Chahlaoui, Younes
(2003)
*Low-rank approximation and model reduction.*
Doctoral thesis, University Catholic of Louvain, Louvain-La-Neuve, Belgium.

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## Abstract

The basic idea of model reduction is to represent a complex linear dynamical system by a much simpler one. This may refer to many different techniques, but in this dissertation we focus on projection-based model reduction of linear systems. The projection is based on the dominant eigen-spaces of energy functions for ingoing and outgoing signals of the system. These energy functions are called Gramians of the system and can be obtained as the solutions of Stein equations. When the system matrices are large and sparse, it is not obvious how to compute efficiently these solutions or their dominant eigen-spaces. In fact, direct methods ignore sparsity in the Stein equations and are not very attractive for parallelization. Their use is then limited if the state dimension N of the system is large. The complexity of these methods is roughly O(N3) floating point operations and they require about O(N2) words of memory. This thesis provides some new ideas of recursive projection-based model reduction for time-varying systems as well as time-invariant systems. We present three algorithms for the recursive computation of the projection. These algorithms combine ideas of two classical methods — namely Balanced Truncation and Krylov subspaces — to produce a low-rank approximation of the Gramians or the input/output map of the system. We show the practical relevance of our results with real world benchmark examples. We also present some new ideas for second order systems. Such systems have a special structure which one wants to preserve in the reduced order model. We show how to adapt our projection based method to such systems.

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