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Subspace iteration and variants, revisited

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Auteurs : Saad, Yousef (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : Computing invariant subspaces is at the core of many applications, from machine learning to signal processing, and control theory, to name just a few examples. Often one wishes to com- pute the subspace associated with eigenvalues located at one end of the spectrum, i.e., either the largest or the smallest eigenvalues. In addition, it is quite common that the data at hand undergoes frequent changes and one is required to keep updating or tracking the target invariant subspace. The talk will present standard tools for computing invariant subspaces, with a focus on methods that do not require solving linear systems. One of the best known techniques for computing invariant subspaces is the subspace iteration algorithm [2]. While this algorithm tends to be slower than a Krylov subspace approach such as the Lanczos algorithm, it has many attributes that make it the method of choice in many applications. One of these attributes is its tolerance of changes in the matrix. An alternative framework that will be emphasized is that of Grassmann manifolds [1]. We will derive gradient-type methods and show the many connections that exist between different viewpoints adopted by practitioners, e.g., the TraceMin algorithm [3]. The talk will end with a few illustrative examples.

Keywords : Invariant subspaces; eigenvalues; Grassmann

Codes MSC :
15A18 - Eigenvalues, singular values, and eigenvectors
15A23 - Factorization of matrices
65F15 - Eigenvalues, eigenvectors

    Informations sur la Vidéo

    Réalisateur : Recanzone, Luca
    Langue : Anglais
    Date de publication : 26/11/2021
    Date de captation : 09/11/2021
    Sous collection : Research talks
    arXiv category : Numerical Analysis
    Domaine : Numerical Analysis & Scientific Computing
    Format : MP4 (.mp4) - HD
    Durée : 00:28:41
    Audience : Researchers
    Download : https://videos.cirm-math.fr/2021-11-9_Saad.mp4

Informations sur la Rencontre

Nom de la rencontre : Numerical Methods and Scientific Computing / Méthodes numériques et calcul scientifique
Organisateurs de la rencontre : Beckermann, Bernhard ; Brezinski, Claude ; da Rocha, Zélia ; Redivo-Zaglia, Michela ; Rodriguez, Giuseppe
Dates : 08/11/2021 - 12/11/2021
Année de la rencontre : 2021
URL Congrès : https://conferences.cirm-math.fr/2431.html

Données de citation

DOI : 10.24350/CIRM.V.19829703
Citer cette vidéo: Saad, Yousef (2021). Subspace iteration and variants, revisited. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19829703
URI : http://dx.doi.org/10.24350/CIRM.V.19829703

Voir aussi

Bibliographie

  • [1] EDELMAN, Alan, ARIAS, Tomás A., et SMITH, Steven T. The geometry of algorithms with orthogonality constraints. SIAM journal on Matrix Analysis and Applications, 1998, vol. 20, no 2, p. 303-353. - https://doi.org/10.1137/S0895479895290954

  • [2] SAAD, Yousef. Numerical methods for large eigenvalue problems, revised ed. SIAM, Philadelphia, 2011. -

  • [3] SAMEH, Ahmed H. et WISNIEWSKI, John A. A trace minimization algorithm for the generalized eigenvalue problem. SIAM Journal on Numerical Analysis, 1982, vol. 19, no 6, p. 1243-1259. - https://doi.org/10.1137/0719089



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