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有效的克雷洛夫子空间方法来减少模型阶数，该方法适用于自动简化甚至非常大规模的状态空间模型。它不需要用户干预，并确保符合给定的近似质量要求。这种大型数学模型通常由偏微分方程的空间离散化产生，它允许描述各种工程领域的动态系统；该过程对于模拟、控制和优化目的通常是必不可少的。然而，模型的维度随着对其准确性的要求的提高而增加；为了完成上述任务，模型的简化往往是不可避免的。文献中描述了用于此目的的许多方法（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保留系统属性的特点，而克雷洛夫子空间方法需要较少的数值工作（在计算时间和存储方面），因此对于约简非常大的原始模型来说更实用。该项目从约简产生的近似误差的新颖公式开始，旨在弥补克雷洛夫子空间方法的主要缺点。其中之一是可能失去稳定性，这可以通过（最佳）极点放置来避免。其次，克雷洛夫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执行；与 IRKA 等已建立的方法不同，该过程包括自适应确定减小的系统维度。最后，插值方法通常不能提供有关所实现的近似质量的可靠信息。然而，关于通用系统规范的全局上限新可用于克雷洛夫子空间方法，并为至少某一类状态空间模型提供严格的误差信息。在预期项目期间，这一概念应进一步发展为完整的模型简化方法。主要目标是扩展点的自动和累积选择（无需与用户交互）、最大程度地减少上限对真实误差的高估、对多变量（MIMO）情况的泛化以及二阶系统方法的定制。使用学术实例以及联合项目模型的案例研究验证了新方法，特别是其工业适用性。

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.