Inverse Modeling

Preface......Page 14
Acknowledgements......Page 16
 Roland Potthast......Page 17
ch18.pdf......Page 0
Chapter 1 Introduction......Page 19
1.1 A tour through theory and applications......Page 20
1.2 Types of inverse problems......Page 32
1.2.1 The general inverse problem......Page 33
1.2.2 Source problems......Page 34
1.2.3 Scattering from obstacles......Page 35
1.2.4 Dynamical systems inversion......Page 37
1.2.5 Spectral inverse problems......Page 39
 Bibliography......Page 40
2.1.1 Norms, convergence and the equivalence of norms......Page 43
2.1.2 Open and closed sets, Cauchy sequences and completeness......Page 47
2.1.3 Compact and relatively compact sets......Page 49
2.2.1 Scalar products and orthonormal systems......Page 52
2.2.2 Best approximations and Fourier expansion......Page 55
2.3.1 Bounded and linear operators......Page 61
2.3.2 The solution of equations of the second kind and the Neumann series......Page 67
2.3.3 Compact operators and integral operators......Page 69
2.3.4 The solution of equations of the second kind and Riesz theory......Page 74
2.4.1 Riesz representation theorem and adjoint operators......Page 75
2.4.2 Weak compactness of Hilbert spaces......Page 78
2.4.3 Eigenvalues, spectrum and the spectral radius of an operator......Page 80
2.4.4 Spectral theorem for compact self-adjoint operators......Page 82
2.4.5 Singular value decomposition......Page 88
2.5 Lax–Milgram and weak solutions to boundary value problems......Page 90
2.6 The Fréchet derivative and calculus in normed spaces......Page 92
 Bibliography......Page 97
3.1.1 Ill-posed problems......Page 98
3.1.2 Regularization schemes......Page 99
3.1.3 Spectral damping......Page 101
3.1.4 Tikhonov regularization and spectral cut-off......Page 104
3.1.5 The minimum norm solution and its properties......Page 107
3.1.6 Methods for choosing the regularization parameter......Page 111
3.2 The Moore–Penrose pseudo-inverse and Tikhonov regularization......Page 117
3.3 Iterative approaches to inverse problems......Page 119
3.3.1 Newton and quasi-Newton methods......Page 120
3.3.2 The gradient or Landweber method......Page 122
3.3.3 Stopping rules and convergence order......Page 128
 Bibliography......Page 131
4.1 Stochastic estimators based on ensembles and particles......Page 132
4.2 Bayesian methods......Page 135
4.3 Markov chain Monte Carlo methods......Page 137
4.4 Metropolis–Hastings and Gibbs sampler......Page 142
4.5 Basic stochastic concepts......Page 145
 Bibliography......Page 150
Chapter 5 Dynamical systems inversion and data assimilation......Page 151
5.1 Set-up for data assimilation......Page 153
5.2 Three-dimensional variational data assimilation (3D-VAR)......Page 155
5.3.1 Classical 4D-VAR......Page 158
5.3.2 Ensemble-based 4D-VAR......Page 163
5.4 The Kalman filter and Kalman smoother......Page 166
5.5 Ensemble Kalman filters (EnKFs)......Page 172
5.6 Particle filters and nonlinear Bayesian data assimilation......Page 179
 Bibliography......Page 183
http://dx.doi.org/10.1093/biomet/ast020\",\",......Page 184
6.1 MATLAB or OCTAVE programming: the butterfly......Page 185
6.2 Data assimilation made simple......Page 188
6.3 Ensemble data assimilation in a nutshell......Page 192
6.4 An integral equation of the first kind, regularization and atmospheric radiance retrievals......Page 193
6.5 Integro-differential equations and neural fields......Page 196
6.6 Image processing operators......Page 199
 Bibliography......Page 202
Chapter 7 Neural field inversion and kernel reconstruction......Page 203
7.1 Simulating neural fields......Page 206
7.2 Integral kernel reconstruction......Page 210
7.3 A collocation method for kernel reconstruction......Page 219
7.4 Traveling neural pulses and homogeneous kernels......Page 222
7.5 Bi-orthogonal basis functions and integral operator inversion......Page 225
7.6 Dimensional reduction and localization......Page 228
 Bibliography......Page 232
8.1 Potentials and potential operators......Page 234
8.2 Simulation of wave scattering......Page 244
8.3 The far field and the far field operator......Page 248
8.4 Reciprocity relations......Page 254
8.5 The Lax–Phillips method to calculate scattered waves......Page 256
 Bibliography......Page 259
9.1 Domain derivatives for boundary integral operators......Page 260
9.2 Domain derivatives for boundary value problems......Page 267
9.3.1 The variational approach......Page 270
