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| 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 |
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