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

该项目的目标是开发一种高效的Krylov子空间模型降阶方法，该方法适用于即使是超大规模状态空间模型的自动简化。它不需要用户干预，并确保符合给定的逼近质量要求。这类大型数学模型通常是由偏微分方程组的空间离散化产生的，允许描述不同工程领域的动态系统；这一过程通常是仿真、控制和优化目的不可缺少的。然而，模型的维度随着对其准确性的日益要求而增长；因此，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，模态法或平衡截断法、POD法和Krylov子空间法)，这些方法显示出特定的优点和缺点。例如，平衡截断具有先验误差界和保持系统性质的特点，而Krylov子空间方法需要较少的数值工作量(关于计算时间和存储)，因此对于非常大的原始模型的约简更实用。其中之一是可能会失去稳定性，这可以通过(最佳)极点配置来避免。其次，Krylov子空间方法需要选择所谓的移位(或扩展点)，这现在是由迭代框架(“Salami技术”)以累积和自动的方式执行的；与IRKA等已建立的方法不同，这一过程包括自适应地确定降低的系统维度。最后，插值法通常不能提供关于所实现的逼近质量的可靠信息。然而，对于Krylov子空间方法，关于公共系统规范的全局上界是新可用的，并且至少对于某一类状态空间模型提供了严格的误差信息。在预期的项目中，这一概念将进一步发展为一种完整的模型降阶方法。主要目标是自动和累积地选择扩展点(不与用户交互)、最小化上界对真实误差的高估、对多变量(MIMO)情况的推广以及二阶系统方法的定制化。使用学术实例的案例研究以及来自联合项目的模型提供了对新方法的验证，特别是在其工业适用性方面。

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.