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A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part I

Author

Listed:
  • Giovanni Fasano

    (Department of Management, Università Ca' Foscari Venezia)

  • Massimo Roma

    (Dipartimento di Informatica e Sistemistica "A. Ruberti", Università Sapienza Roma)

Abstract

We propose a class of preconditioners, which are also tailored for symmetric linear systems from linear algebra and nonconvex optimization. Our preconditioners are specifically suited for large linear systems and may be obtained as by-product of Krylov subspace solvers. Each preconditioner in our class is identified by setting the values of a pair of parameters and a scaling matrix, which are user-dependent, and may be chosen according with the structure of the problem in hand. We provide theoretical properties for our preconditioners. In particular, we show that our preconditioners both shift some eigenvalues of the system matrix to controlled values, and they tend to reduce the modulus of most of the other eigenvalues. In a companion paper we study some structural properties of our class of preconditioners, and report the results on a significant numerical experience.

Suggested Citation

  • Giovanni Fasano & Massimo Roma, 2011. "A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part I," Working Papers 4, Venice School of Management - Department of Management, Università Ca' Foscari Venezia.
    • Handle: RePEc:vnm:wpdman:4
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    1. Giovanni Fasano & Massimo Roma, 2011. "A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part II," Working Papers 5, Venice School of Management - Department of Management, Università Ca' Foscari Venezia.
    2. Renato Leone & Giovanni Fasano & Massimo Roma & Yaroslav D. Sergeyev, 2020. "Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 554-589, August.
    3. G. Fasano, 2005. "Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 1: Theory," Journal of Optimization Theory and Applications, Springer, vol. 125(3), pages 523-541, June.
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    1. Giovanni Fasano & Massimo Roma, 2011. "A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part II," Working Papers 5, Venice School of Management - Department of Management, Università Ca' Foscari Venezia.

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    More about this item

    Keywords

    preconditioners; large indefinite linear systems; large scale nonconvex optimization; Krylov subspace methods;
    All these keywords.

    JEL classification:

    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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