Methods and Applications for PDE-Constrained Optimization

本项目涉及一个为期三年的研究项目，研究与偏微分方程（PDE）约束下优化问题并行隐式解有关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，因为物理现实通常是通过涉及偏微分方程的模型来表达的。偏微分方程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。研究方法的目标包括，但不限于，在四个基本主题上取得最新进展：(1)大规模非线性优化算法的制定和分析；(2) PDEs自适应网格生成；(3)多级PDE求解器；(4)并行计算。尽管每个主题都可以单独进行研究，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。这项研究的动机和指导是由三个特别具有挑战性的应用：(i)地球物理反演问题；（ii）有约束的演化偏微分方程的投影方法；（三）约束水平集方法。这些主题涵盖了计算科学的许多重要应用，包括海上石油勘探，黑洞的数值模拟，晶体生长和生物膜的建模，捕获波的衍射效应和路径规划。所有这些应用程序的两个共同特点是，pde约束必须使用现代自适应多层技术来处理，并且潜在的优化问题是高度非线性的，因此是非凸的。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、数值偏微分方程和并行计算方面具有综合的专业知识。实现上述算法的软件开发是一个重要的目标。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，并将扩展由加州大学圣地亚哥分校的研究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即时访问最先进的方法。在计算和应用数学组内，研究人员提供了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，建模和解决方案中的作用。与此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，这在研究制造、工程和自然科学中的系统中起着至关重要的作用。

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.