Analytical/Computational Methods for Rare Events in Lightwave Systems

摘要DMS-046513，西北大学William Kath，题目：光波系统中罕见事件的分析/计算方法本项目的目标是构建基于方差减少技术的新方法，例如重要性采样，这些方法可以用于有效地模拟光波系统中由噪声引起的罕见事件。重要性抽样是一种偏向(或改变)用于生成随机蒙特卡罗试验的概率分布的方法，从而使模拟误差比其他情况下更频繁地发生。这里的目的是利用控制信号传播的方程的数学结构，以允许执行重要采样。例如，在光纤中，控制方程是非线性薛定谔(NLS)方程，它是一个完全可积的哈密顿系统。NLS方程的逆散射解表明，每个脉冲都有一组与其相关联的模式；这些模式对应于脉冲的幅度、相位、位置和频率的变化。由于脉冲参数的任何值都会产生NLS方程的有效解，因此如果它们中的任何一个被噪声改变，则不会遇到任何阻力。因此，这四种模式为构造故意偏置噪声以产生大信号波动的方法提供了自然基础。光纤上高比特率数据传输的发展是20世纪末的主要技术成就之一。光纤推动了全球互联网的发展，并正在彻底改变信息的通信和处理方式。由于现代光波系统对性能要求的提高水平，传统的分析或计算方法本身不足以准确地模拟决定这些高度复杂系统的总体性能的罕见事件。同时，他们的开发成本往往很高，因此对他们的行为和性能的准确预测至关重要。然而，最近的工作表明，分析/计算混合方法不仅可以使完成这一任务成为可能，而且是可行的。概念验证的例子表明，在这种小到万亿分之一的系统中，错误概率不应再被考虑到超出估计的范围。将作为该项目的一部分开发的方法预计将为计算工具提供基础，这些工具可以大大减少确定这类系统的性能所需的时间。

AbstractDMS-046513William Kath, Northwestern UniversityTitle: Analytical/Computational Methods for Rare Events in Lightwave SystemsThe goal of this project is to construct new methods based upon variance reduction techniques, such as importance sampling, that can be used to efficiently simulate rare events caused by noise in lightwave systems. Importance sampling is a method of biasing (or, altering) the probability distributions used to generate random Monte-Carlo trials so that simulated errors occur more frequently than would be the case otherwise. The aim here is to exploit the mathematical structure of the equations governing the propagation of signals to allow importance sampling to be performed. For example, in optical fibers, the governing equation is the nonlinear Schroedinger (NLS) equation, which is a completely integrable Hamiltonian system. The inverse scattering solution of the NLS equation shows that each pulse has a set of modes associated with it; these modes correspond to changes in the pulse's amplitude, phase, position and frequency. Since anyvalue of the pulse parameters yields a valid solution of the NLS equation, no resistance is encountered if any of them is changed by noise. These four modes thus provide a natural basis upon which to construct methods by which the noise is intentionally biased to produce large signal fluctuations.The development of high-bit-rate data transmission over optical fibers is one of the major technological achievements of the late 20th century. Optical fibers have fueled the growth of the global internet and are revolutionizing the ways in which information is communicated and processed. Because of the enhanced levels of performance demanded of modern lightwave systems, traditional analytical or computational methods are by themselves insufficient to accurately model the rare events that determine the overallperformance of these highly complex systems. At the same time, their often large development costs makes the accurate prediction of their behavior and performance essential. Recent work has demonstrated, however, that hybrid analytical/computational approaches can make accomplishing this task not only possible, but also practical. Proof-of-concept examples have shown that error probabilities in such systems as small as one part in a trillion should no longer be considered beyond the reach of estimation. The methods that will be developed as part of this project are expected to provide the basis for computational tools that can yield large reductions in the time required to determine the performance of such systems.