Multi-level Substructuring Methods for Model Order Reduction

Beschreibung

This PhD thesis works out two ideas of combining model order reduction with techniques from other fields of numerical linear algebra. Since the local behaviour of many physical processes is basically linear they can be sufficiently modelled by linear, time-invariant, dynamical systems (LTI-systems). This includes for instance fluiddynamical processes, elastodynamical studies or electric circuits. Often the numerical treatment requires the application of model reduction methods which reduce the complexity of the LTI-systems. To this end the original system is replaced by a system with much less state space variables. As a matter of course the reduced order system should reproduce the fundamental input/output behaviour and important system characteristics. Nowadays there exist a variety of suitable methods such as SVD-based or Krylov-based algorithms. SVD-based methods determine global approximations of the LTI-system and are capable to preserve important systems characteristics. Moreover they provide computable a priori error bounds. However, their outstanding disadvantage is the high computational effort. Hence their application is restricted to relatively small systems in general. Krylov-based methods are local approximation methods which are in particular suitable for large LTI-systems for the price of the lack of global error bounds and preservation of system characteristics. Nevertheless, also their applicability is prevented if the order of the system is sufficiently large. To remedy this problem we combine model reduction methods with recent multi-level substructuring (MLS) techniques which are successfully used in very large eigenvalue computations. We derive multi-level substructuring for LTI systems both algebraically and in a continuous context. A general reduction algorithm which is capable to reduce very large LTI-systems is presented. Within this framework one is free to choose a reduction method. Therefore we benefit from the properties of SVD-based methods, particularly, but also strategies to apply Krylov-based methods are given. Furthermore we examine the error sources of the MLS reduction method and adjust the algorithm to systems which depend on higher order differential equations. The second part of this thesis adresses large eigenvalue problems which are perturbed by a rational term whose rank is much smaller compared to the problem dimension. Since it is in general preferable to develop problem specific algorithms we exploit the given form. Due to the special structure it is possible to solve a much smaller rational eigenvalue problem instead of the original one. But the application of appropriate nonlinear eigensolvers requires the approximation of parts of the eigenvalue problem by model reduction methods. In particular we show that it is possible to benefit from a variational characterization of the nonlinear eigenvalues in the symmetric case. Therefore it is guaranteed to find all eigenvalues of interest and we can apply very efficient eigensolvers such as safeguarded iteration.

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Frank Blömeling

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large-scale dynamical systems model oder reduction multi-level substructuring rationally perturbed eigenvalue problems

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