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

项目目标是开发一种有效的Krylov子空间方法用于模型降阶，该方法适用于甚至非常大规模的状态空间模型的自动简化。它不需要用户干预，并确保符合给定的近似质量要求。这类大型数学模型通常是由偏微分方程的空间离散化产生的，它允许在各种工程领域描述动态系统；这一过程对于仿真、控制和优化来说是必不可少的。然而，随着对模型精度要求的增加，模型的尺寸也在增加；为了完成上述任务，模型的简化往往是不可避免的。文献中已经描述了许多用于此目的的方法（例如模态或平衡截断，POD和Krylov子空间方法），这些方法具有特定的优点和缺点。例如，平衡截断具有先验误差界限和系统属性的保留，而Krylov子空间方法需要较少的数值努力（关于计算时间和存储），因此对于缩减非常大的原始模型更实用。从一种新的近似误差公式开始，该项目旨在弥补Krylov子空间方法的主要缺点。其中可能的稳定性损失是可以通过（最佳）极点配置来避免的。其次，Krylov子空间方法需要选择所谓的移位（或扩展点），现在由迭代框架（“萨拉米技术”）以累积和自动的方式进行；与IRKA等已建立的方法不同，该过程包括自适应确定降维系统。最后，插值方法通常不能提供关于获得的近似质量的可靠信息。然而，对于Krylov子空间方法来说，关于公共系统规范的全局上界是新的可用的，并且至少为某类状态空间模型提供严格的错误信息。在预期的项目中，这一概念应进一步发展成为一个完整的模型简化方法。主要目标是自动和累积选择扩展点（无需与用户交互），最大限度地减少上界对真实误差的高估，对多变量（MIMO）情况的泛化以及二阶系统的方法定制。使用学术实例和联合项目模型的案例研究验证了新方法，特别是在工业应用方面。

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.