Table of Contents
Preface xiii
Acknowledgements xv
Author biographies xvi
1 Introduction 1-1
1.1 A tour through theory and applications 1-2
1.2 Types of inverse problems 1-14
1.2.1 The general inverse problem 1-15
1.2.2 Source problems 1-16
1.2.3 Scattering from obstacles 1-17
1.2.4 Dynamical systems inversion 1-19
1.2.5 Spectral inverse problems 1-21
Bibliography 1-22
2 Functional analytic tools 2-1
2.1 Normed spaces, elementary topology and compactness 2-1
2.1.1 Norms, convergence and the equivalence of norms 2-1
2.1.2 Open and closed sets, Cauchy sequences and completeness 2-5
2.1.3 Compact and relatively compact sets 2-7
2.2 Hilbert spaces, orthogonal systems and Fourier expansion 2-10
2.2.1 Scalar products and orthonormal systems 2-10
2.2.2 Best approximations and Fourier expansion 2-13
2.3 Bounded operators, Neumann series and compactness 2-19
2.3.1 Bounded and linear operators 2-19
2.3.2 The solution of equations of the second kind and the Neumann series 2-25
2.3.3 Compact operators and integral operators 2-27
2.3.4 The solution of equations of the second kind and Riesz theory 2-32
2.4 Adjoint operators, eigenvalues and singular values 2-33
2.4.1 Riesz representation theorem and adjoint operators 2-33
2.4.2 Weak, compactness of Hilbert spaces 2-36
2.4.3 Eigenvalues, spectrum and the spectral radius of an operator 2-38
2.4.4 Spectral theorem for compact self-adjoint operators 2-40
2.4.5 Singular value decomposition 2-46
2.5 Lax-Milgram and weak solutions to boundary value problems 2-48
2.6 The Fréchet derivative and calculus in normed spaces 2-50
Bibliography 2-55
3 Approaches to regularization 3-1
3.1 Classical regularization methods 3-1
3.1.1 Ill-posed problems 3-1
3.1.2 Regularization schemes 3-2
3.1.3 Spectral damping 3-4
3.1.4 Tikhonov regularization and spectral cut-off 3-7
3.1.5 The minimum norm solution and its properties 3-10
3.1.6 Methods for choosing the regularization parameter 3-14
3.2 The Moore-Penrose pseudo-inverse and Tikhonov regularization 3-20
3.3 Iterative approaches to inverse problems 3-22
3.3.1 Newton and quasi-Newton methods 3-23
3.3.2 The gradient or Landweber method 3-25
3.3.3 Stopping rules and convergence order 3-31
Bibliography 3-34
4 A stochastic view of inverse problems 4-1
4.1 Stochastic estimators based on ensembles and particles 4-1
4.2 Bayesian methods 4-4
4.3 Markov chain Monte Carlo methods 4-6
4.4 Metropolis-Hastings and Gibbs sampler 4-11
4.5 Basic stochastic concepts 4-14
Bibliography 4-19
5 Dynamical systems inversion and data assimilation 5-1
5.1 Set-up for data assimilation 5-3
5.2 Three-dimensional variational data assimilation (3D-VAR) 5-5
5.3 Four-dimensional variational data assimilation (4D-VAR) 5-8
5.3.1 Classical 4D-VAR 5-8
5.3.2 Ensemble-Based 4D-VAR 5-13
5.4 The Kalman filter and Kalman smoother 5-16
5.5 Ensemble Kalman filters (EnKFs) 5-22
5.6 Particle filters and nonlinear Bayesian data assimilation 5-29
Bibliography 5-33
6 Programming of numerical algorithms and useful tools 6-1
6.1 MATLAB or OCTAVE programming: the butterfly 6-1
6.2 Data assimilation made simple 6-4
6.3 Ensemble data assimilation in a nutshell 6-8
6.4 An integral equation of the first kind, regularization and atmospheric radiance retrievals 6-9
6.5 Integro-differential equations and neural fields 6-12
6.6 Image processing operators 6-15
Bibliography 6-18
7 Neural field inversion and kernel reconstruction 7-1
7.1 Simulating neural fields 7-4
7.2 Integral kernel reconstruction 7-8
7.3 A collocation method for kernel reconstruction 7-17
7.4 Traveling neural pulses and homogeneous kernels 7-20
7.5 Bi-orthogonal basis functions and integral operator inversion 7-23
7.6 Dimensional reduction and localization 7-26
Bibliography 7-30
8 Simulation of waves and fields 8-1
8.1 Potentials and potential operators 8-1
8.2 Simulation of wave scattering 8-11
8.3 The far field and the far field operator 8-15
8.4 Reciprocity relations 8-21
8.5 The Lax-Phillips method to calculate scattered waves 8-23
Bibliography 8-26
9 Nonlinear operators 9-1
