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石油储层建模、天气预报和飞机机翼形状优化设计等越来越多的实际应用中起着重要的作用。虽然传统的PDECO使用确定性模型，但该项目针对的是PDECO，其中微分方程还包括不确定性，例如地面成分的不规则波动或湍流风速。其最终目的是显着提高解决方案的质量和计算时间的最优控制问题。从这个项目产生的新算法将影响在地球物理学，天气建模等优化问题，这些问题是通用的，解决方案技术的进步也将有利于其他科学。开源软件将被创建并与社区共享。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。该项目的重点是开发强大的，可扩展的多层次求解器，主要是两类潜在的大规模PDECO问题：随机偏微分方程（PDE）和非局部PDE约束的PDECO。另一个推力是开发多级求解器，以支持基于优化的区域分解-另一种PDECO -用于正向PDE模型本身。多层/多重网格求解器是已知的许多类的正演模型的最佳。然而，他们的应用，以解决PDECO问题仍处于起步阶段。天真的应用程序的多级方法来解决这样的优化问题，可能会导致依赖于分辨率（网格依赖）和其他参数的问题，如随机尺寸或子域的数量。此外，由于每个迭代都涉及至少一个PDE求解，因此求解此类优化问题的成本对于大规模、高分辨率问题来说可能是过高的，特别是对于比传统的确定性问题昂贵得多的问题。在这个项目中开发的算法旨在建立新的标准的效率和鲁棒性。 新的数学工具将进一步推进数值分析和优化的知识。新的专题课程将根据项目中产生的研究开发，并将与社区分享笔记。研究成果将通过技术研究论文和在国家和国际会议上的演讲积极传播。该奖项反映了NSF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。

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.