标题： 双线性迭代有理Krylov算法的稳定性分析 显示英文标题

标题： Stability Analysis of Bilinear Iterative Rational Krylov Algorithm

摘要： 一个现实世界的物理或抽象现象可以通过动力系统进行数学描述。 动力系统出现在许多应用中，如天气预测、空气质量数据同化、生物系统、工业最优冷却、液体流动模型等。 一般来说，这些动力系统在行为上是非线性的。 过去，已经对线性动力系统进行了大量研究。 在这里，我们研究一种特殊的非线性动力系统，称为双线性动力系统。 双线性动力系统是线性和非线性动力系统之间的桥梁。 来自不同物理应用的模型规模非常大。 使用这样的系统进行仿真成本很高，因此通常通过模型降阶得到一个简化模型，该模型能够复制原始完整模型的输入输出行为。 一种最近提出的用于双线性动力系统模型降阶的算法，双线性迭代有理Krylov算法（BIRKA），以最优的方式进行。 该算法需要求解非常大的线性方程组。 通常这些系统通过直接方法（例如，LU）求解，这非常昂贵。 更好的选择是迭代方法（例如，Krylov）。 然而，迭代方法在求解线性方程组时会引入误差，因为它们不是精确的。 它们将线性系统求解到一定的容差范围内。 本文的主要贡献是我们证明了BIRKA在不精确线性求解引入的误差方面是稳定的。 我们的结果通过实验得到了支持。 显示英文摘要

摘要： A real world physical or abstract phenomenon can be described mathematically by a dynamical system. Dynamical systems arise in many applications like weather prediction, air-quality data assimilation, biological systems, industrial optimal cooling, liquid flow model, etc. Generally these dynamical systems are nonlinear in behaviour. In the past, substantial work has been done on linear dynamical systems. Here, we work on a special case of nonlinear dynamical system called bilinear dynamical system. Bilinear dynamical systems are a bridge between linear and nonlinear dynamical systems. Models coming from different physical applications are very large in size. Simulation with such systems is expensive so one usually obtains a reduced model (by model reduction) that replicates the input-output behaviour of the original full model. A recently proposed algorithm for model reduction of bilinear dynamical systems, Bilinear Iterative Rational Krylov Algorithm (BIRKA), does so in an optimal way. The algorithm requires solving very large linear systems of equations. Usually these systems are solved by direct methods (e.g., LU), which are very expensive. A better choice is iterative methods (e.g., Krylov). However, iterative methods introduce errors in linear solves because they are not exact. They solve the linear system up to a certain tolerance. The main contribution of this work is that we prove that BIRKA is stable with respect to the error introduced by inexact linear solves. We support our result by experiments.