Stochastic Optimization for Design under Uncertainty with Dependent Probability Measures

许多复杂的系统和工程结构都受到制造过程和运行环境中的不确定性的困扰。传统的设计方法依赖于启发式得出的安全系数，并且不能定量地考虑系统响应的统计变化。在这个项目中，主要研究人员将对存在统计相关不确定性的复杂系统的设计优化进行基础研究。考虑到系统的行为是不确定的，并且由相关的输入变量驱动，将开发新的方法来确定最佳设计方案。潜在的工程应用包括提高耐用性和耐撞性的地面车辆设计，民用和航空航天应用的抗疲劳和抗断裂设计，以及在恶劣环境下的微电子封装的可靠设计。除了工程，这项研究的结果将通过在能源、金融、管理、调度以及运输和物流等领域的潜在应用而造福美国经济和社会，在这些领域，不确定性下的优化发挥着至关重要的作用。这项研究涉及多个学科，包括工程学、计算机科学、数学和统计学。这个项目的目标是建立坚实的数学基础，设计有效的数值算法，并开发实用工具，用于以相关概率分布为特征的不确定性条件下的设计优化。这项工作将涉及(1)高维随机响应广义多项式维分解方法的新理论发展；(2)用于计算统计矩和可靠性的新公式和可扩展算法，以及随后的设计灵敏度分析；(3)形状和拓扑设计的新的基于可靠性的稳健优化算法。由于对展开系数的创新计算，无论随机设计问题的大小如何，广义分解方法都将被有效地实现。统计矩、可靠性分析和设计灵敏度的创新公式要求对所有可能的设计进行一次或至多几次随机模拟，这将显著加快优化过程，潜在地产生随机设计问题的突破性解决方案。

Many complex systems and engineering structures are plagued by uncertainties in manufacturing processes and operating environments. Conventional design approaches rely on heuristically derived safety factors and do not account quantitatively for the statistical variation of a system response. In this project, the principal investigator will conduct fundamental research on design optimization of complex systems in the presence of statistically dependent uncertainty. Novel methods will be developed to determine the best design alternative considering that the system behavior is uncertain and driven by dependent input variables. Potential engineering applications include ground vehicle design for improved durability and crashworthiness, fatigue- and fracture-resistant design for civil and aerospace applications, and reliable design of microelectronic packaging under harsh environments. Beyond engineering, the results from this research will benefit the U.S. economy and society through potential application in areas such as energy, finance, management, scheduling, and transportation and logistics, where optimization under uncertainty plays a vital role. This research is multi-disciplinary, encompassing several disciplines, including engineering, computer science, mathematics, and statistics. It will help broaden participation of underrepresented groups in research and positively impact engineering education.The objectives of this project are to build a solid mathematical foundation, devise efficient numerical algorithms, and develop practical tools for design optimization subject to uncertainty characterized by dependent probability distributions. The effort will involve (1) a new theoretical development of the generalized polynomial dimensional decomposition method for a high-dimensional stochastic response; (2) new formulae and scalable algorithms for calculating the statistical moments and reliability, followed by design sensitivity analysis; and (3) new reliability-based and robust optimization algorithms for shape and topology designs. Due to innovative calculation of the expansion coefficients, the generalized decomposition method will be efficiently implemented regardless of the size of the stochastic design problem. The innovative formulation of the statistical moment and reliability analyses and design sensitivities, which requires a single or at most a few stochastic simulations for all possible designs, will markedly accelerate the optimization process, potentially producing breakthrough solutions to stochastic design problems.