Theoretical and Numerical Studies of Nonlocal Equations Derived from Stochastic Differential Equations with Levy Noises

带Levy噪声的随机微分方程推导的非局部方程的理论与数值研究

• 批准号: 1620449
• 负责人: Xiaofan Li
• 金额: $ 19.96万
• 依托单位: Illinois Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-15 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620449&HistoricalAwards=false
• 关键词: Theoretical; Numerical; Studies; Nonlocal; Equations

Stochastic effects are ubiquitous in complex systems in science and engineering. Although random mechanisms may appear to be very small or very fast, their long time impact on the system evolution may be delicate or even profound, which has been observed in, for example, stochastic bifurcation, stochastic resonance and noise-induced pattern formation. The research team will study the complex systems under uncertainty by developing numerical methods to be simulated on computers and answer fundamental questions about the average quantities of the systems. The investigators will deliver the following broader impact outcomes: (1) Two graduate students (including one female underrepresented minority) will receive education and training and (2) they will continue to recruit and nurture underrepresented students in STEM. Additionally, the resulting computational tools have broad applications in areas ranging from biology to geophysics. The software resulting from the proposed project will be made publicly available.Mathematical modeling of complex systems under uncertainty often leads to stochastic differential equations (SDEs). Fluctuations appeared in the SDEs are often non-Gaussian (e.g., Levy motions) rather than Gaussian (e.g., Brownian motion). Compared with systems with Gaussian noises, quantifying the impacts of non-Gaussian Levy fluctuations are much less understood. The researchers will develop convergent and efficient numerical techniques for investigating the deterministic macroscopic quantities that can help understand the dynamics of SDEs with Levy noises, in particular the mean exit time, escape probability, and probability density function. They will also answer theoretical questions with regard to the well-posedness and the regularity of the solutions to these nonlocal equations. Building upon previously developed methods in the one-dimensional case, the team will focus its effort on two-dimensional systems. The proposed project will provide not only broadly applicable computational techniques to solve integro-differential equations with singular integrands, but also theoretically address the central questions pertaining to the solutions. The theory will help to provide insight into fundamental issues in quantifying the impacts of non-Gaussian Levy fluctuations in dynamical systems.

随机效应在科学和工程的复杂系统中普遍存在。 尽管随机机制可能看起来非常小或非常快，但它们对系统演化的长期影响可能是微妙的甚至是深远的，这已经在例如随机分叉、随机共振和噪声引起的模式形成中观察到。 研究小组将通过开发在计算机上模拟的数值方法来研究不确定性下的复杂系统，并回答有关系统平均量的基本问题。 调查人员将实现以下更广泛的影响成果：(1) 两名研究生（包括一名代表性不足的少数族裔女性）将接受教育和培训；(2) 他们将继续招募和培养 STEM 领域代表性不足的学生。 此外，由此产生的计算工具在生物学到地球物理学等领域具有广泛的应用。 该项目产生的软件将公开提供。不确定条件下复杂系统的数学建模通常会产生随机微分方程（SDE）。 SDE 中出现的波动通常是非高斯的（例如利维运动）而不是高斯的（例如布朗运动）。 与具有高斯噪声的系统相比，量化非高斯征波动的影响的了解要少得多。 研究人员将开发收敛且高效的数值技术来研究确定性宏观量，这有助于理解带有 Levy 噪声的 SDE 动力学，特别是平均退出时间、逃逸概率和概率密度函数。 他们还将回答有关这些非局部方程解的适定性和规律性的理论问题。 基于之前在一维情况下开发的方法，该团队将把精力集中在二维系统上。 拟议的项目不仅将提供广泛适用的计算技术来求解具有奇异被积函数的积分微分方程，而且从理论上解决与解有关的核心问题。 该理论将有助于深入了解量化动力系统中非高斯利维波动影响的基本问题。