n-Widths and Singular Perturbation Problems

奇异摄动偏微分方程用于模拟传输和流体流动以及高频波传播中的边界层和内部层。由于解中存在快速变化的区域，因此很难近似求解这些方程。将研究一些奇摄动线性偏微分方程解的最佳近似性质。这项研究将建立在凯洛格和斯泰恩斯最近的一些研究成果的基础上，这些研究成果处理了奇异扰动的自伴随问题。将研究模拟对流和波传播的方程。目标是当问题的数据被限制为具有指定的平滑度时获得最佳近似属性（在柯尔莫哥洛夫意义上）。这些最佳逼近问题的成功解决将导致解的渐近展开，对数据的平滑度要求最小，并有望为这些解的数值计算提供更好的数值方法。涉及边界层和内部层的物理系统在工业应用中非常重要。 一个典型的例子是飞机机翼附近的湍流空气层。 出现的波传播问题，例如雷达或超声波在医疗和军事应用中都很重要。这些看似不同的问题的相似之处在于，现象都是在很短的距离或时间内发生的。这导致计算此类系统的行为变得困难。该奖项的研究将研究控制这些系统的方程的基本近似性质。

Singularly perturbed partial differential equations are used to model boundary and interior layers in transport and fluid flow, and in high frequency wave propagation. It is difficult to approximate the solution to these equations because of regions of rapid change in the solution. The best approximation properties of solutions to some singularly perturbed linear partial differential equations will be studied. The research will build on some recent work of Kellogg and Stynes that treats asingularly perturbed self adjoint problem. Equations that model convection and wave propagation will be studied. The goal is to obtain the best approximation properties (in the sense of Kolmogorov) when the data of the problem are restricted to have a specified smoothness. A successful resolution of these best approximation problems will lead to asymptotic expansions of the solution with minimal smoothness requirements on the data, and hopefully to better numerical methods for the numerical computation of these solutions.Physical systems that involve boundary and interior layers are of considerable importance in industrial applications. A typical example is the turbulent layer of air near an aircraft wing. Wave propagation problems, arising e.g. in radar or ultrasound are important in both medical and military applications. These seemingly different problems are similar in that there are phenomena taking place over a very short span of distance or time. This leads to difficulties in calculating the behavior of such systems. The research under this award will study the basic approximation properties of the equations governing these systems.