Introduction ................................................... ix
1 Concepts ..................................................... 1
1.1 Inverse problems and ill-posedness ...................... 1
1.2 Regularizing inverse problems ........................... 2
1.3 Convex analysis ......................................... 8
1.4 Geometry of Banach spaces .............................. 12
1.5 Sparsity and iterated soft-shrinkage ................... 15
1.5.1 Contraction property and minimization scheme .... 17
1.5.2 Regularization property and convergence rates ... 22
2 Inexact operator evaluations ................................ 27
2.1 Adaptive operator evaluations: l < p ≤ 2 ............... 28
2.1.1 Adaptive soft-shrinkage iteration ............... 29
2.1.2 Approaching the minimizer ....................... 30
2.1.3 Regularization property ......................... 34
2.1.4 Convergence rates ............................... 38
2.2 Operator approximations ................................ 40
2.2.1 Basic assumptions and estimates ................. 40
2.2.2 Regularization property ......................... 43
2.3 Adaptive operator evaluations: p = 1 ................... 45
3 Morozov's discrepancy principle ............................. 53
3.1 Exact operator evaluations ............................. 54
3.1.1 Regularization property ......................... 55
3.1.2 Convergence rates ............................... 62
3.2 Adaptive operator evaluations .......................... 72
4 Numerical investigations .................................... 77
4.1 An inverse heat conduction problem ..................... 77
4.2 Numerical implementation ............................... 80
4.3 Discussion of numerical results ........................ 82
Concluding remarks ............................................. 89
Bibliography ................................................... 91
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