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中的复杂系统中普遍存在。 虽然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，甚至是深刻的，这已经在例如随机分叉，随机共振和噪声诱导的图案形成中观察到。 该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数量的基本问题。 研究人员将提供以下更广泛的影响成果：（1）两名研究生（包括一名女性代表性不足的少数民族）将接受教育和培训，（2）他们将继续招募和培养STEM中代表性不足的学生。 此外，由此产生的计算工具在从生物学到物理学的各个领域都有广泛的应用。 拟议项目产生的软件将公开提供。在不确定性下对复杂系统进行数学建模往往会导致随机微分方程。 在SDEs中出现的波动通常是非高斯的（例如，Levy运动）而不是高斯（例如，布朗运动）。 与高斯噪声系统相比，量化非高斯Levy波动的影响要少得多。 研究人员将开发收敛和有效的数值技术，用于研究确定性宏观量，这可以帮助理解带有Levy噪声的SDES的动力学，特别是平均退出时间，逃逸概率和概率密度函数。 他们还将回答理论问题方面的适定性和规律性的解决方案，这些非局部方程。 在以前开发的一维情况下的方法的基础上，该团队将集中精力在二维系统。 拟议的项目将不仅提供广泛适用的计算技术来解决积分微分方程的奇异被积函数，但也从理论上解决有关的解决方案的核心问题。 该理论将有助于深入了解量化非高斯Levy波动在动力系统中的影响的基本问题。