Computational Methods for Stochastic Eigenvalue Problems

这个项目涉及数值算法的发展，以有效地计算从物理过程的数学模型导出的方程的代数系统的特征值。特征值的研究和理解在许多工程应用中具有重要的基础意义。例子包括建筑物和桥梁的设计，其特征值对应于结构振动的共振频率，以及确定流动液体或气体的性质以确定流动是否湍流。这个项目的重点是在底层模型是随机的情况下解决特征值问题。当模型的组成部分，如结构中使用的材料的弹性特性或发生流动的介质的渗透率，不是确定的，而是作为随机变量处理时，就会出现这种情况。得到的特征值解本身也是随机变量。有了这样的解决方案，科学家和工程师就可以将新的概率方法纳入设计中，例如，通过使用新的数学技术来分析结构弯曲的可能性，并设计出预防方法。PI将在项目中使用的技术方法包括研究随机特征值问题的解决方案，以加深对动力系统稳定性和不确定性对数学模型的影响的理解。例如，众所周知，伪光谱分析揭示了传统线性稳定性分析所没有显示的稳定性方面，但很难使用伪光谱方法进行定量陈述。当伪光谱表明线性稳定过程不稳定时，PI期望能够评估线性稳定过程不稳定的概率，以及这种评估如何取决于模型随机成分的统计特性。他还将开发新的高效的计算算法来解决特征值问题，包括为随机伽辽金有限元离散和随机搭配方法设计的算法，以及为降阶模型设计的策略。

This project concerns the development of numerical algorithms for efficiently computing eigenvalues of algebraic systems of equations derived from mathematical models of physical processes. The study and understanding of eigenvalues is of fundamental importance in numerous engineering applications. Examples include the design of buildings and bridges, for which eigenvalues correspond to resonant frequencies at which structures vibrate, and identification of qualities of flowing liquids or gases that establish whether or not the flows are turbulent. The focus of the project is on solving eigenvalue problems in cases where the underlying model is stochastic. This scenario arises when components of the model, such as elastic properties of the materials used in structures or permeabilities of the media in which flows take place, are not known with certainty but instead are treated as random variables. The resulting eigenvalue solutions are themselves also random variables. Having such solutions will enable scientists and engineers to incorporate new probabilistic methods into design, for example, by using new mathematical techniques to analyze the likelihood that a structure will buckle and devise methods to prevent it.The technical approaches the PI will use in the project include the study of solutions of stochastic eigenvalue problems to develop an enhanced understanding of stability of dynamical systems and the impact of uncertainty on mathematical models. For example, it is known that pseudospectral analysis reveals aspects of stability not shown by traditional linear stability analysis, but it is difficult to use pseudospectral methods to make quantitative statements. The PI expects to be able to assess the probability of a linearly stable process being unstable when pseudospectra suggest it is, and how such an assessment depends on the statistical properties of the random components of the model. He will also develop new and efficient computational algorithms for solving the eigenvalue problems, including algorithms designed for stochastic Galerkin finite element discretizations and stochastic collocation methods, and strategies for reduced-order models.