AF EAGER: Minimum Sobolev Norm techniques for systems of elliptic PDEs

AF EAGER：椭圆偏微分方程组的最小 Sobolev 范数技术

• 批准号: 1450321
• 负责人: Shivkumar Chandrasekaran
• 金额: $ 5万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1450321&HistoricalAwards=false
• 关键词: AF; EAGER; Minimum; Sobolev; Norm

Designing aircraft, skyscrapers, jet engines, medical imaging equipment, computer chips, communication equipment, etc., have one thing in common: they are expensive to prototype. It is significantly cheaper and faster to be able to simulate new designs on a computer before building them. However this requires the ability to rapidly solve the equations of physics, almost all of which are posed as partial differential equations. Unfortunately the current state of the art for the numerical solution of partial differential equations is just too slow to meet the majority of industrial needs.This project is investigating a new promising numerical technique. First it uses a clever variant of the Golomb--Weinberger principle to deal with the infinite number of unknowns. Second, it picks a finite number of equations but insists that the error be exactly zero. Since there are more unknowns than equations, there are many potential solutions, and it uses the Golomb--Weinberger principle to pick the smoothest solution. Deep mathematical techniques can be used to prove that the computed solution will be close to the true solution for a very wide class of partial differential equations, wider than that for current state of the art methods, and this is what makes the project's approach so significant. Sophisticated modern numerical techniques are needed to make this into a practical approach. The initial goal of this project is a working software system for two dimensional problems, along with a detailed mathematical analysis to increase confidence in the approach.Broader impacts of this research include software development for public use and supporting and mentoring a female graduate student in this interdisciplinary field.

设计飞机、摩天大楼、喷气发动机、医疗成像设备、计算机芯片、通信设备等，都有一个共同点：它们的原型成本都很高。在制造新设计之前，能够在计算机上模拟它们，这要便宜得多，速度也快得多。然而，这需要快速求解物理方程的能力，几乎所有的物理方程都被假定为偏微分方程。不幸的是，目前偏微分方程数值求解的技术水平太慢，不能满足大多数工业需要。这个项目正在研究一种新的有前途的数值技术。首先，它使用了Golomb的一个巧妙变体--Weinberger原理来处理无限数量的未知数。其次，它选择了有限数量的方程，但坚持误差恰好为零。因为未知数比方程多，所以有很多潜在的解，它使用Golomb-Weinberger原理来挑选最平滑的解。深奥的数学技巧可以被用来证明，对于非常广泛的一类偏微分方程组，计算的解将接近真实的解，比目前最先进的方法的解更宽，这就是该项目的方法如此重要的原因。需要复杂的现代数值技术才能使其成为一种实用的方法。这个项目的最初目标是一个二维问题的工作软件系统，以及一个详细的数学分析，以增加对方法的信心。这项研究的广泛影响包括公共使用的软件开发，以及在这个跨学科领域支持和指导一名女研究生。