Model Reduction by a Cross-gramian Approach for Data-sparse Systems

We consider linear time-invariant (LTI) systems of the following form Σ : { ẋ(t) = Ax(t) + Bu(t), t > 0, x(0) = x, y(t) = Cx(t) + Du(t), t ≥ 0, with stable state matrix A ∈ Rn×n and B ∈ Rn×m, C ∈ Rp×n, D ∈ Rp×m, arising, e.g., from the discretization and linearization of parabolic PDEs. Typically, in practical applications, we have a large state-space dimension n = O(105) and a small input and output space, n À m, p. We further assume that the system is square, i.e., p = m. We show how to compute an approximate reduced-order system Σ̂ of order r ¿ n with a balancing-related model reduction method. The method is based on the computation of the cross-Gramian X, which is the solution of the Sylvester equation AX + XA + BC = 0. As standard algorithms for the solution of Sylvester equations are of limited use for large-scale systems, we investigate approaches based on the matrix sign function method [2]. To make this iterative method applicable in the large-scale setting, we propose a modified iteration scheme for computing lowrank factors of the solution X and 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 [1]. We show that the reduced-order model can then be computed from the low-rank factors directly. Note this continues the talk submitted by P. Benner.