Nonlinear multigrid for the solution of large-scale Riccati equations in low-rank and H-matrix format

The algebraic rnatrix Riccati equation AX+XA(T)-XFX+C=0, where matrices A, B,C, F is an element of R-n (x) (n) are given and a solution X is an element of R-n x n is sought, plays a fundamental role in optimal control problems. Large-scale systems typically appear if the constraint is described by a partial differential equation (PDE). We provide a nonlinear multigrid algorithm that computes the solution X in a data-sparse, low-rank format and has a complexity of O(n), subject to the condition that F and C are of low rank and A is the finite element or finite difference discretization of an elliptic PDE. We indicate how to generalize the method to H-matrices C, F and X that are only blockwise of low rank and thus allow a broader applicability with a complexity of O(n log(n)(p)), p being a small constant. The method can also be applied to unstructured and dense matrices C and X in order to solve the Riccati equation in O(n(2)). Copyright (c) 2008 John Wiley & Sons, Ltd.