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Data Assimilation and Inverse Problems

Stuart, Andrew and Taeb, Armeen (2018) Data Assimilation and Inverse Problems. [Teaching Resource] (Unpublished)

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These notes are designed with the aim of providing a clear and concise introduction to the subjects of Inverse Problems and Data Assimilation, and their inter-relations, together with citations to some relevant literature in this area. The first half of the notes is dedicated to studying the Bayesian framework for inverse problems. Techniques such as importance sampling and Markov Chain Monte Carlo (MCMC) methods are introduced; these methods have the desirable property that in the limit of an infinite number of samples they reproduce the full posterior distribution. Since it is often computationally intensive to implement these methods, especially in high dimensional problems, approximate techniques such as approximating the posterior by a Dirac or a Gaussian distribution are discussed. The second half of the notes cover data assimilation. This refers to a particular class of inverse problems in which the unknown parameter is the initial condition of a dynamical system, and in the stochastic dynamics case the subsequent states of the system, and the data comprises partial and noisy observations of that (possibly stochastic) dynamical system. We will also demonstrate that methods developed in data assimilation may be employed to study generic inverse problems, by introducing an artificial time to generate a sequence of probability measures interpolating from the prior to the posterior.

Item Type:Teaching Resource
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URLURL TypeDescription Paper
Taeb, Armeen0000-0002-5647-3160
Alternate Title:Inverse Problems and Data Assimilation
Additional Information:These notes were developed out of Caltech course ACM 159 in Fall 2017. The notes were created in latex by the students in the class, based on lectures presented by the instructor Andrew Stuart, and on input from the course TA Armeen Taeb. The individuals responsible for the notes listed in alphabetic order are: Blancquart, Paul; Cai, Karena; Chen, Jiajie; Cheng, Richard; Cheng, Rui; Feldstein, Jonathan; Huang, De; Idíni, Benjamin; Kovachki, Nikola; Lee, Marcus; Levy, Gabriel; Li, Liuchi; Muir, Jack; Ren, Cindy; Seylabi, Elnaz, Schäfer, Florian; Singhal, Vipul; Stephenson, Oliver; Song, Yichuan; Su, Yu; Teke, Oguzhan; Williams, Ethan; Wray, Parker; Zhan, Eric; Zhang, Shumao; Xiao, Fangzhou. Furthermore, the following students have added content beyond the class materials: Parker Wray – the Overview, Jiajie Chen – alternative proof of Theorem 1.10 and proof idea for Theorem 14.3, Fangzhou Xiao – numerical simulation of prior, likelihood & posterior, Elnaz Seylabi & Fangzhou Xiao – catching typographical errors in a draft of these notes, Cindy Ren – numerical simulations to enhance understanding of importance sampling in Examples 6.2 and 6.5, Cindy Ren & De Huang – improving the constants in Theorem 6.3 regarding the approximation error of importance sampling, Richard Cheng & Florian Schäfer – illustrations to enhance understanding of the coupling argument used to study convergence of MCMC algorithms by presenting the finite state-space case, and Ethan Williams & Jack Muir – numerical simulations and illustrations of Ensemble Kalman Filter and Extended Kalman Filter. In addition to the students who developed the notes, we would also like to Tapio Helin (Helsinki) who used the notes in his own course and provided very helpful feedback on an early draft. The work of AS has been funded by the EPSRC (UK), ERC (Europe) and by AFOSR, ARL, NIH, NSF and ONR (USA). This funded research has helped to shape the presentation of the material here and is gratefully acknowledged. Warning: These are rough notes, far from being perfected. They are likely to contain mathematical errors, incomplete bibliographical information, inconsistencies in notation and typographical errors. We hope that the notes are nonetheless useful. Please contact Armeen Taeb at with any feedback from typos, through mathematical errors and bibliographical omissions, to comments on the structural organization of the material.
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Air Force Office of Scientific Research (AFOSR)UNSPECIFIED
Army Research LaboratoryUNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Record Number:CaltechAUTHORS:20190404-111038658
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:94461
Deposited By: George Porter
Deposited On:04 Apr 2019 18:21
Last Modified:03 Oct 2019 21:04

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