On a family of low-rank algorithms for large-scale algebraic Riccati equations
Christian Bertram, Heike Faßbender
TL;DR
The paper develops a unified, low-rank framework for large-scale continuous-time algebraic Riccati equations based on a block rational Arnoldi decomposition and block rational Krylov subspaces. It shows that the proposed family yields the same iterates as the classic Riccati-ADI methods (RADI, Cayley, qADI, inv) when using identical shifts, and presents two concrete algorithmic realizations, including a more efficient variant that leverages Sherman–Morrison–Woodbury for faster linear solves. The approach supports expanding the Krylov subspace with multiple shifts at once and extends naturally to generalized and nonsymmetric Riccati equations, with a projection view clarifying why these projections preserve the low-rank structure. Numerical experiments demonstrate that the new methods reproduce existing iterates and can outperform existing implementations in scenarios with large numbers of inputs relative to outputs. Overall, the work provides a flexible, parallelizable, and theoretically grounded pathway to compute low-rank CARE solutions in very large-scale settings, with substantial practical impact for control and model reduction tasks.
Abstract
In [3] it was shown that four seemingly different algorithms for computing low-rank approximate solutions $X_j$ to the solution $X$ of large-scale continuous-time algebraic Riccati equations (CAREs) $0 = \mathcal{R}(X) := A^HX+XA+C^HC-XBB^HX $ generate the same sequence $X_j$ when used with the same parameters. The Hermitian low-rank approximations $X_j$ are of the form $X_j = Z_jY_jZ_j^H,$ where $Z_j$ is a matrix with only few columns and $Y_j$ is a small square Hermitian matrix. Each $X_j$ generates a low-rank Riccati residual $\mathcal{R}(X_j)$ such that the norm of the residual can be evaluated easily allowing for an efficient termination criterion. Here a new family of methods to generate such low-rank approximate solutions $X_j$ of CAREs is proposed. Each member of this family of algorithms proposed here generates the same sequence of $X_j$ as the four previously known algorithms. The approach is based on a block rational Arnoldi decomposition and an associated block rational Krylov subspace spanned by $A^H$ and $C^H.$ Two specific versions of the general algorithm will be considered; one will turn out to be a rediscovery of the RADI algorithm, the other one allows for a slightly more efficient implementation compared to the RADI algorithm (in case the Sherman-Morrision-Woodbury formula and a direct solver is used to solve the linear systems that occur). Moreover, our approach allows for adding more than one shift at a time.