Analysis of Krylov subspace approximation to large-scale differential Riccati equations

Abstract

We consider a Krylov subspace approximation method for the symmetric differential Riccati equation$\dot{X} = AX + XA^T + Q - XSX$, $X(0)=X_0$. The method we consider is based on projectingthe 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 the sense that it preservestwo important properties of the exact flow, namely the positivity of the exact flowand also the property of monotonicity. We provide a theoretical a priori error analysis thatshows superlinear convergence of the method.Moreover, we derive an a posteriori error estimate that is shown to be effective in numerical examples.