inhomogeneous media, namely, magnetotelluric sounding, near-infrared optical sensing of layered biotissues and computational time reversal.

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 differential equations. The second part of the book is devoted to the construction of globally convergent numerical methods for coefficient inverse problems and concrete applied problems in geophysics of exploration, medical optical imaging and computational time reversal.

The book consists of 7 chapters. In Chapter 1 some historical remarks, mathematical background and the concept of overdetermination are given. Further, a brief overview of uniqueness results is presented. In Chapter 2 the authors apply the Carleman estimates to establishing the uniqueness and HĂ¶lder stability results for ill-posed Cauchy problems for general partial differential equations and verify these for some second order parabolic, elliptic and hyperbolic equations. In this section the Lipschitz stability estimates also established. Besides, the authors use the method of Carleman estimates for establishing the stability estimates and rates of convergence of the method of quasireversibility. Section 3 is devoted to the global uniqueness in high dimensions by using the Carleman estimates and Section 4 to that of an inverse problem for a nonlinear parabolic equation in high dimensions.

In Chapter 5 the authors review some methods for the numerical solution of coefficient inverse problems and introduce their convexification approach. The core of this concept is the approximation of the original inverse problems by a family of Cauchy-like problems for a second order operator differential equation, whose operator does not contain explicitly an unknown coefficient. Once such an approximating problem is derived, the method of quasisolutions is applied to this problem to derive inverse formulas. To ensure the strict convexity of the resulting objective functional, the Carleman weighted functions are used. The authors claim (p. 171) that a minimizer of the introduced objective functionals is a quasisolution. The main feature of convexification is that it allows for constructing the strictly convex functionals on a correctness set. This ensures the uniqueness and stability of both quasisolution of the auxiliary problem and the approximate solution of the original coefficient inverse problem. Both the problem of local minima and ill-posedness of an original coefficient inverse problem are, as the authors claim, simultaneously treated within the framework of convexification.

In Chapter 6 the authors present the recurrence minimization method for constructing the convexification algorithms for a broad class of coefficient inverse problems. This method reduces computing the quasisolution of the auxiliary problem to solving more simple nonlinear convex minimization problems and provides the global convergence. The last chapter is the application of the convexification approach to high resolution imaging of inhomogeneous media, namely, magnetotelluric sounding, near-infrared optical sensing of layered biotissues and computational time reversal.

Reviewer: Dinh Nho Hao (Brussels)

### MSC:

65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

35R30 | Inverse problems for PDEs |

35G25 | Initial value problems for nonlinear higher-order PDEs |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35K55 | Nonlinear parabolic equations |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |