On the numerical solution of large-scale sparse discrete-time Riccati equations

Authors:
Peter Benner;Heike Faβbender
Affiliations:
Research Group Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany 39106;Carl-Friedrich-Gauβ-Fakultät, Institut Computational Mathematics AG Numerik, TU Braunschweig, Braunschweig, Germany 38092
Venue:
Advances in Computational Mathematics
Year:
2011

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Abstract

We discuss the numerical solution of large-scale discrete-time algebraic Riccati equations (DAREs) as they arise, e.g., in fully discretized linear-quadratic optimal control problems for parabolic partial differential equations (PDEs). We employ variants of Newton's method that allow to compute an approximate low-rank factor of the solution of the DARE. The principal computation in the Newton iteration is the numerical solution of a Stein (aka discrete Lyapunov) equation in each step. For this purpose, we present a low-rank Smith method as well as a low-rank alternating-direction-implicit (ADI) iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.