M. Benner

Chemnitz University of Technology
Mathematics in Industry and Technology
Department of Mathematics ,Germany

mardi 8 juin, 09h30

Titre:
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ADI-based Galerkin-Methods for Algebraic Lyapunov and Riccati Equations

abstract:
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  The efficient numerical solution of large-scale Lyapunov equations and
  algebraic Riccati equations (AREs) is of fundamental importance for
  the efficient implementation of a variety of model reduction methods
  as well as for solving (optimal) control and stabilization
  problems for large-scale control problems.
  In recent years, significant progress has been made for solving
  large-scale Lyapunov equations and AREs for sparse or data-sparse
  coefficient matrices.  We will survey these developments and highlight
  in particular approaches based on the ADI iteration for Lyapunov
  equations with low-rank right-hand side.  If used as Lyapunov solver
  within the Newton-Kleinman framework for AREs, the Newton-ADI method
  results.  We will address recent approaches to improve on the
  convergence of the ADI iteration for Lyapunov equations by employing a
  cyclic Galerkin projection.  We will discuss the same idea for the
  quadratic ADI method for AREs.  Moreover, we will address two issues
  that arise often in practical applications: high-rank constant terms
  or indefinite quadratic terms in the ARE prevent the application of
  the usual Newton-Kleinman iteration: in both situations, the Lyapunov
  equations to be solved in each iteration step have high-rank
  right-hand side.  We will present recent ideas to overcome these
  difficulties.