Regularity and Conditioning of Variational Problems

变分问题的规律性和条件作用

• 批准号: 1008341
• 负责人: Assen Dontchev
• 金额: $ 7.49万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1008341&HistoricalAwards=false
• 关键词: Regularity; Conditioning; Variational; Problems; Regularity; Conditioning; Variational; Problems

While the concept of conditioning is well understood in numerical linear algebra, still little is known in that direction for problems beyond equations, and in particular for variational problems, finite- and infinite-dimensional, where the presence of constraints complicates the analysis considerably. In recent years it has become clear that the basic paradigm behind conditioning, the "distance to good behavior," can be extended to a vast variety of problems, linking it in one general picture with the development of error bounds measuring the effect of perturbations and approximations of a problem on its solutions as well as with the convergence rate of algorithms. The "good behavior" of a problem is usually understood as a regularity property of a mapping related to the problem at hand which describes desirable features of its solutions. In this project we will extend the conditioning paradigm to broad classes of variational problems, including problems in nonlinear programming, calculus of variations, and optimal control. This task requires background work in the general area of variational analysis as well as the development of highly technical tools for tackling specific problems.This project aims to establish theoretical foundations for conditioning of variational problems through rigorous analysis of regularity properties of mappings associated with such problems. The project will lead to a better understanding of the interplay between the theoretical features of a problem and the actual computation of solutions. Furthermore, it has the potential to have a direct impact on the scientific computing of variational problems, by providing tools for the development of preconditioning techniques and, consequently, new fast and efficient algorithms for solving large-scale problems that appear in science and technology.

虽然在数值线性代数中可以很好地理解调节的概念，但对于等式以外的问题，尤其是在变化问题，有限的和无限二维的方向上，仍然鲜为人知，其中约束的存在使分析变得复杂。 近年来，很明显，调理背后的基本范式，“到良好行为的距离”可以扩展到各种各样的问题，将其与误差界的发展联系起来，以衡量扰动效果以及在其溶液中的效果以及算法的融合率以及算法的融合率。 问题的“良好行为”通常被理解为与当前问题相关的映射的规律性属性，该映射描述了其解决方案的理想特征。 在这个项目中，我们将把调节范式扩展到各种变异问题类别，包括非线性编程中的问题，变化的计算和最佳控制。 这项任务需要在变异分析的一般领域中的背景工作，以及开发用于解决特定问题的高科技工具。该项目旨在通过严格分析与此类问题相关的规律性映射的严格分析来建立理论基础来调节变异问题。 该项目将更好地理解问题的理论特征与解决方案的实际计算之间的相互作用。 此外，它有可能通过为开发预处理技术的开发以及新的快速，有效算法来解决解决科学和技术中出现的大规模问题的新算法，从而直接影响变异问题的科学计算。