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H 1 Compressed sensing and high-dimensional approximation: progress and challenges

Auteurs : Adcock, Ben (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : Many problems in computational science require the approximation of a high-dimensional function from limited amounts of data. For instance, a common task in Uncertainty Quantification (UQ) involves building a surrogate model for a parametrized computational model. Complex physical systems involve computational models with many parameters, resulting in multivariate functions of many variables. Although the amount of data may be large, the curse of dimensionality essentially prohibits collecting or processing enough data to reconstruct such a function using classical approximation techniques. Over the last five years, spurred by its successful application in signal and image processing, compressed sensing has begun to emerge as potential tool for surrogate model construction UQ. In this talk, I will give an overview of application of compressed sensing to high-dimensional approximation. I will demonstrate how the appropriate implementation of compressed sensing overcomes the curse of dimensionality (up to a log factor). This is based on weighted l1 regularizers, and structured sparsity in so-called lower sets. If time, I will also discuss several variations and extensions relevant to UQ applications, many of which have links to the standard compressed sensing theory. These include dealing with corrupted data, the effect of model error, functions defined on irregular domains and incorporating additional information such as gradient data. I will also highlight several challenges and open problems.

    Keywords : high-dimensional approximation; compressed sensing; structured sparsity; interpolation; uncertainty quantification

    Codes MSC :
    41A05 - Interpolation (approximations and expansions)
    41A10 - Approximation by polynomials
    65N12 - Stability and convergence of numerical methods (BVP of PDE)
    65N15 - Error bounds (BVP of PDE)
    94A12 - Signal theory (characterization, reconstruction, filtering, etc.)

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 28/11/2018
      Date de captation : 21/11/2018
      Collection : Research schools
      Format : MP4
      Durée : 01:00:34
      Domaine : Numerical Analysis & Scientific Computing
      Audience : Chercheurs ; Doctorants , Post - Doctorants ; Etudiants Science Cycle 2
      Download : https://videos.cirm-math.fr/2018-11-21_Adcok.mp4

    Informations sur la rencontre

    Nom de la rencontre : International traveling workshop on interactions between low-complexity data models and sensing techniques / Colloque international et itinérant sur les interactions entre modèles de faible complexité et acquis
    Organisateurs de la rencontre : Anthoine, Sandrine ; Boursier, Yannick ; Jacques, Laurent
    Dates : 19/11/2018 - 23/11/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1865.html

    Citation Data

    DOI : 10.24350/CIRM.V.19477203
    Cite this video as: Adcock, Ben (2018). Compressed sensing and high-dimensional approximation: progress and challenges. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19477203
    URI : http://dx.doi.org/10.24350/CIRM.V.19477203


    Voir aussi

    Bibliographie

    1. Adcock, B., Brugiapaglia, S., & Webster, C.G. (2018). Compressed sensing approaches for polynomial approximation of high-dimensional functions. In H. Boche, G. Caire, R. Calderbank, M. März, G. Kutyniok, R. Mathar (Eds.), Compresses sensing and its applications (pp. 93-124). Cham: Birkhäuser - http://dx.doi.org/10.1007/978-3-319-69802-1_3

    2. Adcock, B., Bao, A., Jakeman, J.D., & Narayan, A. (2018). Compressed sensing with sparse corruptions: fault-tolerant sparse collocation approximations. - https://arxiv.org/abs/1703.00135

    3. Adcock, B., & Huybrechs, D. (2018). Approximating smooth, multivariate functions on irregular domains. - https://arxiv.org/abs/1802.00602

    4. Adcock, B., Bao, A., & Brugiapaglia, S. (2017). Correcting for unknown errors in sparse high-dimensional function approximation. - https://arxiv.org/abs/1711.07622

    5. Adcock, B. (2017). Infinite-dimensional compressed sensing and function interpolation. Foundations of Computational Mathematics, 18(3), 661–701 - http://dx.doi.org/10.1007/s10208-017-9350-3

    6. Brugiapaglia, S., & Adcock, B. (2017). Robustness to unknown error in sparse regularization. - https://arxiv.org/abs/1705.10299

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