A Multi-Grid Method for Generalized Lyapunov Equations

We present a multi-grid method for a class of structured generalized Lya-punov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial diierential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The eeciency of the method is demonstrated by numerical experiments.