IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2020i1p34-d468173.html
   My bibliography  Save this article

Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)

Author

Listed:
  • Antonio Falcó

    (ESI International Chair@CEU-UCH, Departamento de Matemáticas, Física y Ciencias Tecnológicas, Universidad Cardenal Herrera-CEU, CEU Universities San Bartolomé 55, 46115 Alfara del Patriarca, Spain)

  • Lucía Hilario

    (ESI International Chair@CEU-UCH, Departamento de Matemáticas, Física y Ciencias Tecnológicas, Universidad Cardenal Herrera-CEU, CEU Universities San Bartolomé 55, 46115 Alfara del Patriarca, Spain)

  • Nicolás Montés

    (ESI International Chair@CEU-UCH, Departamento de Matemáticas, Física y Ciencias Tecnológicas, Universidad Cardenal Herrera-CEU, CEU Universities San Bartolomé 55, 46115 Alfara del Patriarca, Spain)

  • Marta C. Mora

    (Departamento de Ingeniería Mecánica y Construcción, Universitat Jaume I, Avd. Vicent Sos Baynat s/n, 12071 Castellón, Spain)

  • Enrique Nadal

    (Departamento de Ingeniería Mecánica y de Materiales, Universitat Politècnica de València Camino de Vera, s/n, 46022 Valencia, Spain)

Abstract

A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure. More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit geometric structure.

Suggested Citation

  • Antonio Falcó & Lucía Hilario & Nicolás Montés & Marta C. Mora & Enrique Nadal, 2020. "Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)," Mathematics, MDPI, vol. 9(1), pages 1-14, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:34-:d:468173
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/1/34/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/1/34/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. A. El Hamidi & H. Ossman & M. Jazar, 2017. "On the convergence of alternating minimization methods in variational PGD," Computational Optimization and Applications, Springer, vol. 68(2), pages 455-472, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.

      Corrections

      All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:34-:d:468173. See general information about how to correct material in RePEc.

      If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

      If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

      If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

      For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

      Please note that corrections may take a couple of weeks to filter through the various RePEc services.

      IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.