Kalman Filters: from Bayes to Inverse Problems

Welcome

This book contains a presentation of Kalman filters, from basics to nonlinear and ensemble filters. To understand these well, examples are provided in the form of jupyter notebooks. Then the notion of Bayesian inverse problems (BIP) is introduced. Finally, there is a detailed presentation of the use of the ensemble Kalman filter as a basis for the solution of inverse problems. This is denoted EKI, or ensemble Kalman inversion, following the magnificent work of Andrew Stuart and his collaborators.

This book is based upon a number of sources. The original Bayesian formulation for inverse problems (Dashti and Stuart 2015) was the basis for the later ensemble Kalman inversion, presented in a series of papers (Iglesias, Law, and Stuart 2013; Calvello, Reich, and Stuart 2022; Huang et al. 2022). Bayesian data assimilation is presented in detail in (Reich and Cotter 2015) and a general approach to Bayesian filtering can be found in (Särkkä and Svensson 2023).

In (Asch, Bocquet, and Nodet 2016) and (Law, Stuart, and Zygalakis 2015) the reader can find detailed presentations of Kalman filter approaches for data assimilation. In (Asch 2022) there are basic explanations of uncertainty quantification, inverse problems and their use for digital twins.

Author

Mark Asch is Emeritus Professor at the Université de Picardie Jules Verne in the mathematics department.

https://markasch.github.io/DT-tbx-v1/,

http://masch.perso.math.cnrs.fr/

Citation

Asch, Mark. Kalman Filters: from Bayes to Inverse Problems. Online (2024) https://markasch.github.io/kfBIPq/

@book{Asch2024
    title = {Kalman {F}ilters: from {B}ayes to {I}nverse {P}roblems},
    author = {Asch, Mark},
    url = {https://markasch.github.io/kfBIPq/},
    year = {2024},
    publisher = {Online}
}

License

This online book is frequently updated and edited. It’s content is free to use, licensed under a Creative Commons licence, and the code can be found on GitHub. A physical copy of the book will be available at a later date.

License: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

References

Asch, Mark. 2022. A Toolbox for Digital Twins: From Model-Based to Data-Driven. Philadelphia, PA: Society for Industrial; Applied Mathematics. https://doi.org/10.1137/1.9781611976977.

Asch, Mark, Marc Bocquet, and Maëlle Nodet. 2016. Data Assimilation: Methods, Algorithms, and Applications. Philadelphia, PA: Society for Industrial; Applied Mathematics. https://doi.org/10.1137/1.9781611974546.

Calvello, Edoardo, Sebastian Reich, and Andrew M. Stuart. 2022. “Ensemble Kalman Methods: A Mean Field Perspective.” arXiv (to appear in Acta Numerica 2025). http://arxiv.org/abs/2209.11371.

Carrillo, J. A., F. Hoffmann, A. M. Stuart, and U. Vaes. 2024a. “Statistical Accuracy of Approximate Filtering Methods.” https://arxiv.org/abs/2402.01593.

———. 2024b. “The Mean Field Ensemble Kalman Filter: Near-Gaussian Setting.” https://arxiv.org/abs/2212.13239.

Dashti, Masoumeh, and Andrew M. Stuart. 2015. “The Bayesian Approach to Inverse Problems.” In Handbook of Uncertainty Quantification, edited by Roger Ghanem, David Higdon, and Houman Owhadi, 1–118. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-11259-6_7-1.

Huang, Daniel Zhengyu, Jiaoyang Huang, Sebastian Reich, and Andrew M Stuart. 2022. “Efficient Derivative-Free Bayesian Inference for Large-Scale Inverse Problems.” Inverse Problems 38 (12): 125006. https://doi.org/10.1088/1361-6420/ac99fa.

Iglesias, Marco A, Kody J H Law, and Andrew M Stuart. 2013. “Ensemble Kalman Methods for Inverse Problems.” Inverse Problems 29 (4): 045001. https://doi.org/10.1088/0266-5611/29/4/045001.

James, G., D. Witten, T. Hastie, and R. Tibshirani. 2021. An Introduction to Statistical Learning with Applications in R. Second Edition. Springer-Verlag New York. https://doi.org/10.1007/978-1-0716-1418-1.

Law, Kody, Andrew Stuart, and Konstantinos Zygalakis. 2015. Data Assimilation: A Mathematical Introduction. Vol. 62. Texts in Applied Mathematics. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-20325-6.

Reich, Sebastian, and Colin Cotter. 2015. Probabilistic Forecasting and Bayesian Data Assimilation. Cambridge University Press.

Sanita Vetra-Carvalho, Lars Nerger, Peter Jan van Leeuwen, and Jean-Marie Beckers. 2018. “State-of-the-Art Stochastic Data Assimilation Methods for High-Dimensional Non-Gaussian Problems.” Tellus A: Dynamic Meteorology and Oceanography 70 (1): 1–43. https://doi.org/10.1080/16000870.2018.1445364.

Särkkä, S., and L. Svensson. 2023. Bayesian Filtering and Smoothing. 2nd ed. Institute of Mathematical Statistics Textbooks. Cambridge University Press. https://doi.org/10.1017/9781108917407.