Multivariate Interpolation Methods for Parametric Model Reduction (MIM4PMOR)

在工程学、经济学、自然科学和社会科学中，动力系统是物理过程建模和控制的主要工具。在许多领域，直接数值模拟(DNS)已经成为研究这些现象的丰富复杂性和设计过程的关键。由于基础数学模型的日益复杂，计算能力的增加无法与之相匹配，模型简化已成为促进甚至实现动态系统的仿真(特别是参数研究和不确定性分析)、控制和优化的不可或缺的工具。在这里，我们将重点介绍参数化模型，其中在降阶模型中将参数保留为符号量是必不可少的。我们将采取两种相辅相成的方法。第一种方法从经验数据出发，使用先进的内插技术，克服了标准投影方法的局限性。经验数据可以通过物理实验或由域名系统提供。第二种方法是基于模型的，利用双线性和线性变参数控制系统之间的联系来获得最优内插条件。我们的共同主题是构造满足内插和某些最优性条件的降阶模型。我们建议将这些技术建立在坚实的数学基础上，以确保准确性并推导出计算高效的模型简化算法。

Dynamical systems are a principal tool in modeling and control of physical processes in engineering, economics, the natural and social sciences. In many areas, direct numerical simulation (DNS) has become essential for studying the rich complexity of these phenomena and for the design process. Due to the increasing complexity of the underlying mathematical models, unmatched by the increase in computing power, model reduction has become an indispensable tool in order to facilitate or even enable simulation (in particular parameter studies and uncertainty analysis), control, and optimization of dynamical systems. Here, we will focus on parametrized models where the preservation of parameters as symbolic quantities in the reduced-order model is essential. We will pursue two complementary approaches. The first one starts from empirical data and employs advanced interpolation techniques, overcoming limitations of standard projection methods. The empirical data may be provided by physical experimentation or by DNS. The second approach is model-based and employs a connection between bilinear and linear parameter-varying control systems to derive optimal interpolation conditions. Our common theme is to construct reduced-order models satisfying interpolation and certain optimality conditions. We are proposing to put these techniques on firm mathematical foundations in order to assure accuracy and to derive computationally efficient model reduction algorithms.