Optimal Control Problems with Linear Bellman Equations

该提案的目标是通过开发一种创新的方法，称为维分解方法，来解决随机动态系统建模和模拟中的一般随机特征值问题，从而在爱荷华大学实现计算力学领域的核心竞争力。所提出的工作将基于：(1)随机本征值问题一般复值本征解的低维近似的新分解方法；(2)用于本征解的概率特征的新的多点分解和单项预条件；以及(3)随机本征解的概率度量的解析梯度的新的设计灵敏度公式。拟议的研究是雄心勃勃和新颖的，与这一领域的大多数先前研究在根本上不同。将要开发的方法将解决高度非线性的输入-输出转换、无限数量的随机变量或场以及随机输入的任意大不确定性。由于解析推导的随机设计灵敏度的创新公式，动态系统的后续优化可以使用任何标准的基于梯度的算法来进行。这种分解方法将有助于解决工程和科学中大规模、多学科、随机的特征值问题，所提出的研究将对许多商业和工业应用具有重要意义，如民用、汽车和航空航天基础设施。潜在的工程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-刺激性；航空航天结构的疲劳耐久性；以及微电子和微电子机械系统的可靠性。除了工程学，潜在的应用还包括核物理、数论、计算生物学和计算金融学等。因此，这里提出的研究将对一些具有国家意义的领域产生积极影响。该项目创造的知识的转让和传播将通过继续与业界合作、在ASME会议上组织研讨会、期刊出版物、在主要会议和机构上的演讲和出版物以及学生教育来进行。与两个政府和工业实验室的伙伴关系将使该项目中开发的基本方法得以实施，以解决几个大规模的工业问题。教育目标包括从代表性不足的少数民族或女性群体中招聘一名博士生，利用该项目的软件工具升级爱荷华大学主要工程项目的课程，并撰写一本研究专著。

The objective of the proposal is to achieve a core competency in the field of computational mechanics at The University of Iowa through the development of an innovative method, referred to as the dimensional decomposition method, for solving a general random eigenvalue problem in modeling and simulation of stochastic dynamic systems. The proposed effort will be based on: (1) new decomposition method for lower-dimensional approximations of general complex-valued eigensolutions of random eigenvalue problems; (2) new multipoint decomposition and monomial preconditioner for probabilistic characteristics of eigensolutions; and (3) new design sensitivity formulation for analytic gradients of probabilistic measures of random eigensolutions. The proposed research is ambitious and novel, differing in fundamental ways from most prior research in this area. The methods to be developed will address highly nonlinear input-output transformations, an unlimited number of random variables or fields, and arbitrarily large uncertainty of random input. Due to innovative formulation of the analytically derived stochastic design sensitivities, subsequent optimization of dynamic systems can be conducted employing any standard gradient-based algorithm. The decomposition method will aid in solving large-scale, multidisciplinary, stochastic eigenvalue problems in engineering and science.The proposed research will be of significant benefit to numerous commercial and industrial applications, such as civil, automotive, and aerospace infrastructure. Potential engineering applications include analysis and design of civil structures; noise-vibration-harshness of ground vehicle systems; fatigue durability of aerospace structures; and reliability of microelectronics and micro-electro-mechanical systems. Beyond engineering, potential applications include nuclear physics, number theory, computational biology, and computational finance, among others. Therefore, the research proposed here will positively impact a number of areas of national significance. The transfer and dissemination of knowledge created by this project will take place through continued collaboration with industries, organization of symposia in ASME conferences, journal publications, presentations and publications at major conferences and institutions, and student education. Partnerships with two government and industrial laboratories will enable implementation of the basic methods developed in this project to resolve several large-scale industrial problems. The educational goals comprise recruitment of a Ph. D. student from underrepresented minority or women groups, implementation of software tools from this project in upgrading courses in The University of Iowa's principal engineering programs, and authoring a research monograph.