Title: A Framework for Reduced Order Modeling with Mixed Moment Matching and Peak Error Objectives

Speaker: Keith Santarelli, Sandia National Laboratories

Date/Time: Wednesday, January 28, 2009 1:00 – 2:00 pm

Location: CSRI Building, Room 90 (Sandia NM)

Brief Abstract: Traditional projection-based model order reduction (MOR) tools for linear systems generally exhibit the following tradeoff:  either the moments of the original and reduced order systems can be matched at a specified set of frequencies, or a bound on the error between the original and reduced order system responses can be made small.  In general, however, the problem of producing a reduced order system which has a guaranteed small error bound subject to a number of moment-matching constraints cannot be handled by typical MOR tools. 

In this talk, we present a new non-projection-based method of creating reduced order models for linear systems which minimizes an error bound subject to a set of moment matching constraints.  The method (which minimizes the L1 norm of the corresponding error system) can be written as a parametric linear program (LP) and, hence, can be solved using standard LP software packages.  Because all optimization is performed in the time domain, the methods we describe can be applied directly to infinite dimensional systems (e.g., systems with time delays) rather than being limited to finite order state-space descriptions.  Moreover, we argue that the way that error is measured (peak-to-peak error, as opposed to the standard measure of power-to-power error of SVD-based projection methods) is more suitable for simulation-based applications.

We begin by showing how the L1 norm minimization problem subject to moment-matching constraints can be represented as an LP whenever a reduced order model is constrained to be a linear combination of a fixed set of basis functions.  We then show how one particularly simple set of basis functions (the Ritz basis) can be used to produce reduced order models of arbitrary accuracy in the L1 norm.  The aforementioned technique will then be applied to two examples (one from the circuits world, and one from the solution of the one-dimensional heat equation) to illustrate the method's utility.