Abstract ...................................................... vit
Acknowledgments ................................................ ix
Introduction .................................................... 1
1 Inverse problems ............................................. 7
1.1 Ill-posed inverse problems .............................. 7
1.2 Classical regularization methods ........................ 9
1.2.1 Variational regularization methods .............. 11
1.2.2 Iterative regularization methods ................ 13
1.3 Incorporating sparsity constraints ..................... 16
1.3.1 Tikhonov regularization with sparsity
constraints ..................................... 16
1.3.2 Greedy iteration ................................ 18
1.4 Convex Junctionals ..................................... 19
2 Tikhonov regularization in Besov scales ..................... 23
2.1 Introduction ........................................... 23
2.2 Besov spaces and notation .............................. 26
2.3 Convergence rates in Besov spaces ...................... 29
2.3.1 Tikhonov regularization in Banach spaces ........ 29
2.3.2 Tikhonov regularization in Besov spaces ......... 30
2.4 Regularization in Besov scales ......................... 36
2.5 Examples ............................................... 40
2.6 Conclusion ............................................. 44
2 Tikhonov regularization in Besov scales ..................... 23
2.1 Introduction ........................................... 23
2.2 Besov spaces and notation .............................. 26
2.3 Convergence rates in Besov spaces ...................... 29
2.3.1 Tikhonov regularization in Banach spaces ........ 29
2.3.2 Tikhonov regularization in Besov spaces ......... 30
2.4 Regularization in Besov scales ......................... 36
2.5 Examples ............................................... 40
2.6 Conclusion ............................................. 44
3 Tikhonov regularization with sparsity constraints ........... 47
3.1 Introduction ........................................... 47
3.2 The ℓ 1 -penalized Tikhonov functional .................. 49
3.3 Stability and convergence rates ........................ 54
3.4 Beyond convergence rates: exact recovery ............... 55
3.4.1 Exact recovery for exact data ................... 56
3.4.2 Exact recovery in the presence of noise ......... 58
3.5 Conclusion ............................................. 66
4 Greedy solution by means of the orthogonal matching
pursuit ..................................................... 69
4.1 Introduction ........................................... 69
4.2 Exact recovery for exact data .......................... 72
4.3 Exact recovery in the presence of noise ................ 78
4.4 Conclusion ............................................. 85
5 Application of exact recovery conditions .................... 87
5.1 Introduction ........................................... 87
5.2 Mass spectrometry ...................................... 88
5.2.1 Introduction .................................... 88
5.2.2 Data model ...................................... 89
5.2.3 Resolution bounds for mass spectrometry ......... 90
5.2.4 Numerical examples .............................. 95
5.3 Digital holography ..................................... 97
5.3.1 Introduction .................................... 97
5.3.2 Data model ...................................... 98
5.3.3 Resolution bounds for digital holography ........ 99
5.3.4 Numerical examples ............................. 107
5.4 Conclusion ............................................ 108
Conclusion .................................................... 111
Bibliography .................................................. 115
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