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

通过 Krylov 子空间方法中扩展点的自适应和累积选择来降低误差控制模型阶数

基本信息

批准号：
256173540
负责人：
Professor Dr.-Ing. Boris Lohmann
金额：
--
依托单位：
Lehrstuhl für Regelungstechnik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2017-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/256173540?language=en
关键词：
Error controlled model order reduction

项目摘要

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.

该项目的目标是开发一个有效的Krylov子空间方法模型降阶，这是适合于自动简化，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。这类大型数学模型通常来自偏微分方程的空间离散化，它允许描述各种工程领域的动态系统;这一过程通常是不可缺少的仿真，控制和优化目的。然而，模型的尺寸随着对其准确性的要求的增加而增加;为了完成上述任务，模型的简化因此常常是不可避免的。在文献中已经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），这些方法表现出特定的优点和缺点。平衡截断，例如，具有先验误差界和系统属性的保护，而Krylov子空间方法需要更少的数值工作（在计算时间和存储方面），因此更实用的减少非常大的原始models.Starting从一个新的配方的近似误差，从减少的结果，该项目旨在弥补Krylov子空间方法的主要缺点。其中之一是可能的稳定性损失，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位移（或扩展点），这现在是通过迭代框架（“萨拉米技术”）以累积和自动的方式进行的;与IRKA等已建立的方法不同，该过程包括自适应确定降低的系统维数。最后，插值方法通常不提供关于所实现的近似质量的可靠信息。然而，Krylov子空间方法最近可获得关于共同系统范数的全局上界，并为至少某类状态空间模型提供严格的误差信息，在预期项目期间，这一概念将进一步发展为完整的模型降阶方法。主要目标是扩展点的自动和累积选择（无需与用户交互），最小化上界对真实误差的高估，对多变量（MIMO）情况下的推广以及二阶系统方法的定制。案例研究使用学术的例子，以及从联合项目的模型提供了新方法的验证，特别是在其工业适用性。