CEMRACS 2021: Data Assimilation and Model Reduction in High Dimensional Problems / CEMRACS 2021: Assimilation de données et réduction de modèle pour des problêmes en grande dimension

Collection CEMRACS 2021: Data Assimilation and Model Reduction in High Dimensional Problems / CEMRACS 2021: Assimilation de données et réduction de modèle pour des problêmes en grande dimension

Organizer(s) Ehrlacher, Virginie ; Lombardi, Damiano ; Mula Hernandez, Olga ; Nobile, Fabio ; Taddei, Tommaso

Date(s) 19/07/2021 - 23/07/2021

linked URL https://conferences.cirm-math.fr/2412.html

00:00:00 / 00:00:00

8 9

Bayesian methods for inverse problems - lecture 2

By Masoumeh Dashti

We consider the inverse problem of recovering an unknown parameter from a finite set of indirect measurements. We start with reviewing the formulation of the Bayesian approach to inverse problems. In this approach the data and the unknown parameter are modelled as random variables, the distribution of the data is given and the unknown is assumed to be drawn from a given prior distribution. The solution, called the posterior distribution, is the probability distribution of the unknown given the data, obtained through the Bayes rule. We will talk about the conditions under which this formulation leads to well-posedness of the inverse problem at the level of probability distributions. We then discuss the connection of the Bayesian approach to inverse problems with the variational regularization. This will also help us to study the properties of the modes of the posterior distribution as point estimators for the unknown parameter. We will also briefly talk about the Markov chain Monte Carlo methods in this context.

Information about the video

Date of recording 19/07/2021
Date of publication 16/08/2021
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19779903
Cite this video Dashti, Masoumeh (19/07/2021). Bayesian methods for inverse problems - lecture 2. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19779903
URL https://dx.doi.org/10.24350/CIRM.V.19779903

Domain(s)

Bibliography

S. Cotter, M. Dashti, and A. Stuart, Variational data assimilation using targetted random walks, International Journal for Numerical Methods in Fluids, 68 (2012), pp. 403–421. - https://doi.org/10.1002/fld.2510
M. Dashti and A. M. Stuart. The Bayesian Approach to Inverse Problems. In Handbook of Uncertainty Quantification, pages 311–428. 2017
M.Dashti, K.J.H.Law, A.M.Stuart, and J.Voss. MAP estimators and their consistency in Bayesian nonparametric inverse problems. Inverse Problems, 29(9):095017, 27, 2013 - https://doi.org/10.1088/0266-5611/29/9/095017
J. Latz. On the well-posedness of Bayesian inverse problems. SIAM/ASA Journal on Uncertainty Quantification, 8(1):451?482, 2020. - https://doi.org/10.1137/19M1247176
D. Sanz-Alonso, A. M. Stuart and A. Taeb, Inverse Problems and Data Assimilation, arXiv:1810.06191 - https://arxiv.org/abs/1810.06191
A. Stuart, Inverse Problems: A Bayesian Perspective, Acta Numerica, 19 (2010), pp. 451–559. - https://doi.org/10.1017/S0962492910000061

MSC codes

Document(s)

http://smai.emath.fr/cemracs/cemracs21/data/presentation-speakers/dashti.pdf

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