H 2 Condition: the geometry of numerical algorithms - Lecture 1

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

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analysis of algorithms condition number smoothed analysis ill-posed problems finite precision conjugate gradients eigenvalue distribution volume of tubes Poincaré formula of integral geometry Gaussian distribution eigenvalue computation questions of the audience

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.)
68W01 90C51 - Interior-point methods

    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 26/01/17
    Date de captation : 16/01/17
    Collection : Research talks ; Computer Science ; Numerical Analysis and Scientific Computing
    Format : MP4 (.mp4) - HD
    Durée : 01:32:36
    Domaine : Computer Science ; Numerical Analysis & Scientific Computing
    Audience : Chercheurs ; Doctorants , Post - Doctorants
    Download : https://videos.cirm-math.fr/2017-01-16_Burgisser_part1.mp4

Informations sur la rencontre

Nom de la rencontre : French computer algebra days / Journées nationales de calcul formel
Organisateurs de la rencontre : 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.19110103
Cite this video as: Bürgisser, Peter (2017). Condition: the geometry of numerical algorithms - Lecture 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19110103
URI : http://dx.doi.org/10.24350/CIRM.V.19110103

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