Collaborative Research: Multilevel Methods for Optimal Control of Partial Differential Equations and Optimization-Based Domain Decomposition

微分方程最优控制在油藏建模、天气预报、飞机机翼形状优化设计等实际应用中发挥着重要的作用。虽然传统的PDECO使用确定性模型，但该项目针对的是PDECO，其中的微分方程式也包括不确定因素，如地面成分的不规则波动或湍流风速。其最终目的是显著提高此类最优控制问题的解质量和计算时间。该项目产生的新算法将影响地球物理、天气模拟等领域中出现的优化问题。这些问题是通用的，求解技术的进步也将造福于其他科学。将创建开放源码软件并与社区共享。四名研究生将从该项目中受益。将特别注意从代表性不足的群体中招收学生。该项目致力于为两类潜在的大规模PDECO问题开发健壮的、可伸缩的多层求解器：由随机偏微分方程组(PDE)约束的PDECO问题和由非局部偏微分方程约束的PDECO问题。另一个推动力是为正向PDE模型本身开发支持基于优化的区域分解的多级求解器--另一种PDECO。已知的多层/多重网格解算器对于许多类型的正演模式都是最优的。然而，它们在解决PDECO问题上的应用还处于起步阶段。简单地应用多层方法来解决这类优化问题可能会导致对分辨率的依赖(网格依赖)和问题的其他参数，如随机维度或子域的数量。此外，由于每次迭代至少涉及一个偏微分方程组的解，因此对于大规模、高分辨率的问题，尤其是对于比传统的确定性问题昂贵得多的问题，求解此类优化问题的成本可能会高得令人望而却步。该项目中开发的算法旨在设定新的效率和健壮性标准。新的数学工具将进一步提高数值分析和优化方面的知识。将根据项目中产生的研究开发新的专题课程，并将与社区共享笔记。这项研究的结果将通过技术研究论文和在国内和国际会议上的演讲积极传播。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。

Optimal control of differential equations (PDECO) plays an important role in an ever increasing number of real-life applications ranging from petroleum reservoir modeling to weather prediction and the optimal shape design of airplane wings. While traditional PDECO uses deterministic models, this project targets PDECO where the differential equations also include uncertainties, such as irregular fluctuations in the ground composition, or turbulent wind speeds. The ultimate aim is to dramatically improve the solution quality and the computing time of such optimal control problems. The novel algorithms resulted from this project will impact optimization problems arising in geophysics, weather modeling etc. These problems are generic and advances in solution techniques will also benefit other sciences. Open source software will be created and shared with the community. Four graduate students will benefit from the project. Special attention will be given to recruit students from underrepresented groups. The project is focused on developing robust, scalable multilevel solvers for mainly two classes of potentially large-scale PDECO problems: PDECOs constrained by stochastic partial differential equations (PDEs) and by nonlocal PDEs. An additional thrust is to develop multilevel solvers in support of optimization-based domain decomposition - another kind of PDECO - for the forward PDE-models themselves. Multilevel/multigrid solvers are known to be optimal for many classes of forward models. However, their application to solve PDECO problems is still in its infancy. A naive application of multilevel methods to solve such optimization problems can lead to dependence on resolution (mesh-dependence) and on other parameters of the problem such as the stochastic dimension or the number of subdomains. In addition, since each iterate involves at least one PDE solve, the cost of solving such optimization problems can be prohibitive for large-scale, high-resolution problems, especially for problems that are significantly more expensive than the traditional, deterministic ones. The algorithms developed in this project aim to set new standards of efficiency and robustness. Novel mathematical tools will further advance the knowledge in numerical analysis and optimization. New special topics courses will be developed based on the research generated in the project and the notes will be shared with the community. The results of the research will be actively disseminated via technical research papers and talks at national and international conferences.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.