Computational Methods for Stochastic Eigenvalue Problems

这个项目涉及有效地计算从物理过程的数学模型导出的代数方程组的特征值的数值算法的发展。本征值的研究和理解在众多的工程应用中具有重要意义。例如，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，以及确定流动的液体或气体的质量，以确定流动是否为湍流。该项目的重点是在基础模型是随机的情况下解决特征值问题。当模型的组成部分，如发生流动的介质的结构或渗透率中使用的材料的弹性性质，不确定地已知，而被视为随机变量时，就会出现这种情况。由此得到的特征值解本身也是随机变量。拥有这样的解决方案将使科学家和工程师能够将新的概率方法融入设计中，例如，通过使用新的数学技术来分析结构屈曲的可能性并设计防止它的方法。PI将在该项目中使用的技术方法包括研究随机特征值问题的解，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，伪谱分析揭示了传统线性稳定性分析所不能显示的稳定性方面，但很难使用伪谱方法进行定量陈述。PI期望能够评估当伪谱表明线性稳定过程不稳定的概率，以及这种评估如何依赖于模型的随机分量的统计特性。他还将开发解决本征值问题的新的高效计算算法，包括为随机Galerkin有限元离散和随机配置方法设计的算法，以及降阶模型的策略。

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.