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Low-rank approximate solutions to large-scale differential matrix Riccati equations

Volume 45 / 2018

Y. Guldoğan, M. Hached, K. Jbilou, M. Kurulay Applicationes Mathematicae 45 (2018), 233-254 MSC: Primary 65F10; Secondary 65F30. DOI: 10.4064/am2355-1-2018 Published online: 3 August 2018

Abstract

We consider large-scale continuous-time differential matrix Riccati equations. The two main approaches proposed in the literature are based on a splitting scheme or on Rosenbrock / Backward Differentiation Formula (BDF) methods. The approach we propose is based on the reduction of the problem dimension prior to integration. We project the initial problem onto an extended block Krylov subspace and obtain a low-dimensional differential matrix Riccati equation. The latter matrix differential problem is then solved by the BDF method and the solution obtained is used to reconstruct an approximate solution of the original problem. This process is repeated with increasing dimension of the projection subspace until achieving a chosen accuracy. We give some theoretical results and a simple expression of the residual allowing the implementation of a stop test in order to limit the dimension of the projection space. Some numerical experiments are reported.

Authors

  • Y. GuldoğanDepartment of Mathematical Engineering
    Yıldız Technical University
    Istanbul, Turkey
  • M. HachedLaboratoire P. Painlevé UMR 8524
    UFR de Mathématiques
    Université des Sciences
    et Technologies de Lille, IUT A
    59653 Villeneuve d’Ascq Cedex, France
  • K. JbilouUniversité Littoral Côte d’Opale
    LMPA, 50 Rue F. Buisson, Calais, France
    e-mail
  • M. KurulayDepartment of Mathematical Engineering
    Yıldız Technical University
    Istanbul, Turkey

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