Efficient Solution of Algebraic Bernoulli Equations Using ℋ-Matrix Arithmetic

Abstract

The algebraic Bernoulli equation (ABE) has several applications in control and system theory, e.g., the stabilization of linear dynamical systems and model reduction of unstable systems arising from the discretization and linearization of parabolic partial differential equations (PDEs). As standard methods for the solution of ABEs are of limited use for large-scale systems, we investigate approaches based on the matrix sign function method. This includes the solution of a linear least-squares (LLS) problem. Due to the large-scale setting we propose to solve this LLS problem via normal equations. To make the whole approach applicable in the large-scale setting, we incorporate structural information from the underlying PDE model into the approach. By using data-sparse matrix approximations, hierarchical matrix formats, and the corresponding formatted arithmetic we obtain an efficient solver having linear-polylogarithmic complexity. The proposed solver computes a low-rank representation of the ABE solution.