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

偏微分方程约束优化：算法、数值和应用

基本信息

批准号：
1818772
负责人：
Harbir Antil
金额：
$ 20万
依托单位：
George Mason University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1818772&HistoricalAwards=false
关键词：
Partial Differential Equation Constrained Optimization

项目摘要

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.

科学和工程中物理过程建模的最新技术水平使我们面临着在存在约束的情况下解决日益复杂的优化问题。想想飞机机翼的设计，它必须能够承受巨大的压力。这只是带有约束的优化问题的一个例子。由于受到约束，这类问题本质上是非线性的。这使得开发和分析用于解决这些问题的算法和数值技术具有极大的挑战性，但也是有益的。该项目将创建新的优化算法，使我们能够以系统和稳健的方式纳入约束。目标应用范围从下一代微米和纳米级芯片实验室设备到新的材料和结构设计。将创建开放源码软件，以便在其他研究领域工作的科学家可以轻松使用这些研究。所考虑的应用可以用偏微分方程组(PDE)来建模。这些偏微分方程组是非局部的(分数阶的)、非光滑的(接触问题)、几何的、非线性的、具有未知域的多尺度的，即自由边界问题。本项目主要研究具有这种偏微分方程约束的优化问题，即所谓的偏微分方程组约束优化问题。它的目标是创建新的优化方案，使之能够解决目前难以解决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、接触问题和风险厌恶的偏微分方程约束优化问题。由此产生的优化算法将为研究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，在微纳尺度上有很大的潜力来加深我们对表面效应占主导地位的体效应的理解。这将影响下一代芯片实验室、法医学和天文望远镜中的液体透镜的设计。将开发开源软件，这不仅有利于优化、FBP和非局部问题的科学家，也将有利于研究非线性偏微分方程组和数据驱动优化问题的科学家。结果将通过两个专题课程传播：(I)不确定条件下的偏微分方程约束最优化；(Ii)深度学习和偏微分方程约束最优化。一名女学生将获得博士学位。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。