High-Dimensional Stochastic Design Optimization by Spline Dimensional Decomposition

该奖项将通过建立复杂工程系统和产品开发的新方法，为国家繁荣做出贡献。现实生活中的工程系统和产品的设计涉及到施加的力、材料特性和制造过程中的不确定性。如果没有适当的考虑，这些不确定性会直接影响产品的预期性能，导致重大的收入损失甚至灾难性的失败。该奖项支持在明确考虑不确定系统行为的同时发现最佳可能设计解决方案的设计优化基础研究。这些方法将导致计算工程设计工具的改进，这些工具将支持更耐用、更健壮和更可靠的商业产品的设计。在这个项目中开发的算法和工具将适用于广泛的多学科工程设计界。该奖项将支持一名研究生，而教育推广活动将有助于扩大未被充分代表的群体在研究和工程教育中的参与。本研究将为存在不确定性的高维复杂系统的设计优化建立良好的数学基础，创建稳健的数值算法，并构建实用的计算工具。这一努力将涉及：(1)新的多元正交基样条（b样条）和非均匀理性b样条（NURBS），导致一种新的样条维数分解（SDD）方法；(2)计算统计矩的SDD方法的新公式和可扩展算法，包括从分数函数估计矩的设计灵敏度；(3)新的计算上方便的稳健设计优化（RDO）算法，包括单个或最多几个随机模拟。这项研究是新颖的，首次推出了b样条和NURBS的随机版本。用于估计统计矩的新公式和可扩展算法将考虑不连续或非光滑随机响应。与分数函数的集成将从相同的计算量中同时确定随机响应特征和设计灵敏度，从而解决来自少数随机模拟的RDO问题。因此，设计过程的速度将大大提高，为大规模随机设计优化问题提供罕见或可能前所未有的解决方案。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award will contribute to national prosperity by establishing new methods for the development of complex engineered systems and products. Design of real-life engineered systems and products involves uncertainties in applied forces, material properties, and manufacturing processes. If not appropriately accounted for, these uncertainties can directly impact the anticipated performance of a product, resulting in a significant loss of revenue or even catastrophic failure. This award supports fundamental research on design optimization for discovering the best possible design solution while explicitly considering the uncertain system behavior. These methods will lead to improvements in computational engineering design tools that will support the design of commercial products that are more durable, robust and reliable. The algorithms and tools developed in this project will be applicable to a broad multidisciplinary engineering design community. The award will support a graduate student, and the educational outreach activities will help broaden participation of underrepresented groups in research and engineering education.This research will establish a sound mathematical foundation, create robust numerical algorithms, and build practical computational tools for design optimization of high-dimensional complex systems in the presence of uncertainty. This effort will involve: (1) new multivariate orthonormal basis splines (B-splines) and non-uniform rational B-splines (NURBS), leading to a new spline dimensional decomposition (SDD) method; (2) new formulae and scalable algorithms of the SDD method for calculating the statistical moments, including estimation of design sensitivities of moments from score functions; and (3) new computationally expedient robust design optimization (RDO) algorithms comprising a single or at most a few stochastic simulations. The research is novel, debuting the stochastic version of B-splines and NURBS for the very first time. New formulae and scalable algorithms generated for estimating the statistical moments will account for discontinuous or nonsmooth stochastic responses. The integration with score functions will concurrently determine both the stochastic response characteristics and design sensitivities from the same computational effort, thereby solving RDO problems from a few stochastic simulations. As a consequence, the speed of design process will be substantially enhanced, producing rare or potentially unprecedented solutions to large-scale stochastic design optimization problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.