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，用于解决高维随机动态系统建模和仿真中常见的随机特征值问题。所提出的努力将包括：（1）一个新的多项式维数分解（PDD）的方法来预测随机本征解的统计矩和概率分布;（2）一个坚实的理论基础，量化近似误差的PDD和多项式混沌展开（PCE）的方法;和（3）新的乘法PDD，杂交，和各自的错误分析。新的PDD方法将挑战或破坏现有的计算思维，特别是通过解决高度非线性的输入-输出转换，大量的随机变量和随机输入的任意大的不确定性。因此，一个长期存在的随机问题与维数灾难将在一定程度上得到缓解，在工程和科学的积极影响。这些成果包括开发快速可靠的随机方法和算法、生成建模和仿真工具、记录研究结果、工程专业学生教育以及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。潜在的应用包括民用结构的分析和设计;地面车辆系统的噪声-振动-粗糙度和耐撞性;以及航空航天结构的疲劳耐久性。除了工程，潜在的应用包括核物理，数论，计算生物学等，其中随机本征值分析起着至关重要的作用。该奖项所创造的知识转移将通过组织与动力学相关的研讨会、同行评审的期刊出版物、在重大会议上的演讲、软件开发和学生教育来进行。 教育目标包括从代表性不足的少数群体中招收研究生，实施软件工具来升级爱荷华州大学的现有课程，以及积极参与爱荷华州的活动。的项目带路计划，为初中和高中学生。

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.