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Multi angle Condition: the geometry of numerical algorithms - Lecture 2

Auteurs : Bürgisser, Peter (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
    A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
    In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.

    Codes MSC :
    15A12 - Conditioning of matrices
    65F10 - Iterative methods for linear systems
    65F35 - Matrix norms, conditioning, scaling (numerical linear algebra)
    65H10 - Systems of equations
    65K05 - Mathematical programming methods
    68Q15 - Complexity classes (hierarchies, relations among complexity classes, etc.)
    90C51 - Interior-point methods

      Informations sur la Vidéo

      Réalisateur : Hennenfent, Guillaume
      Langue : Anglais
      Date de publication : 26/01/17
      Date de captation : 17/01/17
      Collection : Research talks
      Format : MP4
      Durée : 01:32:30
      Domaine : Computer Science ; Numerical Analysis & Scientific Computing
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : http://videos.cirm-math.fr/2017-01-17_Burgisser_Part2.mp4

    Informations sur la rencontre

    Nom du congrès : French computer algebra days / Journées nationales de calcul formel
    Organisteurs Congrès : Brisebarre, Nicolas ; El Bacha, Carole ; Giorgi, Pascal ; Mezzarobba, Marc ; Moroz, Guillaume
    Dates : 16/01/17 - 20/01/17
    Année de la rencontre : 2017
    URL Congrès : http://conferences.cirm-math.fr/1584.html

    Citation Data

    DOI : 10.24350/CIRM.V.19110503
    Cite this video as: Bürgisser, Peter (2017). Condition: the geometry of numerical algorithms - Lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19110503
    URI : http://dx.doi.org/10.24350/CIRM.V.19110503


    Voir aussi

    Bibliographie

    1. Bürgisser, P., Cucker, F. (2013). Condition. The geometry of numerical algorithms. Berlin: Springer - http://dx.doi.org/10.1007/978-3-642-38896-5

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