##
**An introduction to the mathematical theory of inverse problems.**
*(English)*
Zbl 0865.35004

Applied Mathematical Sciences. 120. Berlin: Springer-Verlag. x, 282 p. (1996).

This well-written book introduces the reader to the mathematical theory of inverse problems.

Starting at ‘basic notions and difficulties encountered with ill-posed problems’ and continuing with ‘basic properties of regularization methods for linear ill-posed problems’, the book finally ‘gives a first insight into two special fields of nonlinear inverse problems, the inverse spectral theory and the inverse scattering theory’. It is intended to ‘present a fairly elementary and complex introduction to the field of inverse problems, accessible not only to mathematicians but also to physicists and engineers’.

In Chapter I, ‘Introduction and basic concepts’, the author describes examples of inverse problems, in particular several first kind integral equations, and considers the notions of ill-posedness and worst case error. In Chapter II, ‘Regularization theory for equations of the first kind’, the operator equation with a linear, compact operator is considered. The author studies a general concept of regularization, optimality conditions, parameter strategies (a-priori, a-posteriori) and concrete regularization methods as Tikhonov’s regularization, Landweber’s iteration and the conjugate gradient method. Chapter III, ‘Regularization by discretization’, is engaged with projection methods, in particular the least squares and dual least squares methods and collocation methods for linear ill-posed problems. Moreover, numerical results for various regularization techniques are presented and compared, where the logarithmic kernel integral equation (Symm’s integral equation) serves as a model equation. Finally, the Backus-Gilbert method is treated in this chapter.

In Chapters IV and V the author studies two important classes of nonlinear inverse problems. In Chapter IV, ‘Inverse eigenvalue problems’, first the canonical Sturm-Liouville eigenvalue problem is considered, such as countability and asymptotics of eigenvalues. Then, the inverse problem of determining the Sturm-Liouville operator from the eigenvalues, its connection to parameter identification problems in parabolic differential equations, and numerical procedures for the reconstruction are investigated.

Finally, in Chapter V, ‘An inverse scattering problem’, the author formulates the direct scattering problem for the Helmholtz equation and proves existence and uniqueness. Then, after describing the far field pattern, he proves uniqueness for the inverse problem and presents numerical algorithms for its solution.

Appendix A, ‘Basic facts from functional analysis’, is a short account of functional analysis as far it is used in the book. This is convenient to the reader and makes the book accessible to a broader auditorium.

Each chapter concludes with several instructive exercises. The useful bibliography encloses more than 220 items. This is a valuable book not only for graduate students of applied and industrial mathematics with basic knowledge in advanced calculus and functional analysis, but also for physicists, engineers and researchers being engaged with the application of mathematics to the solution of real world problems.

Starting at ‘basic notions and difficulties encountered with ill-posed problems’ and continuing with ‘basic properties of regularization methods for linear ill-posed problems’, the book finally ‘gives a first insight into two special fields of nonlinear inverse problems, the inverse spectral theory and the inverse scattering theory’. It is intended to ‘present a fairly elementary and complex introduction to the field of inverse problems, accessible not only to mathematicians but also to physicists and engineers’.

In Chapter I, ‘Introduction and basic concepts’, the author describes examples of inverse problems, in particular several first kind integral equations, and considers the notions of ill-posedness and worst case error. In Chapter II, ‘Regularization theory for equations of the first kind’, the operator equation with a linear, compact operator is considered. The author studies a general concept of regularization, optimality conditions, parameter strategies (a-priori, a-posteriori) and concrete regularization methods as Tikhonov’s regularization, Landweber’s iteration and the conjugate gradient method. Chapter III, ‘Regularization by discretization’, is engaged with projection methods, in particular the least squares and dual least squares methods and collocation methods for linear ill-posed problems. Moreover, numerical results for various regularization techniques are presented and compared, where the logarithmic kernel integral equation (Symm’s integral equation) serves as a model equation. Finally, the Backus-Gilbert method is treated in this chapter.

In Chapters IV and V the author studies two important classes of nonlinear inverse problems. In Chapter IV, ‘Inverse eigenvalue problems’, first the canonical Sturm-Liouville eigenvalue problem is considered, such as countability and asymptotics of eigenvalues. Then, the inverse problem of determining the Sturm-Liouville operator from the eigenvalues, its connection to parameter identification problems in parabolic differential equations, and numerical procedures for the reconstruction are investigated.

Finally, in Chapter V, ‘An inverse scattering problem’, the author formulates the direct scattering problem for the Helmholtz equation and proves existence and uniqueness. Then, after describing the far field pattern, he proves uniqueness for the inverse problem and presents numerical algorithms for its solution.

Appendix A, ‘Basic facts from functional analysis’, is a short account of functional analysis as far it is used in the book. This is convenient to the reader and makes the book accessible to a broader auditorium.

Each chapter concludes with several instructive exercises. The useful bibliography encloses more than 220 items. This is a valuable book not only for graduate students of applied and industrial mathematics with basic knowledge in advanced calculus and functional analysis, but also for physicists, engineers and researchers being engaged with the application of mathematics to the solution of real world problems.

Reviewer: G.Bruckner (Berlin)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35R30 | Inverse problems for PDEs |

65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |

34B24 | Sturm-Liouville theory |

35P25 | Scattering theory for PDEs |

65J10 | Numerical solutions to equations with linear operators |

65J15 | Numerical solutions to equations with nonlinear operators |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |