##
**Inverse acoustic and electromagnetic scattering theory.**
*(English)*
Zbl 0760.35053

Applied Mathematical Sciences. 93. Berlin: Springer-Verlag. x, 305 p. (1992).

This book is a monograph summarizing the mathematical basis of inverse scattering theory developed especially during last ten years. The authors present inverse scattering theory from the perspective of their own research interest. Hence the inverse scattering problems treated in the book are divided into two groups: i) inverse obstacle problems (treated in Ch. 5 and Ch. 7); ii) Inverse medium problems (treated in Ch. 10).

In the inverse obstacle problem, the boundary conditions on the scattering obstacle is supposed to be known (Dirichlet type condition) and the inverse problem consists of determining the shape of the obstacle from a knowledge of the scattered field at infinity (far-field pattern). In the inverse medium problem, the scattering object is an inhomogeneous medium and the inverse problem is to determine one or more of the constitutive parameters (or, refractive index) from the far-field pattern. To treat these problems the authors use to different methods. In the first method one looks for an obstacle or parameters whose far-field pattern best fits the measured data whereas in the second method one looks for an obstacle or parameters whose far field pattern has the same weighted averages as the measured data. The theoretical as well as the numerical developments of these two methods for solving the inverse scattering problems are systematically established. In treating the inverse medium problems a particular dual space method, which has some numerical advantages, is introduced.

As background material the rudiments of the theory of spherical harmonics, spherical Bessel functions, operator-valued analytic functions, Herglotz wave functions, the concepts of ill-posed problems and regularization as well as some basic properties and numerical solutions of the direct scattering problems are included. They compose the six chapters (Ch. 2,3,4,6,8,9) of the book which involves nine main chapters except the introduction. This complementary material provides a solid knowledge about the corresponding direct problems and thus makes the book self-contained and easily readable. The level of the book is suitable for advanced students and researchers who have a basic knowledge of classical and functional analysis.

In the inverse obstacle problem, the boundary conditions on the scattering obstacle is supposed to be known (Dirichlet type condition) and the inverse problem consists of determining the shape of the obstacle from a knowledge of the scattered field at infinity (far-field pattern). In the inverse medium problem, the scattering object is an inhomogeneous medium and the inverse problem is to determine one or more of the constitutive parameters (or, refractive index) from the far-field pattern. To treat these problems the authors use to different methods. In the first method one looks for an obstacle or parameters whose far-field pattern best fits the measured data whereas in the second method one looks for an obstacle or parameters whose far field pattern has the same weighted averages as the measured data. The theoretical as well as the numerical developments of these two methods for solving the inverse scattering problems are systematically established. In treating the inverse medium problems a particular dual space method, which has some numerical advantages, is introduced.

As background material the rudiments of the theory of spherical harmonics, spherical Bessel functions, operator-valued analytic functions, Herglotz wave functions, the concepts of ill-posed problems and regularization as well as some basic properties and numerical solutions of the direct scattering problems are included. They compose the six chapters (Ch. 2,3,4,6,8,9) of the book which involves nine main chapters except the introduction. This complementary material provides a solid knowledge about the corresponding direct problems and thus makes the book self-contained and easily readable. The level of the book is suitable for advanced students and researchers who have a basic knowledge of classical and functional analysis.

Reviewer: M.Idemen (İstanbul)

### MSC:

35R30 | Inverse problems for PDEs |

35P25 | Scattering theory for PDEs |

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

45A05 | Linear integral equations |

78A45 | Diffraction, scattering |

65R30 | Numerical methods for ill-posed problems for integral equations |

76Q05 | Hydro- and aero-acoustics |