Advances in Multilevel Methods for Saddle Point Problems

鞍点问题多级方法的进展

• 批准号: 1522454
• 负责人: Constantin Bacuta
• 金额: $ 15万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522454&HistoricalAwards=false
• 关键词: Advances; Multilevel; Methods; Saddle; Point

In spite of the fast increase in computational power of today's computers, the development of faster algorithms for equational models is paramount for the simulation of phenomena investigated nowadays in practical applications in science and engineering. For a large spectrum of situations in the scientific world, a computer simulation has to replace experimental simulation and data collection. For a computer simulation, in order to obtain accurate approximation of the physical quantities of interest, a powerful computational system has to be combined with a fast and reliable algorithm to test and to confirm the designed models. The project will contribute to the construction and implementation of new and efficient algorithms for solving numerically challenging problems. In particular, the proposed research will focus on constructing fast and robust algorithms for solving computational fluid dynamics and electromagnetic problems. The breakthrough approach of the project is based on recent results in numerical analysis, on new algorithm designing and implementation, and on scientific testing and validation of the new computational tools. The methodology from this project for the time-harmonic Maxwell's equations has a broad range of applications in nano-optics and analog signal packages. The work for solving variational problems has scientific and technical applications in optimization of electrical networks and image restoration. The research will study and develop reliable and efficient numerical algorithms for solving partial differential equations that can be described as variational saddle point systems or mixed variational formulations. The goal of the project is to build robust algorithms for solving such equations in the presence of low regularity of solutions due to discontinuities in data or coefficients. The research will produce a rigorous and systematic analysis of a large class of saddle point problems, focusing on efficient algorithm development and testing. The methods will be based on finite element discretization algorithms in the context of a multilevel and adaptive choice of approximation spaces. The PI's approach uses a new saddle point least-squares type of discretization for systems of PDEs, that takes full advantage of the regularity of the solution and involves an efficient level change criterion that minimizes the running time of the global iterative process. This study will lead to more reliable methods for a variety of applications of the finite element method to science and engineering communities, such as those interested in elasticity, electromagnetism, friction, and computational fluid dynamics. The research findings will be shared within the field and with a more general audience including mathematics and science high-school teachers.

尽管当今计算机的计算能力快速增长，但方程模型的更快算法的发展对于当今科学和工程实际应用中研究的现象的模拟至关重要。对于科学世界中的大量情况，计算机模拟必须取代实验模拟和数据收集。对于计算机模拟，为了获得感兴趣的物理量的精确近似，必须将强大的计算系统与快速可靠的算法相结合，以测试和确认所设计的模型。该项目将有助于建设和实施新的和有效的算法来解决具有挑战性的数值问题。特别是，拟议的研究将集中在构建快速和强大的算法来解决计算流体力学和电磁问题。该项目的突破性方法是基于数值分析的最新结果，新算法的设计和实现，以及新计算工具的科学测试和验证。 从这个项目的时间谐波麦克斯韦方程组的方法在纳米光学和模拟信号包有广泛的应用。求解变分问题的工作在电力网络优化和图像恢复中具有科学和技术应用。该研究将研究和开发可靠和有效的数值算法，用于解决可以描述为变分鞍点系统或混合变分公式的偏微分方程。该项目的目标是建立强大的算法，用于解决由于数据或系数的不连续性而存在低正则性的解。该研究将对一大类鞍点问题进行严格而系统的分析，重点是有效的算法开发和测试。该方法将基于有限元离散化算法的上下文中的多级和自适应选择的近似空间。PI的方法使用了一种新的鞍点最小二乘类型的离散化系统的偏微分方程，充分利用了解决方案的规律性，并涉及一个有效的水平变化的标准，最大限度地减少了全局迭代过程的运行时间。这项研究将导致更可靠的方法，为各种应用的有限元方法的科学和工程社区，如那些感兴趣的弹性，电磁，摩擦和计算流体动力学。研究结果将在该领域内分享，并与包括数学和科学高中教师在内的更广泛的受众分享。