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Bayesian methods for inverse problems - lecture 2

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Virtualconference
Auteurs : Dashti, Masoumeh (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : 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

Codes MSC :
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

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

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de publication : 16/08/2021
    Date de captation : 19/07/2021
    Sous collection : Research School
    arXiv category : Statistics Theory ; Probability
    Domaine : PDE ; Probability & Statistics
    Format : MP4 (.mp4) - HD
    Durée : 01:30:35
    Audience : Researchers
    Download : https://videos.cirm-math.fr/2021-07-19_Dashti_2.mp4

Informations sur la Rencontre

Nom de la rencontre : 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
Organisateurs de la rencontre : Ehrlacher, Virginie ; Lombardi, Damiano ; Mula Hernandez, Olga ; Nobile, Fabio ; Taddei, Tommaso
Dates : 19/07/2021 - 23/07/2021
Année de la rencontre : 2021
URL Congrès : https://conferences.cirm-math.fr/2412.html

Données de citation

DOI : 10.24350/CIRM.V.19779903
Citer cette vidéo: 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

Voir aussi

Bibliographie

  • 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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