Title: Numerical Solutions of Matrix Equations Arising in Dimension Reduction for Linear-Time-Invariant Systems in the Large-Scale Setting       

Speaker: Hung (Ryan) Nong, Rice University

Date/Time: Monday, January 12, 2009, 11:00 – 11:45 am

Location: CSRI Building, Room 279 (Sandia NM)

Brief Abstract: Advances in manufacturing enable the production of complex physical systems, which are commonly described as mathematical models. Demands of accuracy in computational results, as for constructional designs for example, and of speed in performance, as for electronic devices for example, result in sophisticated and large-scale mathematical models. It is crucial to understand how these models behave dynamically under certain constraints in time. In some cases, such as in wave surge forecasting or in weather prediction, this desire becomes essential. However, most of the time, the goal is out of reach due to a discrepancy between the moderate computational capability of the current technology and a desired superior one. Model order reduction (MOR) becomes indispensable since MOR techniques aim to produce low-dimensional systems that capture the same response characteristics as the originals while enabling substantial improvement in simulation time and resulting in greatly reduced storage requirements.

My thesis work is concerned with the existence and efficient computation of low-rank approximations to the solution of matrix equations in the large-scale setting. Matrix equations of interest include the Sylvester and Lyapunov equations, which play an important role in dimension reduction for linear-time-invariant systems. Current techniques to solve the Lyapunov equation include the Smith or Alternate Direction Implicit (ADI) method by Smith, Peaceman and Rachford, Penzl, Li and White, Gugercin et al. and Sabino, and the Approximate Power Iteration (API) by Hodel et al. The former has a well-understood and complete convergence theory but relies on shift parameters. A poor shift selection can result in very slow convergence in practice. The API method is parameter free and tends to be efficient in practice; however, there is little theoretical understanding of its convergence properties.

In this talk, I will introduce the Parameter Free ADI-like (PFADI) method proposed by Sorensen, which is based on a synthesis of the API and ADI methods. This algorithm uses the API method to obtain a basis update and then constructs an appropriate re-weighting of this basis to provide a factorization update that satisfies ADI-like convergence properties. At each step of the PFADI iteration, the solution of the Sylvester equation is required for the basis update. I will then present an elegant method to obtain the solution of the Sylvester equation via the solution of an eigenvalue problem which can be done effectively. The talk will end with more promising and new research directions in light of our results.

CSRI POC: Heidi Thornquist, (505) 284-8426