Iterative Methods for PDE-Constrained Optimization

非线性优化是数值分析的一个基本分支，在科学和工程中普遍存在。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，预测使光线强度最大化的望远镜阵列的形状和布局，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优化问题的求解开辟了新的视角，这些优化问题的约束由离散化的偏微分方程组定义。这类问题具有挑战性，既是因为连续方程中存在的结构性质，也是因为它们的巨大规模。它们之所以重要是因为它们的广泛应用；一个特别重要的例子是流体力学的基本方程--纳维-斯托克斯方程。尽管这些方程可能具有挑战性，但有限元方法的离散导致了一个可以被数值算法利用的一致结构。我们建议的研究是设计和分析求解偏微分方程组约束优化问题的数值方法，并建立高质量的软件以供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，并最终快速收敛到这个解的方法。我们将讨论物理上可能施加的附加副约束和由微分方程引起的自然秩亏问题。

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.