Dimension Reduction of Large-Scale Systems: Proceedings of a by Peter Benner, Volker Mehrmann, Danny C. Sorensen

By Peter Benner, Volker Mehrmann, Danny C. Sorensen

In the earlier many years, version relief has develop into an ubiquitous software in research and simulation of dynamical platforms, keep an eye on layout, circuit simulation, structural dynamics, CFD, and plenty of different disciplines facing advanced actual types. the purpose of this booklet is to survey probably the most winning version relief equipment in instructional sort articles and to give benchmark difficulties from numerous program parts for checking out and evaluating present and new algorithms. because the mentioned tools have usually been constructed in parallel in disconnected program components, the purpose of the mini-workshop in Oberwolfach and its court cases is to make those principles to be had to researchers and practitioners from a lot of these diversified disciplines.

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Extra info for Dimension Reduction of Large-Scale Systems: Proceedings of a Workshop held in Oberwolfach, Germany, October 19-25, 2003

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GH03] The set of hierarchical matrices is defined by H(TI×I , k) := {M ∈ RI×I | rank (M |t×s ) ≤ k for all admissible leaves t × s of TI×I }. Submatrices of M ∈ H(TI×I , k) corresponding to inadmissible leaves are stored as dense blocks whereas those corresponding to admissible leaves are stored in factorized form as rank-k matrices, called Rk -format. 5 shows the H-matrix representation with k = 4 of the stiffness matrix of the FEM discretization for a 2D heat equation with distributed control and isolation boundary conditions using linear elements on a uniform mesh, resulting in n = 1024.

48 Peter Benner and Enrique S. : Mathematical Control Theory. Springer-Verlag, New York, NY, 2nd edition (1998). , I. : Truncated balanced realization of a stable non-minimal state-space system. Internat. J. Control, 46:4, 1319–1330 (1987). : Using PLAPACK: Parallel Linear Algebra Package. MIT Press, Cambridge, MA (1997). : Gramian based model reduction of large-scale dynamical systems. F. A. Watson, editors, Numerical Analysis 1999. Proc. 18th Dundee Biennial Conference on Numerical Analysis, pages 231–247, Chapman & Hall/CRC, London, UK (2000).

This is not true for Algorithm 4 although it does not completely break the O(n2 ) storage and O(n3 ) flops barriers. 2 it will be shown that by reducing the complexity of the first stage of Algorithm 4 down to O(n · q(log n)), where 1 Model Reduction Based on Spectral Projection Methods 29 q is a quadratic or cubic polynomial, it is possible to break this curse of dimensionality for certain problem classes. An analysis of Algorithm 4 reveals the following: assume that A is a full matrix with no further structure to be exploited, and define nco := max{rank (S) , rank (R)} n, where by abuse of notation “rank” denotes the numerical rank of the factors of the Gramians.

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