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Bayesian methods for inverse problems - lecture 2
Virtualconference
Authors : Dashti, Masoumeh (Author of the conference)
CIRM (Publisher )
35R30 - Inverse problems for PDE
60J10 - Markov chains (discrete-time Markov processes on discrete state spaces)
65C05 - Monte Carlo methods
65M12 - Stability and convergence of numerical methods (IVP of PDE)
65M32 - inverse problem
76D07 - Stokes and related (Oseen, etc.) flows
65C50 - Other computational problems in probability
Additional resources :
http://smai.emath.fr/cemracs/cemracs21/data/presentation-speakers/dashti.pdf
CIRM (Publisher )
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Abstract :
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.
Keywords : Inverse problems; bayesian inversion; MAP estimators; Markov chain Monte Carlo
MSC Codes : Keywords : Inverse problems; bayesian inversion; MAP estimators; Markov chain Monte Carlo
35R30 - Inverse problems for PDE
60J10 - Markov chains (discrete-time Markov processes on discrete state spaces)
65C05 - Monte Carlo methods
65M12 - Stability and convergence of numerical methods (IVP of PDE)
65M32 - inverse problem
76D07 - Stokes and related (Oseen, etc.) flows
65C50 - Other computational problems in probability
Additional resources :
http://smai.emath.fr/cemracs/cemracs21/data/presentation-speakers/dashti.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 16/08/2021 Conference Date : 19/07/2021 Subseries : Research School arXiv category : Statistics Theory ; Probability Mathematical Area(s) : PDE ; Probability & Statistics Format : MP4 (.mp4) - HD Video Time : 01:30:35 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2021-07-19_Dashti_2.mp4 
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Information on the Event
Event Title : 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 dimensionEvent Organizers : Ehrlacher, Virginie ; Lombardi, Damiano ; Mula Hernandez, Olga ; Nobile, Fabio ; Taddei, Tommaso Dates : 19/07/2021 - 23/07/2021 Event Year : 2021 Event URL : https://conferences.cirm-math.fr/2412.html
Citation Data
DOI : 10.24350/CIRM.V.19779903Cite this video as: Dashti, Masoumeh (2021). Bayesian methods for inverse problems - lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19779903 URI : http://dx.doi.org/10.24350/CIRM.V.19779903 |
See Also
- [Multi angle] Bayesian data assimilation and filtering - lecture 2 / Author of the conference Schillings, Claudia.
- [Multi angle] How to compute transition times? / Author of the conference Lelièvre, Tony.
- [Multi angle] Coupling models instead of coupling codes: two toy examples / Author of the conference Lafitte, Olivier.
- [Virtualconference] Statistical theory for deep neural networks - lecture 2 / Author of the conference Schmidt-Hieber, Johannes.
- [Virtualconference] Statistical theory for deep neural networks - lecture 1 / Author of the conference Schmidt-Hieber, Johannes.
- [Multi angle] Approximation and learning with tree tensor networks - lecture 2 / Author of the conference Nouy, Anthony.
- [Multi angle] Approximation and learning with tree tensor networks - lecture 1 / Author of the conference Nouy, Anthony.
- [Multi angle] Theory of approximation of high-dimensional functions - lecture 2 / Author of the conference Cohen, Albert.
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
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