Network Traffic Analysis and Multiresolution Schemes for Homogenization

美国国家科学基金会工业博士后研究员Anna C. Gilbert在耶鲁大学数学系和at&t实验室研究中心联合开展研究。耶鲁大学的教师导师是数学教授Ronald Coifman， at&t的导师是Walter Willinger博士。我们设想了两个一般的研究领域，这两个领域都是AT&amp；T直接感兴趣的：(1)用于网络流量分析和性能管理的小波技术，(2)用于微分方程均匀化的小波技术。这两个领域的共同点是使用小波将问题分解为许多尺度，并分析哪些尺度对解决问题至关重要。这项研究将与at&t的Walter Willinger博士一起进行，将包括使用小波来研究Willinger等人的基本物理模型。我们希望在小波项中对队列、交换机和其他组件网络组件进行建模和调优，然后在小波项中推导相应自相似行为的属性。有如此多的可用数据集和如此详细，我们想提取更详细和更基本的信息，而不仅仅是赫斯特参数。我们想要建立一个数据和产生这些数据的网络的结构模型。这项研究将与耶鲁大学的Ronald Coifman教授一起进行，将包括扩展Anna C. Gilbert论文中开发的方法。我们将构建多分辨率方法来简化和均匀化偏微分方程。我们设想在许多不同的应用程序中使用这些技术。特别是，我们想使用这些均匀化技术来重建电磁波散射的粗糙表面。我们还比较了n维椭圆偏微分方程的MRA均匀化方法与经典理论。Anna Gilbert于1997年6月在普林斯顿大学的I. Daubechies的指导下完成了她的论文。在这篇文章中，她比较了用于均匀化的小波技术与经典分析技术，开发了用于均匀化非线性常微分方程的小波方法，并将这些方法应用于反应扩散模型。该GOALI项目由MPS多学科活动办公室（OMA）和数学科学司（DMS）联合支持。

An NSF-Industrial Postdoctoral Fellow, Anna C. Gilbert, carries out research jointly in the Department of Mathematics at Yale University and AT&T Labs-Research. The Faculty Mentor at Yale is Ronald Coifman, Professor of Mathematics, and the supervisor at AT&T is Dr. Walter Willinger. We envisage two general areas of research, both of which are of direct interest to AT&T: (1) wavelet techniques for analysis and performance management of network traffic, (2) wavelet techniques for homogenization of differential equations. The common element of these two areas is the use of wavelets to decompose a problem into many scales and to analyze which scales are crucial to solving the problem. The research which will be conducted with Dr. Walter Willinger at AT&T will include using wavelets to investigate the physical model of Willinger, et al, at a basic level. We want to model and to tune queues, switches, and other component network components in wavelet terms and then to derive the properties, in wavelet terms, of the corresponding self-similar behavior. There are so many data sets available and in such detail that we want to extract more detailed and more basic information than just the Hurst parameter. We want to build a structural model of the data and the networks which produce that data. The research which will be conducted with Prof. Ronald Coifman at Yale University will include extending the methods developed in Anna C. Gilbert's thesis. We will construct multiresolution methods for the reduction and homogenization of partial differential equations. We envision using these techniques in many different applications. In particular, we would like to use these homogenization techniques to reconstruct a rough surface off which electromagnetic waves are scattered. We would also compare the MRA homogenization methods for n-dimensional elliptic PDEs with the classical theory. Anna Gilbert completed her thesis under the supervision of I. Daubechies at Princ eton University in June, 1997. In it she compares the wavelet techniques for homogenization with classical analytical techniques, develops a wavelet method for homogenizing nonlinear ordinary differential equations, and applies these methods to a reaction-diffusion model. This GOALI project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).