Solution of Large Scale Matrix Equations
We investigate the numerical solution of large scale linear and quadratic, algebraic and differential matrix equations. A special focus here lies on the solution of large scale continuous time Sylvester, Lyapunov and Riccati equations as appearing in model reduction and linear quadratic regulator problems for semidiscretized PDEs. The solvers employed are based on the low rank alternating directions implicit iteration, as well as Krylov projection methods, and the outcome directly supports the development of our software library and MATLAB® toolbox M.E.S.S.
In this effort we investigate a broad range of matrix equations appearing in basically all other teams. We exploit the strong connection between the solutions of certain Sylvester equations and moment matching model order reduction to apply our speciallized solvers for so called sparse-dense Sylvester equations. For the different balancing based model reduction approaches we compute factored solutions of (differential) Lyapunov and Riccati equations. Moreover we develop structure preserving and -exploiting solvers for a number of Riccati equations appearing in the optimal control of multi-physics PDE control problems.
In collaboration with the scientific computing team we also develop accelerated solvers for matrix equations with dense coefficient matrices.