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Contents 1 Introduction 1 References 3 2 Direct Acoustic Scattering Problems 5 2.1 Acoustic Scattering From Obstacles 6 2.2 Acoustic Scattering From Mediums 7 2.3 Acoustic Scattering From Complex Scatterers 8 2.4 Green's Formula and Linear Potential Theory for Scattering Problems 12 2.5 Numerical Methods for Acoustic Scattering Problems 18 2.5.1 Nystxom Method 18 2.5.2 Finite Element Method with Perfectly Matched Layer (PML) 24 References 28 3 Numerical Inverse Acoustic Scattering Problems 29 3.1 Overview 29 3.2 Strengthened Linear Sampling Methods 32 3.2.1 Strengthened Linear Sampling Method with a Reference Ball 33 3.2.2 Numerical Experiments and Discussion 41 3.2.3 Conclusion 53 3.3 Single-Shot Method for Multiple Multiscale Scatterers 56 3.3.1 Locating Small Scatterers 57 3.3.2 Locating Scatterers of Regular Size 65 3.3.3 Locating Scatterers of Multiple Scales 75 3.4 Reconstruction by Phaseless Backscattering Measurements 81 3.4.1 Physical Optics Approximation 82 3.4.2 Recovery Scheme 88 3.4.3 Numerical Experiments and Discussions 92 3.4.4 Concluding Remarks 96 References 99 4 Direct Electromagnetic Scattering Problems 105 4.1 Electromagnetic Scattering From Obstacles 106 4.2 Electromagnetic Scattering From Mediums 107 4.3 Electromagnetic Scattering From Complex Scatterers 108 4.4 Green's Theorem and Representation Formulas 112 4.5 Numerical Methods for Electromagnetic Scattering Problems .116 References 124 5 Numerical Inverse Electromagnetic Scattering Problems 125 5.1 Overview 125 5.2 Strengthened Linear Sampling Methods 126 5.3 Single-shot Method for Multiple Multiscale Scatterers 127 5.3.1 The Locating Schemes 129 5.3.2 Proofs of Theorems 5.3.1 and 5.3.2 136 5.3.3 Numerical Experiments and Discussions 145 5.3.4 Concluding Remarks 152 5.4 Scatterers 153 5.4.1 Multi-scale EM Scatterers and Two Locating Schemes ________154 5.4.2 Scheme R with Augmented Reference Spaces 158 5.4.3 Locating Multiple Multi-scale Scatterers 164 5.4.4 Numerical Experiments and Discussions 166 5.4.5 Discussions 175 References 176 6 Direct Elastic Scattering Problems 179 6.1 Elastic Scattering from Obstacles 180 6.2 Elastic Scattering from Mediums 181 6.3 Elastic Scattering from Complex Scatterers 182 6.4 Green's Theorems and Representation Formulas 189 6.5 Numerical Methods for Elastic Scattering Problems 196 References 204 7 Numerical Inverse Elastic Scattering Problems 205 7.1 Overview 205 7.2 Single-Shot Method for Multiple Multiscale Scatterers 206 7.2.1 Locating Multiple Small Scatterers 206 7.2.2 Locating Multiple Extended Scatterers 212 7.2.3 Locating Multiple Multiscale Scatterers 223 7.2.4 Numerical Examples 225 7.2.5 Concluding Remarks 230 7.3 Traction Free Case 231 7.3.1 Elastic Scattering From Multiscale Scatterers 232 7.3.2 Locating Multiple Multiscale Elastic Scatterers 242 7.4 Reconstructing Multiple Small Scatterers 248 7.5 Reconstructing Multiple Extended Scatterers 251 7.6 Reconstructing Multiple Multiscale Scatterers 257 7.7 Two-Stage Fast Imaging of Multiple Multiscale Scatterers 260 References 265 8 Miscellaneous Topics 269 8.1 Multilevel Linear Sampling Methods 269 8.1.1 Multilevel Linear Sampling Method 271 8.1.2 Numerical Experiments and Some Discussions 279 8.2 EMLSM 292 8.2.1 Inverse Acoustic Obstacle Scattering Problem 293 8.2.2 Review of LSM and MLSM 294 8.2.3 Enhanced Multilevel-LSMs 297 8.2.4 Numerical Experiments 304 References 313 9 Others 315 9.1 Ground Detection by a Single Electromagnetic Far-Field Measurement 315 9.1.1 Scattering from Multiscale Ground Objects 318 9.1.2 Locating Multiscale Ground Objects 324 9.1.3 Numerical Experiments and Discussions 326 9.2 Recovering Multiscale Buried Anomalies in a Two-Layered Medium 336 9.2.1 Mathematical Formulation 337 9.2.2 Results on Direct Scattering Problem 339 9.2.3 Recovery Scheme 351 9.2.4 Numerical Experiments 357 References 362 Book list of the Series in Information and Computational Science 365