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The inverse scattering problem is central to many areas of science and technology such as radar and sonar, medical imaging, geophysical exploration and nondestructive testing. This book is devoted to the mathematical and numerical analysis of the inverse scattering problem for acoustic and electromagnetic waves. To this second edition the authors have added new material on Newton's method for the inverse obstacle problem, a new elegant proof of uniqueness for the inverse medium problem, a discussion of the spectral theory of the far field operator and a presentation of a new method for determining the support of an inhomogeneous medium from far field data. In addition the text has been updated in various places.

1 Introduction
1.1 The Direct Scattering Problem
1.2 The Inverse Scattering Problem
2 The Hetmholtz Equation
2.1 Acoustic Waves
2.2 Green's Theorem and Formula
2.3 Spherical Harmonics
2.4 Spherical Bessel Functions
2.5 The Far Field Mapping
3 Direct Acoustic Obstacle Scattering
3.1 Single- and Double-Layer Potentials
3.2 Scattering from a Sound-Soft Obstacle
3.3 The Reciprocity Relation
3.4 The Two-Dimensional Case
3.5 On the Numerical Solution in IR2
3.6 On the Numerical Solution in IR3
4 Ill-Posed Problems
4.1 The Concept of Ill-Posedness
4.2 Regularization Methods
4.3 Singular Value Decomposition
4.4 Tikhonov Regularization
4.5 Nonlinear Operators
5 Inverse Acoustic Obstacle Scattering
5.1 Uniqueness
5.2 Physical Optics Approximation
5.3 Continuity and Differentiability of the Far Field Mapping
5.4 Approximation of the Scattered Field
5.5 Superposition of the Incident Fields
6 The Maxwell Equations
6.1 Electromagnetic Waves
6.2 Green's Theorem and Formula
6.3 Vector Potentials
6.4 Scattering from a Perfect Conductor
6.5 Vector Wave Functions
6.6 The Reciprocity Relation
7 Inverse Electromagnetic Obstacle Scattering
7.1 Uniqueness
7.2 Continuous Dependence on the Boundary
7.3 Approximation of the Scattered Field
7.4 Superposition of the Incident Fields
8 Acoustic Waves in an Inhomogeneous Medium
8.1 Physical Background
8.2 The Lippmann-Schwinger Equation
8.3 The Unique Continuation Principle
8.4 Far Field Patterns
8.5 The Analytic Fredholm Theory
8.6 Transmission Eigenvalues
8.7 Numerical Methods
9 Electromagnetic Waves in an Inhomogeneous Medium
9.1 Physical Background
9.2 Existence and Uniqueness
9.3 Far Field Patterns
9.4 The Spherically Stratified Dielectric Medium
9.5 The Exterior Impedance Boundary Value Problem
10 The Inverse Medium Problem
10.1 The Inverse Medium Problem for Acoustic Waves
10.2 A Uniqueness Theorem
10.3 A Dual Space Method
10.4 A Modified Dual Space Method
10.5 The Inverse Medium Problem for Electromagnetic Waves
10.6 Numerical Examples
10.7 The Two-Dimensional Case
References
Index