Novel Computational Methods for Solving Random Eigenvalue Problems

该奖项的研究目标是开发创新的计算方法来解决在高维随机动态系统建模和仿真中经常遇到的随机特征值问题。提出的工作将包括：(1)一种新的多项式维分解（PDD）方法来预测随机特征解的统计矩和概率分布；(2)通过量化PDD和多项式混沌展开（PCE）方法的近似误差，获得坚实的理论基础；(3)新的乘法PDD、杂交和各自的误差分析。新的PDD方法将挑战或颠覆现有的计算思维，特别是通过解决高度非线性的输入输出转换、大量随机变量和随机输入的任意大不确定性。因此，长期存在的与维数诅咒相关的随机问题将在一定程度上得到缓解，对工程和科学具有积极的影响。交付成果包括快速可靠的随机方法和算法的开发，建模和仿真工具的生成，研究结果的文档，工程学生教育以及K-12学生的实践经验。如果成功，这项研究的结果将适用于广泛的工业应用，如民用、汽车和航空航天基础设施。潜在的应用包括土木结构的分析和设计；地面车辆系统的噪声-振动严酷性和耐撞性以及航空结构的疲劳耐久性。在工程之外，潜在的应用包括核物理，数论，计算生物学等，其中随机特征值分析起着至关重要的作用。该奖项创造的知识转移将通过组织与动力学相关的专题讨论会、同行评议的期刊出版物、在主要会议上的演讲、软件开发和学生教育来实现。教育目标包括从未被充分代表的少数群体中招收研究生，实施软件工具来升级爱荷华大学现有的课程，以及积极参与爱荷华州？为中学生和高中生开设的“领路项目”。

The research objective of this award is to develop innovative computational methods for solving random eigenvalue problems commonly encountered in modeling and simulation of high-dimensional stochastic dynamic systems. The proposed effort will involve: (1) a new polynomial dimensional decomposition (PDD) method for predicting the statistical moments and probability distributions of random eigensolutions; (2) a solid theoretical basis by quantifying approximation errors from the PDD and polynomial chaos expansion (PCE) methods; and (3) new multiplicative PDD, hybridization, and respective error analyses. The new PDD method will challenge or disrupt existing computational thinking, specifically by addressing highly nonlinear input-output transformations, large number of random variables, and arbitrarily large uncertainty of random input. Therefore, a long-standing stochastic problem associated with the curse of dimensionality will be alleviated to some degree, with positive ramifications in engineering and sciences. Deliverables include development of fast and reliable stochastic methods and algorithms, generation of modeling and simulation tools, documentation of research results, engineering student education, and hands-on experience for K-12 students.If successful, the results of this research will be applicable to a broad range of industrial applications, such as civil, automotive, and aerospace infrastructure. Potential applications include analysis and design of civil structures; noise-vibration-harshness and crashworthiness of ground vehicle systems; and fatigue durability of aerospace structures. Beyond engineering, potential applications include nuclear physics, number theory, computational biology, among others, where random eigenvalue analysis plays a vital role. The transfer of knowledge created by this award will take place through organization of dynamics-related symposia, peer-reviewed journal publications, presentations at major conferences, software development, and student education. Educational goals include graduate student recruitment from an underrepresented minority group, implementation of software tools to upgrade existing courses at The University of Iowa, and active participation in Iowa?s Project Lead the Way program for middle- and high-school students.