A clear and concise mathematical introduction to the subjects of inverse problems and data assimilation, and their inter-relations.

Daniel Sanz-Alonso (Author), Andrew Stuart (Author), Armeen Taeb (Author)

9781009414296, Cambridge University Press

Paperback, published 31 August 2023

221 pages
27 x 18 x 1.5 cm, 0.38 kg

This concise introduction provides an entry point to the world of inverse problems and data assimilation for advanced undergraduates and beginning graduate students in the mathematical sciences. It will also appeal to researchers in science and engineering who are interested in the systematic underpinnings of methodologies widely used in their disciplines. The authors examine inverse problems and data assimilation in turn, before exploring the use of data assimilation methods to solve generic inverse problems by introducing an artificial algorithmic time. Topics covered include maximum a posteriori estimation, (stochastic) gradient descent, variational Bayes, Monte Carlo, importance sampling and Markov chain Monte Carlo for inverse problems; and 3DVAR, 4DVAR, extended and ensemble Kalman filters, and particle filters for data assimilation. The book contains a wealth of examples and exercises, and can be used to accompany courses as well as for self-study.

Introduction
Part I. Inverse Problems: 1. Bayesian inverse problems and well-posedness
2. The linear-Gaussian setting
3. Optimization perspective
4. Gaussian approximation
5. Monte Carlo sampling and importance sampling
6. Markov chain Monte Carlo
Exercises for Part I
Part II. Data Assimilation: 7. Filtering and smoothing problems and well-posedness
8. The Kalman filter and smoother
9. Optimization for filtering and smoothing: 3DVAR and 4DVAR
10. The extended and ensemble Kalman filters
11. Particle filter
12. Optimal particle filter
Exercises for Part II
Part III. Kalman Inversion: 13. Blending inverse problems and data assimilation
References
Index.

Subject Areas: Numerical analysis [PBKS]