A Structure Preserving Krylov Subspace Method for Large Scale Differential Riccati Equations

We propose a Krylov subspace approximation method for the symmetric differential Riccati equation $\dot{X} = AX + XA^T + Q + XSX$, $X(0)=X_0$. The method is based on projecting the large scale equation onto a Krylov subspace spanned by the matrix $A$ and the low rank factors of $X_0$ and $Q$. We prove that the method is structure preserving in a sense that it preserves two important properties of the exact flow, namely the positivity of the exact flow, and also the property of monotonicity under certain practically relevant conditions. We also provide theoretical a priori error analysis which shows a superlinear convergence of the method. This behavior is illustrated in the numerical experiments. Moreover, we carry out a derivation of an efficient a posteriori error estimate as well as discuss multiple time stepping combined with a cut of the rank of the numerical solution.