9.1 Domain derivatives for boundary integral operators 9-1
9.2 Domain derivatives for boundary value problems 9-8
9.3 Alternative approaches to domain derivatives 9-11
9.3.1 The variational approach 9-11
9.3.2 Implicit function theorem approach 9-18
9.4 Gradient and Newton methods for inverse scattering 9-21
9.5 Differentiating dynamical systems: tangent linear models 9-27
Bibliography 9-30
10 Analysis: uniqueness, stability and convergence questions 10-1
10.1 Uniqueness of inverse problems 10-3
10.2 Uniqueness and stability for inverse obstacle scattering 10-4
10.3 Discrete versus continuous problems 10-7
10.4 Relation between inverse scattering and inverse boundary value problems 10-9
10.5 Stability of cycled data assimilation 10-14
10.6 Review of convergence concepts for inverse problems 10-18
10.6.1 Convergence concepts in stochastics and in data assimilation 10-19
10.6.2 Convergence concepts for reconstruction methods in inverse scattering 10-21
Bibliography 10-24
11 Source reconstruction and magnetic tomography 11-1
11.1 Current simulation 11-2
11.1.1 Currents based on the conductivity problem 11-2
11.1.2 Simulation via the finite integration technique 11-4
11.2 The Biot-Savart operator and magnetic tomography 11-8
11.2.1 Uniqueness and non-uniqueness results 11-12
11.2.2 Reducing the ill-posedness of the reconstruction by using appropriate subspaces 11-16
11.3 Parameter estimation in dynamic magnetic tomography 11-25
11.4 Classification methods for inverse problems 11-28
Bibliography 11-32
12 Field reconstruction techniques 12-1
12.1 Series expansion methods 12-2
12.1.1 Fourier-Hankel series for field representation 12-2
12.1.2 Field reconstruction via exponential functions with an imaginary argument 12-6
12.2 Fourier plane-wave methods 12-10
12.3 The potential or Kirsch-Kress method 12-12
12.4 The point source method 12-20
12.5 Duality and equivalence for the potential method and the point source method 12-27
Bibliography 12-29
13 Sampling methods 13-1
13.1 Orthogonality or direct sampling 13-2
13.2 The linear sampling method of Colt on and Kirsch 13-4
13.3 Kirsch's factorization method 13-10
Bibliography 13-17
14 Probe methods 14-1
14.1 The SSM 14-2
14.1.1 Basic ideas and principles 14-2
14.1.2 The needle scheme for probe methods 14-7
14.1.3 Domain sampling for probe methods 14-9
14.1.4 The contraction scheme for probe methods 14-10
14.1.5 Convergence analysis for the SSM 14-13
14.2 The probing method for near field data by Ikehata 14-16
14.2.1 Basic idea and principles 14-17
14.2.2 Convergence and equivalence of the probe and SSM 14-20
14.3 The multi-wave no-response and range test of Schulz and Potthast 14-21
14.4 Equivalence results 14-26
14.4.1 Equivalence of SSM and the no-response test 14-27
14.4.2 Equivalence of the no-response test and the range test 14-29
14.5 The multi-wave enclosure method of Ikehata 14-31
Bibliography 14-38
15 Analytic continuation tests 15-1
15.1 The range test 15-1
15.2 The no-response test of Luke-Potthast 15-6
15.3 Duality and equivalence for the range test and no-response test 15-30
15.4 Ikehata's enclosure method 15-11
15.4.1 Oscillating-decaying solutions 15-13
15.4.2 Identification of the singular points 15-18
Bibliography 15-20
16 Dynamical sampling and probe methods 16-1
16.1 Linear sampling method for identifying cavities in a heat conductor 16-2
16.1.1 Tools and theoretical foundation 16-4
16.1.2 Property of potential 16-14
16.1.3 The jump relations of K* 16-16
16.2 Nakamura's dynamical probe method 16-17
16.2.1 Inverse boundary value problem for heat conductors with inclusions 16-17
16.2.2 Tools and theoretical foundation 16-18
16.2.3 Proof of theorem 16.2.6 16-20
16.2.4 Existence of Runge's approximation functions 16-24
16.3 The time-domain probe method 16-26
16.4 The BC method of Belishev for the wave equation 16-29
Bibliography 16-35
17 Targeted observations and meta-inverse problems 17-1
17.1 A framework for meta-inverse problems 17-1
17.2 Framework adaption or zoom 17-7
17.3 Inverse source problems 17-8
Bibliography 17-13
Appendix A A-1
