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Description:
This thesis deals with the numerical solution approximation of large-scale (autonomous) differential Riccati equations. The first part of the thesis focuses on the differential Lyapunov equation. We recapitulate well-known explicit solution formulas and use them to motivate a Galerkin approach for the numerical solution approximation. For the trial space of the Galerkin method, we propose to use a system of orthonormal eigenvectors of the solution of the algebraic Lyapunov equation. We motivate our choice by estimating the projection error on the trial space using the Loewner partial order. Then, the Galerkin condition yields a system of a smaller order, which can be treated numerically more efficiently. Finally, we compare the proposed Galerkin approach with the BDF-ADI method in terms of accuracy and computational time in several numerical experiments. In the second part, we extend the proposed Galerkin method to the differential Riccati equation. First, we review the essential analytical properties of the solution of the differential Riccati equation. Then, we estimate the projection error of the solution of the differential Riccati equation using the Loewner partial order and, therefore, motivating a Galerkin approach based on a system of orthonormal eigenvectors of the solution of the algebraic Riccati equation. The Galerkin condition yields a small-scale differential Riccati equation. We recapitulate the Davison–Maki and the modified Davison–Maki method for the numerical solution of the small-scale differential Riccati equation. We compare the proposed Galerkin approach with different splitting methods in terms of accuracy and computing time in several numerical experiments. Furthermore, we discuss a possible extension of the Galerkin method to the case of non-zero initial conditions. ; Diese Arbeit befasst sich mit der numerischen Lösungsapproximation von großskaligen (autonomen) Riccati–Differentialgleichungen. Der erste Teil der Arbeit konzentriert sich auf die Lyapunov–Differentialgleichung. Wir ...