9.3.2 Implicit function theorem approach......Page 277
9.4 Gradient and Newton methods for inverse scattering......Page 280
9.5 Differentiating dynamical systems: tangent linear models......Page 286
 Bibliography......Page 289
Chapter 10 Analysis: uniqueness, stability and convergence questions......Page 290
10.1 Uniqueness of inverse problems......Page 292
10.2 Uniqueness and stability for inverse obstacle scattering......Page 293
10.3 Discrete versus continuous problems......Page 296
10.4 Relation between inverse scattering and inverse boundary value problems......Page 298
10.5 Stability of cycled data assimilation......Page 303
10.6 Review of convergence concepts for inverse problems......Page 307
10.6.1 Convergence concepts in stochastics and in data assimilation......Page 308
10.6.2 Convergence concepts for reconstruction methods in inverse scattering......Page 310
 Bibliography......Page 313
Chapter 11 Source reconstruction and magnetic tomography......Page 315
11.1.1 Currents based on the conductivity problem......Page 316
11.1.2 Simulation via the finite integration technique......Page 318
11.2 The Biot–Savart operator and magnetic tomography......Page 322
11.2.1 Uniqueness and non-uniqueness results......Page 326
11.2.2 Reducing the ill-posedness of the reconstruction by using appropriate subspaces......Page 330
11.3 Parameter estimation in dynamic magnetic tomography......Page 339
11.4 Classification methods for inverse problems......Page 342
 Bibliography......Page 346
Chapter 12 Field reconstruction techniques......Page 347
12.1.1 Fourier–Hankel series for field representation......Page 348
12.1.2 Field reconstruction via exponential functions with an imaginary argument......Page 352
12.2 Fourier plane-wave methods......Page 356
12.3 The potential or Kirsch–Kress method......Page 358
12.4 The point source method......Page 366
12.5 Duality and equivalence for the potential method and the point source method......Page 373
 Bibliography......Page 375
Chapter 13 Sampling methods......Page 376
13.1 Orthogonality or direct sampling......Page 377
13.2 The linear sampling method of Colton and Kirsch......Page 379
13.3 Kirsch’s factorization method......Page 385
 Bibliography......Page 392
Chapter 14 Probe methods......Page 394
14.1.1 Basic ideas and principles......Page 395
14.1.2 The needle scheme for probe methods......Page 400
14.1.3 Domain sampling for probe methods......Page 402
14.1.4 The contraction scheme for probe methods......Page 403
14.1.5 Convergence analysis for the SSM......Page 406
14.2 The probing method for near field data by Ikehata......Page 409
14.2.1 Basic idea and principles......Page 410
14.2.2 Convergence and equivalence of the probe and SSM......Page 413
14.3 The multi-wave no-response and range test of Schulz and Potthast......Page 414
14.4 Equivalence results......Page 419
14.4.1 Equivalence of SSM and the no-response test......Page 420
14.4.2 Equivalence of the no-response test and the range test......Page 422
14.5 The multi-wave enclosure method of Ikehata......Page 424
 Bibliography......Page 431
15.1 The range test......Page 433
15.2 The no-response test of Luke–Potthast......Page 438
15.3 Duality and equivalence for the range test and no-response test......Page 442
15.4 Ikehata’s enclosure method......Page 443
15.4.1 Oscillating-decaying solutions......Page 445
15.4.2 Identification of the singular points......Page 450
 Bibliography......Page 452
Chapter 16 Dynamical sampling and probe methods......Page 453
16.1 Linear sampling method for identifying cavities in a heat conductor......Page 454
16.1.1 Tools and theoretical foundation......Page 456
16.1.2 Property of potential......Page 466
16.1.3 The jump relations of *......Page 468
16.2.1 Inverse boundary value problem for heat conductors with inclusions......Page 469
16.2.2 Tools and theoretical foundation......Page 470
16.2.3 Proof of theorem 16.2.6......Page 472
16.2.4 Existence of Runge’s approximation functions......Page 476
16.3 The time-domain probe method......Page 478
16.4 The BC method of Belishev for the wave equation......Page 481
 Bibliography......Page 487
17.1 A framework for meta-inverse problems......Page 490
17.2 Framework adaption or zoom......Page 496
17.3 Inverse source problems......Page 497
 Bibliography......Page 502
A.1 Basic notation and abbreviations......Page 503
A.3 Vector calculus......Page 505
A.4 Some further helpful analytic results......Page 506
 Bibliography......Page 510