Large Scale Inverse Problems (eBook)

Preface 5
Synergy of inverse problems and data assimilation techniques 11
1 Introduction 12
2 Regularization theory 16
3 Cycling, Tikhonov regularization and 3DVar 18
4 Error analysis 22
5 Bayesian approach to inverse problems 24
6 4DVar 29
7 Kalman filter and Kalman smoother 33
8 Ensemble methods 39
9 Numerical examples 44
9.1 Data assimilation for an advection-diffusion system 44
9.2 Data assimilation for the Lorenz-95 system 51
10 Concluding remarks 58
Variational data assimilation for very large environmental problems 65
1 Introduction 65
2 Theory of variational data assimilation 66
2.1 Incremental variational data assimilation 70
3 Practical implementation 72
3.1 Model development 72
3.2 Background error covariances 74
3.3 Observation errors 80
3.4 Optimization methods 83
3.5 Reduced order approaches 85
3.6 Issues for nested models 89
3.7 Weak-constraint variational assimilation 91
4 Summary and future perspectives 93
Ensemble filter techniques for intermittent data assimilation 101
1 Bayesian statistics 101
1.1 Preliminaries 102
1.2 Bayesian inference 105
1.3 Coupling of random variables 108
1.4 Monte Carlo methods 114
2 Stochastic processes 116
2.1 Discrete time Markov processes 117
2.2 Stochastic difference and differential equations 118
2.3 Ensemble prediction and sampling methods 122
3 Data assimilation and filtering 125
3.1 Preliminaries 125
3.2 SequentialMonte Carlo method 126
3.3 Ensemble Kalman filter (EnKF) 129
3.4 Ensemble transform Kalman–Bucy filter 132
3.5 Guided sequential Monte Carlo methods 136
3.6 Continuous ensemble transform filter formulations 137
4 Concluding remarks 142
Inverse problems in imaging 145
1 Mathematicalmodels for images 146
2 Examples of imaging devices 149
2.1 Optical imaging 149
2.2 Transmission tomography 149
2.3 Emission tomography 151
2.4 MR imaging 153
2.5 Acoustic imaging 153
2.6 Electromagnetic imaging 154
3 Basic image reconstruction 154
3.1 Deblurring and point spread functions 155
3.2 Noise 156
3.3 Reconstruction methods 157
4 Missing data and prior information 159
4.1 Prior information 159
4.2 Undersampling and superresolution 162
4.3 Inpainting 165
4.4 Surface imaging 168
5 Calibration problems 171
5.1 Blind deconvolution 172
5.2 Nonlinear MR imaging 173
5.3 Attenuation correction in SPECT 173
5.4 Blind spectral unmixing 174
6 Model-based dynamic imaging 175
6.1 Kinetic models 176
6.2 Parameter identification 178
6.3 Basis pursuit 180
6.4 Motion and deformation models 182
6.5 Advanced PDE models 184
The lost honor of l2-based regularization 191
1 Introduction 191
2 l1-based regularization 195
3 Poor data 198
4 Large, highly ill-conditioned problems 201
4.1 Inverse potential problem 201
4.2 The effect of ill-conditioning on L1 regularization 204
4.3 Nonlinear, highly ill-posed examples 208
5 Summary 210
List of contributors 214