Direct and Inverse Problems in Wave Propagation and Applications (eBook)

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Ivan Graham, University of Bath, UK; Ulrich Langer, University of Linz, Austria; Jens Markus Melenk, Vienna University of Technology, Austria;Mourad Sini, RICAM, Linz, Austria.

Preface 5
Differential electromagnetic imaging 13
1 Introduction 13
2 Basic theory of electromagnetic waves 15
2.1 The Helmholtz equation 15
2.2 The Maxwell equations 15
2.3 Fundamental solutions and radiation conditions 16
2.4 Transmission and boundary conditions 17
2.5 Dirichlet and Neumann functions and the Hodge decomposition 18
2.6 Trace theorems and first Green identity 19
2.7 Lippman–Schwinger representation formulas 20
2.8 The Helmholtz–Kirchhoff theorems 21
2.9 Limiting models 22
2.10 TheMaxwell equations with axis invariance 23
2.11 The Maxwell equations versus the Helmholtz equation 24
3 Electric and magnetic polarization tensors 24
4 small-volume expansions 25
4.1 The full Maxwell equations 25
4.2 The eddy currents model 29
4.3 The Helmholtz equation 30
4.4 The conductivity equation 30
4.5 Asymptotic formulas in the time domain 30
5 Imaging in the frequency domain 31
5.1 MUSIC-type imaging at a single frequency 32
5.2 Backpropagation type imaging at a single frequency 34
5.3 Imaging with a broad range of frequencies 35
6 Imaging in the time domain 36
6.1 Time-domain imaging with full viewmeasurements 36
6.2 Time-domain imaging in a cavity with limited-view data 37
6.3 Time-domain imaging in dissipative media 40
7 Numerical examples of MUSIC reconstructions for the full Maxwell equations 45
8 Shape representations 50
8.1 High-order polarization tensors 50
8.2 Frequency dependent high-order polarization tensors 53
9 Far-field imaging versus near-field imaging 57
10 Open problems 59
Multitrace boundary integral equations 63
1 Introduction 63
1.1 Geometry 66
1.2 Transmission problems 66
2 Boundary integral operators 69
2.1 Trace spaces and operators 70
2.2 Potentials 73
2.3 Calderón projectors 74
3 Classical single-trace integral equations 76
3.1 Skeleton trace spaces 77
3.2 A first-kind boundary integral equation 81
3.3 Boundary element Galerkin discretization 84
4 Preconditioning 87
4.1 Operator products 88
4.2 Calderón identities 89
4.3 Operator preconditioning 91
4.4 Stable duality pairing for boundary elements 92
4.5 The challenge 93
5 Global multitrace formulation 94
5.1 Separated subdomains 94
5.2 The gap idea 97
5.3 Properties of global MTF 98
5.4 Galerkin discretization 99
6 Local multitrace formulation 100
6.1 Partial transmission conditions 101
6.2 Local MTF: variational formulation 103
6.3 Local MTF: Stability 105
6.4 Boundary element Galerkin discretization 107
Direct and Inverse Elastic Scattering Problems for Diffraction Gratings 113
1 Introduction 113
2 Mathematical formulation of direct and inverse scattering problems 115
3 Solvability results for direct scattering problems: variational method 119
3.1 An equivalent variational formulation and its Fredholmproperty 119
3.2 Uniqueness and existence for direct scattering problems 121
3.3 Uniqueness and existence for transmission gratings 123
4 Uniqueness for inverse scattering problems 127
4.1 Inverse scattering of incident pressure waves 128
4.2 Inverse scattering of incident shear waves 133
5 Numerical solution of direct and inverse scattering problems 134
5.1 A discrete Galerkin method for (DP) 134
5.2 A two-step algorithm for (IP) 137
Multigrid methods for Helmholtz problems: A convergent scheme in 1D using standard components 147
1 Introduction 147
2 Smoothing 151
2.1 Smoothing analysis 152
2.2 Jacobi smoothing 154
2.3 Two-step Jacobi smoothing 157
3 Coarse-grid correction 164
3.1 The Laplacian 170
3.2 The Helmholtz operator 171
4 Two-grid iteration 177
4.1 The Laplacian 177
4.2 The Helmholtz operator 179
5 Numerical examples 187
5.1 Two-grid experiments 188
5.2 Multigrid experiments, complexity 190
6 Conclusions 194
Explicit local time-steppingmethods for time-dependent wave propagation 199
1 Introduction 199
2 Finite element discretizations for the wave equation 202
2.1 Continuous Galerkin formulation 202
2.2 Interior penalty discontinuous Galerkin formulation 203
2.3 Nodal discontinuous Galerkin formulation 205
3 Leap-frog-based LTS methods 206
3.1 Second-order method for undamped waves 207
3.2 Fourth-order method for undamped waves 211
3.3 Second-order leap-frog/Crank–Nicolson-basedmethod for damped waves 214
4 Adams–Bashforth-based LTS methods for damped waves 217
5 Numerical results 223
5.1 Stability 223
5.2 Convergence 224
5.3 Two-dimensional example 226
6 Concluding remarks 227
Absorbing boundary conditions and perfectly matched layers in wave propagation problems 231
1 Introduction 231
2 ABC 232
2.1 Exact ABC 233
2.2 Approximation of the exact ABC 234
3 Plane waves analysis of an ABC 236
4 Perfectly matched layers 237
4.1 Helmholtz equation 238
4.2 The wave equation 240
5 Computation of the reflection coefficient of a PML 241
6 Conclusion 243
Dynamic inverse scattering 245
1 Introduction 245
2 Reconstruction of time-dependent pulses by the point-source method 248
3 Time-domain probe method (TDPM) 250
4 Orthogonality sampling 252
5 Dynamic inversion via data assimilation techniques 252
5.1 Three-dimensional variational data assimilation 254
5.2 Cycled probing and samplingmethod 256
5.3 Partial reconstruction matching scheme 257
6 Numerical examples 259
Boundary integral equations for Helmholtz boundary value and transmission problems 265
1 Introduction 265
2 Boundary integral equations 267
2.1 Boundary integral operators 267
2.2 Coercivity of boundary integral operators 270
2.3 Injectivity of boundary integral operators 272
2.4 Interior Robin boundary value problem 276
2.5 Boundary integral equations for exterior boundary value problems 278
3 Exterior Dirichlet boundary value problem 278
3.1 Direct boundary integral equations 279
3.2 Indirect boundary integral equations 288
3.3 Regularised combined boundary integral equations 291
4 Transmission problems 292
4.1 Steklov–Poincaré operator equations 293
4.2 Combined boundary integral equations 297
5 Conclusions 302
Color plates 305
Index 321