Methods and Applications for PDE-Constrained Optimization

这个项目涉及一个为期三年的计划，研究偏微分方程(PDE)约束优化问题的并行隐式求解所涉及的几个基本计算问题。这样的问题出现在工程和科学计算的许多环境中，因为物理现实通常通过涉及偏微分方程组的模型来表示。偏微分方程组的精确离散化导致了非常大的稀疏约束优化问题，其中至少部分结构反映了离散化。该方法的研究目标包括但不限于以下四个基本主题：(1)大规模非线性优化算法的建立和分析；(2)偏微分方程组的自适应网格生成；(3)多层偏微分方程组求解器；(4)并行计算。尽管这些主题中的每一个都可以单独研究，但研究人员认为，利用它们之间的相互作用对于创建有效的全局算法至关重要。这项研究是由三个特别具有挑战性的应用推动和指导的：(I)地球物理反问题；(Ii)带约束的演化偏微分方程组的投影方法；以及(Iii)约束水平集方法。这些拓扑学涵盖了计算科学的一些重要应用，包括近海石油勘探、黑洞的数值模拟、晶体生长和生物膜的模拟、捕捉波的绕射效应和路径规划。所有这些应用程序的两个共同特征是，必须使用现代自适应多级技术来处理PDE约束，并且基本的优化问题是高度非线性的，因此是非凸性的。研究人员是加州大学圣地亚哥分校数学系计算与应用数学(CAM)小组的成员。他们在应用数学、数值优化、数值偏微分方程式和并行计算方面拥有丰富的专业知识。一个重要的目标是开发体现上述算法的软件。作为该项目的一部分开发的软件将提供一种有效的技术转让方法，并将扩大加州大学圣地亚哥分校调查人员开发的现有代码的范围和有效性。该项目的软件组件将对涉及复杂系统建模的研究产生重大影响，因为它将为科学家和工程师提供对最先进方法的即时访问。在计算和应用数学小组内，研究人员提供一个教学和研究计划，强调计算科学在不同和不断变化的领域的问题的制定、建模和解决中的作用。与该项目相关的活动将有助于吸引高级研究生进入计算科学领域，该领域在制造业、工程学和自然科学中的系统研究中发挥着至关重要的作用。

This project involves a three-year program of research on severalfundamental computational issues involved in the parallel implicitsolution of optimization problems with partial differential equation(PDE) constraints. Such problems arise in many contexts inengineering and scientific computation, since physical reality isoften expressed through models involving PDEs. Accuratediscretizations of PDEs lead to very large sparse constrainedoptimization problems, where at least part of the structure reflectsthe discretization. The goals of the research on methods include, butare not limited to, advancing the state-of-the-art in four fundamentaltopics: (1) the formulation and analysis of algorithms for large-scalenonlinear optimization; (2) adaptive mesh generation for PDEs; (3)multilevel PDE solvers; and (4) parallel computation. Although eachof these topics can be investigated in isolation, the investigatorsbelieve that the exploitation of their interactions is crucial for thecreation of effective global algorithms. The research is motivatedand guided by three particularly challenging applications: (i)geophysical inverse problems; (ii) projection methods for evolutionPDEs with constraints; and (iii) constrained level-set methods. Thesetopics cover a number of important applications of computationalscience, including off-shore petroleum exploration, the numericalmodeling of black holes, the modeling of crystal growth andbiomembranes, capturing diffraction effects of waves and pathplanning. Two features common to all these applications are that thePDE constraints must be handled using modern adaptive multi-leveltechniques and that the underlying optimization problem is highlynonlinear and hence nonconvex.The Investigators are members of the Computational and AppliedMathematics (CAM) Group within the Department of Mathematics at UC SanDiego. They have a combined expertise in applied mathematics,numerical optimization, numerical partial differential equations andparallel computation. An important goal is the development ofsoftware embodying the above algorithms. Software developed as partof the project will provide an effective method of technology transferand will extend the scope and effectiveness of existing codes thathave been developed by the investigators at UC San Diego. Thesoftware component of the project will have a substantial impact onresearch involving the modeling of complex systems as it will providescientists and engineers with instant access to state-of-the-artmethods. Within the Computational and Applied Mathematics Group, theInvestigators offer a program of instruction and research thatemphasizes the role of computational science in the formulation,modeling, and solution of problems from diverse and changing areas.The activities associated with this project will help attract advancedgraduate students into the area of computational science, which playsa vital role in the study of systems arising in manufacturing,engineering and the natural sciences.