This book provides a comprehensive introduction to the techniques, tools and methods for inverse problems and data assimilation, designed for students, researchers and developers in mathematics, physics, engineering, acoustics, electromagnetics, meteorology, biology, environmental and other applied sciences.

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.

1 Introduction 2 Functional analytic tools 3 Approaches to regularization 4 A stochastic view of inverse problems 5 Dynamical systems inversion and data assimilation 6 Programming of numerical algorithms and useful tools 7 Neural field inversion and kernel reconstruction 8 Simulation of waves and fields 9 Nonlinear operators 10 Analysis: uniqueness, stability and convergence questions 11 Source reconstruction and magnetic tomography 12 Field reconstruction techniques 13 Sampling methods 14 Probe methods 15 Analytic continuation tests 16 Dynamical sampling and probe methods 17 Targeted observations and meta-inverse problems

This book provides a comprehensive introduction to the techniques, tools and methods for inverse problems and data assimilation. Inverse problems are widespread today in science and technology, and this book is written at the interface between mathematics and applications, designed 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, among others classical regularization and iterative methods, but also more recent approaches like sampling and probe methods, field reconstruction techniques in acoustics and electromagnetics and source reconstruction. In the framework of data assimilation, stochastic and deterministic methods are becoming more and more intertwined – both variational and ensemble data assimilation methods are introduced, as well as stochastic and deterministic viewpoints and analysis. Applications range from neural kernel reconstruction to inverse acoustic and electromagnetic scattering and thermography, from magnetic tomography for fuel cells or in a medical environment to weather forecasting and earth science. OCTAVE /MATLAB® codes are included, which serve as a first step towards simulations and more sophisticated inversion or data assimilation algorithms.