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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.

Inverse problems (Differential equations) --- Problemes inversos (Equacions diferencials)

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The once esoteric idea of embedding scientific computing into a probabilistic framework, mostly along the lines of the Bayesian paradigm, has recently enjoyed wide popularity and found its way into numerous applications. This book provides an insider’s view of how to combine two mature fields, scientific computing and Bayesian inference, into a powerful language leveraging the capabilities of both components for computational efficiency, high resolution power and uncertainty quantification ability. The impact of Bayesian scientific computing has been particularly significant in the area of computational inverse problems where the data are often scarce or of low quality, but some characteristics of the unknown solution may be available a priori. The ability to combine the flexibility of the Bayesian probabilistic framework with efficient numerical methods has contributed to the popularity of Bayesian inversion, with the prior distribution being the counterpart of classical regularization. However, the interplay between Bayesian inference and numerical analysis is much richer than providing an alternative way to regularize inverse problems, as demonstrated by the discussion of time dependent problems, iterative methods, and sparsity promoting priors in this book. The quantification of uncertainty in computed solutions and model predictions is another area where Bayesian scientific computing plays a critical role. This book demonstrates that Bayesian inference and scientific computing have much more in common than what one may expect, and gradually builds a natural interface between these two areas.

Bayesian statistical decision theory. --- Inverse problems (Differential equations) --- Numerical solutions. --- Numerical analysis --- Bayes' solution --- Bayesian analysis --- Statistical decision --- Estadística bayesiana --- Problemes inversos (Equacions diferencials) --- Solucions numèriques

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Image processing --- Inverse problems (Differential equations) --- Mathematics. --- Differential equations --- Processament d'imatges --- Problemes inversos (Equacions diferencials) --- Equacions diferencials --- Anàlisi d'imatges --- Processament de dades fotogràfiques --- Processament de la imatge --- Tractament d'imatges --- Tractament de la imatge --- Processament òptic de dades --- Processament digital d'imatges --- Visió per ordinador --- Lectors òptics --- Sistemes d'imatges

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Equacions diferencials --- Càlcul --- Funcions de Bessel --- Àlgebra diferencial --- Càlcul integral --- Càlcul operacional --- Equacions d'evolució --- Equacions de camp d'Einstein --- Equacions de Lagrange --- Equacions de Pfaff --- Equacions diferencials algebraiques --- Equacions diferencials ordinàries --- Equacions en derivades parcials --- Funcions de Green --- Funcions de Lyapunov --- Problemes de contorn --- Problemes inversos (Equacions diferencials) --- Teoria d'estabilitat (Matemàtica) --- Transformació de Laplace --- Differential equations. --- 517.91 Differential equations --- Differential equations

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Differential equations --- Equacions diferencials --- Càlcul --- Funcions de Bessel --- Àlgebra diferencial --- Càlcul integral --- Càlcul operacional --- Equacions d'evolució --- Equacions de camp d'Einstein --- Equacions de Lagrange --- Equacions de Pfaff --- Equacions diferencials algebraiques --- Equacions diferencials ordinàries --- Equacions en derivades parcials --- Funcions de Green --- Funcions de Lyapunov --- Problemes de contorn --- Problemes inversos (Equacions diferencials) --- Teoria d'estabilitat (Matemàtica) --- Transformació de Laplace --- 517.91 Differential equations