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On the convergence of alternating minimization methods in variational PGD

Author

Listed:
  • A. El Hamidi

    (University of La Rochelle)

  • H. Ossman

    (University of La Rochelle
    Lebanese University)

  • M. Jazar

    (Lebanese University)

Abstract

The approximation of solutions to partial differential equations by tensorial separated representations is one of the most efficient numerical treatment of high dimensional problems. The key step of such methods is the computation of an optimal low-rank tensor to enrich the obtained iterative tensorial approximation. In variational problems, this step can be carried out by alternating minimization (AM) technics, but the convergence of such methods presents a real challenge. In the present work, the convergence of rank-one AM algorithms for a class of variational linear elliptic equations is studied. More precisely, we show that rank-one AM-sequences are in general bounded in the ambient Hilbert tensor space and are compact if a uniform non-orthogonality condition between iterates and the reaction term is fulfilled. In particular, if a rank-one AM-sequence is weakly convergent then it converges strongly and the common limit is a solution of the rank-one optimization problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:68:y:2017:i:2:d:10.1007_s10589-017-9920-y
    DOI: 10.1007/s10589-017-9920-y
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    References listed on IDEAS

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    1. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
    2. F. Tröltzsch & S. Volkwein, 2009. "POD a-posteriori error estimates for linear-quadratic optimal control problems," Computational Optimization and Applications, Springer, vol. 44(1), pages 83-115, October.
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    Cited by:

    1. 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.

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