Nonlinear Eigenvalue Problems: Theory and Numerics

非线性本征值问题出现在各种科学和工程应用中，例如机械系统的动态分析（其中本征值表示振动频率），流体力学中流动的线性稳定性，时滞系统的稳定性分析以及光子晶体的电子能带结构计算。这些问题提出了许多数学挑战。对一些人来说，缺乏基本的理论。对于其他数值方法则难以提供任何精度或在合理的时间内解决非常大的问题。（例如在微机电（MEMS）设备和超大型喷气式飞机中）意味着这些非线性特征问题通常条件较差（因此很难准确解决）同时还具有应当在数值方法中利用的代数结构，以便确保计算结果的物理意义。作为一个具体的例子，在柏林工业大学的一个项目中，对欧洲高速列车的声音和振动级进行建模，发现标准有限元软件包在计算解中没有提供正确的图形，直到在该项目中开发的线性代数技术在基础二次方程中发挥作用。（2度）特征值问题（见2004年11月SIAM新闻的封面文章）。在资助研究团队的帮助下，我将开发理论和方法，使新出现的特征值问题（例如，合理），更有效、更准确地解决存在的问题。该项目将利用矩阵多项式的结构保持变换的新概念和有理特征值问题的新的基于线性化的方法。对于一般的非线性特征值问题，我们将通过有理函数或多项式函数来设计非线性部分的良好近似，然后用后一问题的技术来处理。这项工作将通过提供算法和软件产生重大影响，这些算法和软件可以是开源的，也可以通过数值库分发，使这些问题的有效计算机解决方案成为可能。

Nonlinear eigenvalue problems arise in a wide variety of science and engineering applications, such as the dynamic analysis of mechanical systems (where the eigenvalues represent vibrational frequencies), the linear stability of flows in fluid mechanics, the stability analysis of time-delay systems, and electronic band structure calculations for photonic crystals. These problems present many mathematical challenges. For some there is a lack of underlying theory. For others numerical methods struggle to provide any accuracy or to solve very large problems in a reasonable time.The trend towards extreme designs (such as in micro-electromechanical (MEMS) devices and superjumbo jets) means that these nonlinear eigenproblems are often poorly conditioned (hence difficult to solve accurately) while also having algebraic structure that should be exploited in a numerical method in order to ensure physical meaning of the computed results. As a specific example, in a project at TU Berlin modelling the sound and vibration levels in European high-speed trains it was found that standard finite element packages provided no correct figures in the computed solutions until linear algebra techniques of the type to be developed in this project were brought into play in the underlying quadratic (degree 2) eigenvalue problem (see the cover article in SIAM News, Nov. 2004).With the help of the funded research team I will develop theory and methods that enable the solution of new classes of emerging eigenproblems (e.g., rational) and more efficient and more accurate solution of existing problems. The project will exploit the new concept of structure preserving transformations for matrix polynomials and a new linearization-based approach for rational eigenproblems. For the general nonlinear eigenproblem, we will to devise good approximations to the nonlinear parts by rational or polynomial functions that will then be handled with techniques for the latter problems.The work will have significant impact through the provision of algorithms and software, either open source or distributed through numerical libraries, that enables efficient computer solution of these problems.