The Department of Mathematics at Virginia Tech

Colloquium April 3

Date: Friday, April 3

Time: 16:00 to 17:00

Place: 455 McBryde (Commons Room)

Speaker: Urike Baur of Technische Uni, Chemitz Germany

Title: Control-oriented model reduction using hierarchical matrix arithmetic

Abstract: We consider linear time-invariant (LTI) stable systems of the following form Σ :

x'(t)	=		Ax(t) + Bu(t),		t > 0, x(0) = x⁰
y(t)	=		Cx(t) + Du(t),		t≥ 0

with A∈R^n×n, B∈R^n×m, and C∈R^p×n arising, e.g., from the discretization and linearization of parabolic PDEs. We will assume that the system Σ is large-scale, having order n = O(10⁵) with n > > m, p. We allow the system matrix A to be dense, provided that a data-sparse representation exists.

In practical applications, e.g. in VLSI chip design and in control of PDEs, the associated large-scale LTI systems must be solved very often. A reduction of the state-space dimension n is absolutely necessary to attack these problems in reasonable time.

To preserve important system properties as stability and to obtain a global error bound, we focus on model order reduction methods based on balanced truncation (BT), the most commonly used technique in control theory. The major computational task in BT and related methods is the solutions of two large-scale Lyapunov equations,

AP + PA^T + BB^T = 0, A^TQ + QA + C^TC = 0.

During the last 17 years, a lot of methods were derived for the solution of large-scale matrix equations. But all methods are of cubic complexity when applied to dense problems.

We have developed an iterative solver based on the sign function method which breaks this curse of dimensionality for the practically relevant class of data-sparse systems. By using data-sparse matrix approximations, hierarchical (H) matrix formats, and the corresponding formatted arithmetic, we obtain efficient implementations of BT and related methods having all linear-polylogarithmic complexity. We will demonstrate the accuracy and applicability of the derived methods on large-scale systems coming from 2D and 3D FEM and BEM discretizations.

Return to