Table of Contents

1. Introduction.
1.1. A tour through theory and applications.
1.2. Types of inverse problems 10. Analysis : uniqueness, stability and convergence questions.
10.1. Uniqueness of inverse problems.
10.2. Uniqueness and stability for inverse obstacle scattering.
10.3. Discrete versus continuous problems.
10.4. Relation between inverse scattering and inverse boundary value problems.
10.5. Stability of cycled data assimilation.
10.6. Review of convergence concepts for inverse problems 11. Source reconstruction and magnetic tomography.
11.1. Current simulation.
11.2. The Biot-Savart operator and magnetic tomography.
11.3. Parameter estimation in dynamic magnetic tomography.
11.4. Classification methods for inverse problems 12. Field reconstruction techniques.
12.1. Series expansion methods.
12.2. Fourier plane-wave methods.
12.3. The potential or Kirsch-Kress method.
12.4. The point source method.
12.5. Duality and equivalence for the potential method and the point source method 13. Sampling methods.
13.1. Orthogonality or direct sampling.
13.2. The linear sampling method of Colton and Kirsch.
13.3. Kirsch's factorization method 14. Probe methods.
14.1. The SSM.
14.2. The probing method for near field data by Ikehata.
14.3. The multi-wave no-response and range test of Schulz and Potthast.
14.4. Equivalence results.
14.5. The multi-wave enclosure method of Ikehata 15. Analytic continuation tests.
15.1. The range test.
15.2. The no-response test of Luke-Potthast.
15.3. Duality and equivalence for the range test and no-response test.
15.4. Ikehata's enclosure method 16. Dynamical sampling and probe methods.
16.1. Linear sampling method for identifying cavities in a heat conductor.
16.2. Nakamura's dynamical probe method.
16.3. The time-domain probe method.
16.4. The BC method of Belishev for the wave equation 17. Targeted observations and meta-inverse problems.
17.1. A framework for meta-inverse problems.
17.2. Framework adaption or zoom.
17.3. Inverse source problems.
Appendix A. 2. Functional analytic tools.
2.1. Normed spaces, elementary topology and compactness.
2.2. Hilbert spaces, orthogonal systems and Fourier expansion.
2.3. Bounded operators, Neumann series and compactness.
2.4. Adjoint operators, eigenvalues and singular values.
2.5. Lax-Milgram and weak solutions to boundary value problems.
2.6. The Fréchet derivative and calculus in normed spaces 3. Approaches to regularization.
3.1. Classical regularization methods.
3.2. The Moore-Penrose pseudo-inverse and Tikhonov regularization.
3.3. Iterative approaches to inverse problems 4. A stochastic view of inverse problems.
4.1. Stochastic estimators based on ensembles and particles.
4.2. Bayesian methods.
4.3. Markov chain Monte Carlo methods.
4.4. Metropolis-Hastings and Gibbs sampler.
4.5. Basic stochastic concepts 5. Dynamical systems inversion and data assimilation.
5.1. Set-up for data assimilation.
5.2. Three-dimensional variational data assimilation (3D-VAR).
5.3. Four-dimensional variational data assimilation (4D-VAR).
5.4. The Kalman filter and Kalman smoother.
5.5. Ensemble Kalman filters (EnKFs).
5.6. Particle filters and nonlinear Bayesian data assimilation 6. Programming of numerical algorithms and useful tools.
6.1. MATLAB or OCTAVE programming : the butterfly.
6.2. Data assimilation made simple.
6.3. Ensemble data assimilation in a nutshell.
6.4. An integral equation of the first kind, regularization and atmospheric radiance retrievals.
6.5. Integro-differential equations and neural fields.
6.6. Image processing operators 7. Neural field inversion and kernel reconstruction.
7.1. Simulating neural fields.
7.2. Integral kernel reconstruction.
7.3. A collocation method for kernel reconstruction.
7.4. Traveling neural pulses and homogeneous kernels.
7.5. Bi-orthogonal basis functions and integral operator inversion.
7.6. Dimensional reduction and localization 8. Simulation of waves and fields.
8.1. Potentials and potential operators.
8.2. Simulation of wave scattering.
8.3. The far field and the far field operator.
8.4. Reciprocity relations.
8.5. The Lax-Phillips method to calculate scattered waves 9. Nonlinear operators.
9.1. Domain derivatives for boundary integral operators.
9.2. Domain derivatives for boundary value problems.
9.3. Alternative approaches to domain derivatives.
9.4. Gradient and Newton methods for inverse scattering.
9.5. Differentiating dynamical systems : tangent linear models This book provides a comprehensive introduction to the techniques, tools and methods for inverse problems and data assimilation, and is written at the interface between mathematics and applications for students, researchers and developers in mathematics, physics, engineering, acoustics, electromagnetics, meteorology, biology, environmental and other applied sciences. Basic analytic questions and tools are introduced, as well as a wide variety of concepts, methods and approaches to formulate and solve inverse problems. OCTAVE /MATLAB codes are included, which serve as a first step towards simulations and more sophisticated inversion or data assimilation algorithms.