Partial Differential Equation Constrained Optimization: Algorithms, Numerics, and Applications

科学和工程中物理过程建模的最新技术使我们在存在约束的情况下解决日益复杂的优化问题。想想设计飞机的机翼，它必须能承受巨大的压力。这只是一个带有约束的优化问题的例子。由于约束，这类问题本质上是非线性的。这使得解决这些问题的算法和数值技术的开发和分析极具挑战性，但也很有价值。该项目将创建新的优化算法，使我们能够以系统和健壮的方式合并约束。目标应用范围从下一代微纳米级芯片实验室设备到新型材料和结构设计。开源软件将被创建，这样在其他研究领域工作的科学家就可以很容易地使用这项研究。所考虑的应用可以用偏微分方程（PDEs）建模。这些偏微分方程是非局部的（分数）；不光滑（接触问题）；具有未知域的几何、非线性、多尺度问题，即自由边界问题。这个项目的重点是具有这种PDE约束的优化问题-所谓的PDE约束优化问题。它旨在创建新的优化方案，以解决当前棘手的具有非光滑特征的优化问题，包括：具有非线性不等式约束的优化问题，接触问题和风险规避PDE约束优化。由此产生的优化算法将为非凸非光滑问题提供新的见解。表面张力优化问题是一个新的研究领域，具有很大的潜力，可以提高我们对微观和纳米尺度的理解，在微观和纳米尺度上，表面效应主导着体效应。这将影响下一代芯片实验室、法医和天文望远镜液体透镜的设计。开源软件的开发，不仅有利于优化、fbp和非局部问题的科学家，也有利于非线性偏微分方程和数据驱动优化问题的科学家。结果将通过两个专题课程进行传播：(i)不确定条件下PDE约束优化；（ii）深度学习与PDE约束优化。一名女学生将获得博士学位。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。

The state of the art in modeling of the physical processes in science and engineering confronts us with solving increasingly complex optimization problems in the presence of constraints. Think of designing the wing of an aircraft where it is essential that it can withstand a tremendous amount of pressure. This is just one of the example of an optimization problem with constrains. Such problems are inherently nonlinear due to constraints. This makes the development and analysis of algorithms and numerical techniques for the solution of these problems extremely challenging but rewarding. This project will create new optimization algorithms that allows us to incorporate constraints in a systematic and robust fashion. The targeted applications range from next generation micro- and nano-scale lab-on-chip devices to novel material and structural designs. Open source software will be created so that the research can be easily used by scientists working in other research areas. The applications under consideration can be modeled using partial differential equations (PDEs). These PDEs are nonlocal (fractional); nonsmooth (contact problems); geometric, nonlinear, multiscale with an unknown domain, i.e. free boundary problems (FBPs). This project focuses on optimization problems with such PDE constraints - the so-called PDE constrained optimization problems. It aims to create new optimization schemes that will enable the solution of currently intractable optimization problems with nonsmooth features, including: optimization problems with nonlinear inequality constraints, contact problems, and risk-averse PDE constrained optimization. The resulting optimization algorithms will provide new insights into nonconvex nonsmooth problems. Optimization problems with surface tension is a new research field with great potential to enhance our understanding at the micro and nano-scales where surface effects dominate bulk effects. This will impact the design of next generation lab-on-chip, forensics and liquid lenses in astronomical telescopes. Open source software will be developed, which will not only benefit scientists in optimization, FBPs, and nonlocal problems but also scientists in nonlinear PDEs and data driven optimization problems. The results will be disseminated via two special topics courses on (i) PDE constrained optimization under uncertainty; (ii) Deep learning and PDE constrained optimization. One female student will get her PhD.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.