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

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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.

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Authors and Affiliations

  1. Research Group Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106, Magdeburg, Germany

    Peter Benner

  2. Carl-Friedrich-Gauß-Fakultät, Institut Computational Mathematics AG Numerik, TU Braunschweig, 38092, Braunschweig, Germany

    Heike Faßbender

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  1. Peter Benner
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  2. Heike Faßbender
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Correspondence to Peter Benner.

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Communicated by Juan Manuel Peña.

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Benner, P., Faßbender, H. On the numerical solution of large-scale sparse discrete-time Riccati equations. Adv Comput Math 35, 119–147 (2011). https://doi.org/10.1007/s10444-011-9174-7

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  • DOI: https://doi.org/10.1007/s10444-011-9174-7

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