Error-controlled model order reduction by adaptive and cumulative choice of expansion points in Krylov subspace methods

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The project goal is to develop an efficient Krylov subspace method for model order reduction, which is suited for the automatic simplification of even very large-scale state space models. It does not require intervention by the user and assures compliance with given requirements on the approximation quality.Large mathematical models of this kind typically result from the spatial discretization of partial differential equations, which allow for the description of dynamic systems in various engineering domains; this procedure is often indispensable for simulation, control and optimization purposes. The dimension of the model, however, grows with increasing demands on its accuracy; to complete the mentioned tasks, a simplification of the model is therefore frequently inevitable. Numerous methods for this purpose have been described in the literature (e.g. modal or balanced truncation, POD and Krylov subspace methods) which exhibit specific advantages and disadvantages. Balanced truncation, for instance, features a priori error bounds and preservation of system properties, while Krylov subspace methods require less numerical effort (with regard to computation time and storage) and are therefore more practical for the reduction of very large original models.Starting from a novel formulation of the approximation error that results from the reduction, the project aims to remedy the main drawbacks of Krylov subspace methods. Among those is the possible loss of stability, which can be avoided by (optimal) pole placement. Secondly, Krylov subspace methods require the choice of so-called shifts (or expansion points), which is now carried out by an iterative framework ("salami technique") in a cumulative and automatic manner; unlike established methods like IRKA, this procedure includes the adaptive determination of the reduced system dimension. Finally, interpolatory methods generally do not deliver reliable information on the achieved approximation quality. Global upper bounds with respect to common system norms are, however, newly available for Krylov subspace methods and deliver rigorous error information for at least a certain class of state space models.During the intended project, this concept shall be further developed into a complete model reduction method. The main goals are the automatic and cumulative choice of expansion points (without interaction with the user), the minimization of the overestimation of the true error by the upper bounds, the generalization towards the multi-variable (MIMO) case as well as the customization of the method for second order systems. Case studies using academic examples as well as models from joint projects provide the validation of the new method, in particular with respect to its industrial applicability.