Optimization with PDE Constraints

本文主要研究了偏微分方程最优化问题的并行隐式求解中的几个基本问题，并在以下四个方面取得了一定的进展：（1）大规模最优化算法的公式化和分析;（2）偏微分方程自适应网格生成;（3）偏微分方程多层求解器;（4）偏微分方程并行隐式求解器。（4）并行计算。 虽然这些主题中的每一个都可以单独研究，但它们之间的相互作用对于创建有效的全局算法至关重要。 调查人员是科学计算小组的成员，该小组提供了一个教学和研究计划，强调科学计算在制定，建模和解决来自不同和不断变化的领域的问题中的作用。 该项目的主要部分涉及软件的开发及其在制造业、工程和科学界的传播。 作为项目一部分开发的软件提供了一种有效的技术转让方法，并扩大了调查人员开发的现有代码PLTMG、MC和SNOPT的范围和有效性。 由于偏微分方程严格地描述了科学和工程中许多复杂系统的物理定律，它们也是用于模拟和预测这些系统行为的数学模型的核心。 优化这些系统性能的需求是实际应用的共同特征，这些应用涵盖了不同的领域，如美洲杯游艇船体的优化设计，有毒废物场地的清理，组织工程中生物人工动脉的建造，以及股票投资组合和对冲基金的管理。 该项目开发的软件为工程师和科学家提供了即时访问最先进的方法，用于对涉及偏微分方程约束的复杂系统进行建模和优化。 这些模型在效率、准确性和鲁棒性方面的改进对美国全球竞争力至关重要的制造业和工程领域产生了重大影响。

The investigator and his colleagues focus on severalfundamental computational issues involved in the parallelimplicit solution of optimization problems with partialdifferential equation The goal of the research is to advance thestate of the art in four fundamental topics: (1) the formulationand analysis of algorithms for large-scale optimization; (2)adaptive mesh generation for partial differential equations; (3)multilevel partial differential equation solvers; and (4)parallel computation. Although each of these topics can beinvestigated in isolation, the exploitation of their interactionsis crucial for the creation of effective global algorithms. Theinvestigators are members of a Scientific Computation Group thatoffers a program of instruction and research emphasizing the roleof scientific computation in the formulation, modeling, andsolution of problems from diverse and changing areas. A majorpart of the project involves the development of software and itsdissemination within the manufacturing, engineering, andscientific community. Software developed as part of the projectprovides an effective method of technology transfer and extendsthe scope and effectiveness of the existing codes PLTMG, MC, andSNOPT developed by the investigators. Because partial differential equations convenientlycharacterize the physical laws of many complex systems occurringin science and engineering, they also lie at the heart of themathematical models used to simulate and predict the behavior ofthese systems. The need to optimize the performance of suchsystems is the common feature of practical applications thatrange over such diverse areas as the optimal design of the hullof an America's Cup yacht, the cleanup of toxic waste sites, theconstruction of bioartificial arteries in tissue engineering, andthe management of stock portfolios and hedge funds. Softwaredeveloped in this project provides engineers and scientists withinstant access to state-of-the-art methods for the modeling andoptimization of complex systems involving partial differentialequation constraints. The resulting improvements in theefficiency, accuracy and robustness of these models have asubstantial impact in areas of manufacturing and engineering thatare vital to US global competitiveness.