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

设计飞机、摩天大楼、喷气发动机、医学成像设备、计算机芯片、通信设备等，都有一个共同点：它们的原型制作成本很高。在构建新设计之前，能够在计算机上模拟新设计是非常便宜和快速的。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是偏微分方程。不幸的是，偏微分方程的数值解的现有技术水平太慢，不能满足大多数工业需求。本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-Weinberger原理的一个巧妙变体来处理无限多的未知数。第二，它选择了有限个方程，但坚持误差恰好为零。由于未知数比方程多，因此有许多潜在的解，它使用Golomb-Weinberger原理来选择最平滑的解。深度数学技术可以用来证明计算的解决方案将接近一类非常广泛的偏微分方程的真实解，比目前最先进的方法更广泛，这就是为什么该项目的方法如此重要。需要复杂的现代数值技术，使之成为一种实用的方法。该项目的最初目标是一个二维问题的工作软件系统，沿着详细的数学分析，以增加对该方法的信心。该研究的更广泛的影响包括公共使用的软件开发和支持和指导这一跨学科领域的女研究生。

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.