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Iterative Methods for PDE-Constrained Optimization

偏微分方程约束优化的迭代方法

基本信息

批准号：
EP/H026053/1
负责人：
Nicholas Ian Mark Gould
金额：
$ 1.54万
依托单位：
Science and Technology Facilities Council
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH026053%2F1
关键词：
Iterative Methods PDE Constrained Optimization

项目摘要

Nonlinear optimization is a fundamental branch of numerical analysis andis ubiquitous in science and engineering. It allows one, for example, todetermine an optimal flight plan under specified circumstances, to findthe shortest paths between fixed positions on a given manifold(geodesics), to predict the shape and layout of telescope arrays thatmaximize light intensity, or to find an optimal design among infinitelymany possible elastic structures. Recent progress in large-scalenonlinear optimization and linear algebra opens new perspectives on thesolution of optimization problems whose constraints are defined bydiscretized sets of partial-differential equations (PDEs). Such problemsare challenging both because of the structural properties present in thecontinuous equations and because of their sheer intimidating size. Theyare important because of their wide-ranging application; a particularlyvital instance is the Navier-Stokes equations, the basic equations offluid mechanics. Challenging as such equations may be, theirdiscretization by the finite-element method leads to a consistentstructure that can be exploited by numerical algorithms.Our proposed research is to design and analyse numerical methodsfor solving PDE-constrained optimization problems, and to buildhigh-quality software to allow others to use our work. We shallparticularly be concerned with methods which are guaranteed toconverge to a locally-best solution, and which ultimately convergerapidly to this solution fast. We shall address the issues ofadditional side constraints that might be imposed physically andof natural rank-deficiencies that arise from the differential equations.

非线性优化是数值分析的一个基本分支，在科学和工程中无处不在。例如，它允许人们在特定情况下确定最佳飞行计划，找到给定流形（测地线）上固定位置之间的最短路径，预测望远镜阵列的形状和布局以最大化光强度，或者在无限多可能的弹性结构中找到最佳设计。大规模非线性优化和线性代数的最新进展为求解约束由离散偏微分方程组定义的优化问题开辟了新的视角。这类问题具有挑战性，一方面是因为连续方程的结构特性，另一方面是因为它们的大小令人生畏。它们之所以重要，是因为它们的应用范围很广；一个特别重要的例子是Navier-Stokes方程，流体力学的基本方程。尽管这样的方程可能具有挑战性，但通过有限元方法将其离散化会导致可以通过数值算法利用的一致结构。我们提出的研究是设计和分析解决pde约束优化问题的数值方法，并构建高质量的软件，以允许其他人使用我们的工作。我们将特别关注那些保证收敛到局部最优解，并最终快速收敛到该解的方法。我们将讨论可能在物理上施加的附加侧约束和由微分方程引起的自然秩缺陷的问题。