Low-rank approximate solutions to large-scale differential matrix Riccati equations
We consider large-scale continuous-time differential matrix Riccati equations. The two main approaches proposed in the literature are based on a splitting scheme or on Rosenbrock / Backward Differentiation Formula (BDF) methods. The approach we propose is based on the reduction of the problem dimension prior to integration. We project the initial problem onto an extended block Krylov subspace and obtain a low-dimensional differential matrix Riccati equation. The latter matrix differential problem is then solved by the BDF method and the solution obtained is used to reconstruct an approximate solution of the original problem. This process is repeated with increasing dimension of the projection subspace until achieving a chosen accuracy. We give some theoretical results and a simple expression of the residual allowing the implementation of a stop test in order to limit the dimension of the projection space. Some numerical experiments are reported.