Collaborative Research: Mathematical and computational methods for stochastic systems in nonlinear optics

这个合作研究项目的目标是开发方法的分析，建模和计算的随机偏微分方程中出现的非线性光学。这些系统中的噪声的研究由于几个因素而变得复杂。特别是：（i）由于许多物理效应的相互作用而产生的复杂性，（ii）存在多个长度和时间尺度，以及（iii）代表系统故障的关键事件在设计上极其罕见。 我们将通过（a）创建一组分析方法来解决这些挑战，以获得从控制系统行为的随机偏微分方程到一组低维随机微分方程的简化;（B）开发跨多个时间和空间尺度的噪声研究的有效方法;及（c）实施混合分析-计算方法，可以有效地针对感兴趣的罕见事件，并最终估计这些系统的可靠性。光波系统具有广泛的激光器和光通信线路的出现在过去的五十年里对我们的生活产生了巨大的影响。激光器在低功率下用于与扫描和通信相关的应用;在较高功率下，它们用于外科手术和工业应用。由于噪声是光学系统的主要限制因素，进一步的技术进步将主要取决于对噪声影响的理解以及控制噪声引起的系统故障的能力。这项研究将提供新的分析和计算工具，支持下一代光学器件的分析和设计。

The objective of this collaborative research project is to develop methods for analysis, modeling, and computation for stochastic partial differential equations arising in nonlinear optics. The study of noise in these systems is complicated by several factors. In particular:(i) the complexity due to the interplay of many physical effects,(ii) the presence of multiple length and time scales, and(iii) the fact that the critical events that represent system failures are, by design, extremely rare. We will address these challenges by (a) creating a set of analytical methods to obtain reductions from the stochastic partial differential equations that govern the system's behavior to a set of low-dimensional stochastic differential equations;(b) developing efficient methods for the study of noise across multiple time and space scales; and (c) implementing hybrid analytical-computational methods that can efficiently target the rare events of interest and ultimately estimate the reliability of these systems.Lightwave systems have a wide range of applications, in particular as lasers and optical communication lines, and their emergence has had a dramatic impact on our lives over the last fifty years. Lasers are used at low powers in applications related to scanning and communications; at higher powers they are employed in surgical and industrial applications. As noise is a major limiting factor in optical systems, further technological progress will depend critically on understanding the impact of noise and on being able to control noise-induced system failures. This research will provide new analytical and computational tools that support the analysis and design of the next generation of optical devices.