Preface ........................................................ xi
I Linear Inverse Problems ...................................... 1
1 Introduction ................................................. 3
2 Naive reconstructions and inverse crimes ..................... 7
2.1 Convolution ............................................. 7
2.2 Heat propagation ....................................... 14
2.3 Tomographic X-ray projection data ...................... 21
3 Ill-posedness in inverse problems ........................... 35
3.1 Forward map and Hadamard's conditions .................. 35
3.2 Ill-posedness of the backward heat equation ............ 36
3.3 Ill-posedness in the continuous case ................... 40
3.4 Regularized inversion .................................. 47
3.5 The SVD for matrices ................................... 49
3.6 SVD for the guiding examples ........................... 51
4 Truncated singular value decomposition ...................... 53
4.1 Minimum norm solution .................................. 53
4.2 Truncated SVD .......................................... 55
4.3 Measuring the quality of reconstructions ............... 56
4.4 TSVD for the guiding examples .......................... 57
5 Tikhonov regularization ..................................... 63
5.1 Classical Tikhonov regularization ...................... 63
5.2 Normal equations and stacked form ...................... 67
5.3 Generalized Tikhonov regularization .................... 69
5.4 Choosing the regularization parameter .................. 72
5.5 Large-scale implementation ............................. 78
6 Total variation regularization .............................. 83
6.1 What is total variation? ............................... 83
6.2 Quadratic programming .................................. 86
6.3 Sparsity-based parameter choice ........................ 88
6.4 Large-scale implementation ............................. 90
7 Besov space regularization using wavelets ................... 95
7.1 An introduction to wavelets ............................ 95
7.2 Besov spaces and wavelets .............................. 98
7.3 Using В111 regularization to promote sparsity .......... 99
8 Discretization-invariance .................................. 103
8.1 Tikhonov regularization and discretizations ........... 104
8.2 Total variation regularization and discretizations .... 107
8.3 Besov norm regularization and discretizations ......... 108
9 Practical X-ray tomography with limited data ............... 111
9.1 Sparse full-angle tomography .......................... 114
9.2 Limited-angle tomography .............................. 119
9.3 Low-dose three-dimensional dental X-ray imaging ....... 122
10 Projects ................................................... 131
10.1 Image deblurring ...................................... 132
10.2 Inversion of the Laplace transform .................... 133
10.3 Backward parabolic problem ............................ 133
II Nonlinear Inverse Problems ................................. 137
11 Nonlinear inversion ........................................ 139
11.1 Analysis of nonlinear Ill-posedness ................... 140
11.2 Nonlinear regularization .............................. 143
11.3 Computational inversion ............................... 144
11.4 Examples of nonlinear inverse problems ................ 145
12 Electrical impedance tomography ............................ 159
12.1 Applications of EIT ................................... 160
12.2 Derivation from Maxwell's equations ................... 162
12.3 Continuum model boundary measurements ................. 163
12.4 Nonlinearity of EIT ................................... 165
12.5 Ill-posedness of EIT .................................. 165
12.6 Electrode models ...................................... 170
12.7 Current patterns and distinguishability ............... 173
12.8 Further reading ....................................... 180
13 Simulation of noisy EIT data ............................... 185
13.1 Eigenvalue data for symmetric σ ....................... 185
13.2 Continuum model data and FEM .......................... 187
13.3 Complete electrode model and FEM ...................... 191
13.4 Adding noise to EIT data matrices ..................... 197
14 Complex geometrical optics solutions ....................... 199
14.1 Calderön's pioneering work ............................ 199
14.2 The  operator and its kin ............................ 203
14.3 CGO solutions for the Schrцdinger equation ............ 205
14.4 CGO solutions for the Beltrami equation ............... 215
15 A regularized D-bar method for direct EIT .................. 223
15.1 Reconstruction with infinite-precision data ........... 224
15.2 Regularization via nonlinear low-pass filtering ....... 231
15.3 Numerical solution of the boundary integral equation .. 234
15.4 Numerical solution of the D-bar equation .............. 237
15.5 Regularized reconstructions ........................... 242
16 Other direct solution methods for EIT ...................... 249
16.1 D-bar methods with approximate scattering transforms .. 249
16.2 Calderón's method ..................................... 259
16.3 The Astala-Päivärinta method .......................... 266
16.4 The enclosure method of Ikehata ....................... 277
17 Projects ................................................... 281
17.1 Enclosure method for EIT .............................. 281
17.2 The D-bar method with Born approximation .............. 283
17.3 Calderцn's method ..................................... 285
17.4 Inverse obstacle scattering ........................... 286
A Banach spaces and Hilbert spaces ........................... 291
В Mappings and compact operators ............................. 293
С Fourier transform and Sobolev spaces ....................... 297
С.1 Sobolev spaces on domains Ω   ....................... 297
C.2 Fourier series and spaces  s(∂Ω) ...................... 301
C.3 Traces of functions in m(Ω) .......................... 306
D Iterative solution of linear equations ..................... 307
Bibliography .................................................. 311
Index ......................................................... 349
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