Geodesic
On the parameter selection problem in the Newton-ADI iteration for large-scale Riccati equations
Electronic transactions on numerical analysis, Tome 29 (2008).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The numerical treatment of linear-quadratic regulator (LQR) problems for parabolic partial differential equations (PDEs) on infinite-time horizons requires the solution of large-scale algebraic Riccati equations (AREs). The Newton-ADI iteration is an efficient numerical method for this task. It includes the solution of a Lyapunov equation by the alternating direction implicit (ADI) algorithm at each iteration step. Here, we study the selection of shift parameters for the ADI method. This leads to a rational min-max problem which has been considered by many authors. Since knowledge about the exact shape of the complex spectrum is crucial for computing the optimal solution, this is often infeasible for the large-scale systems arising from finite element discretization of PDEs. Therefore, several methods for computing suboptimal parameters are discussed and compared on numerical examples.
Classification : 15A24, 30E10, 65B99
Keywords: algebraic Riccati equation, Newton-ADI, shift parameters, Lyapunov equation, rational min-Max problem, zolotarev problem 
@article{ETNA_2008__29__a5,
     author = {Benner, Peter and Mena, Hermann and Saak, Jens},
     title = {On the parameter selection problem in the {Newton-ADI} iteration for large-scale {Riccati} equations},
     journal = {Electronic transactions on numerical analysis},
     publisher = {mathdoc},
     volume = {29},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2008__29__a5/}
}
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Benner, Peter; Mena, Hermann; Saak, Jens. On the parameter selection problem in the Newton-ADI iteration for large-scale Riccati equations. Electronic transactions on numerical analysis, Tome 29 (2008). http://geodesic.mathdoc.fr/item/ETNA_2008__29__a5/