Parameterization and Reduction for Nonlinear Stochastic Systems with Applications to Fluid Dynamics

非线性随机系统的参数化和简化及其在流体动力学中的应用

• 批准号: 2108856
• 负责人: Honghu Liu
• 金额: $ 11.38万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2108856&HistoricalAwards=false
• 关键词: Parameterization; Reduction; Nonlinear; Stochastic; Systems

The dynamics of the oceans exhibits several large-scale persistent currents, including the Gulf stream and the Kuroshio in the middle latitudes as prominent examples. Together with other currents in low and high latitudes, they transfer substantial amounts of heat and momentum from the tropics to the polar regions, influencing local and global climate. The balmy jet of seawater also carries a great potential for producing clean offshore carbon-free energy. To understand the spatial and time variabilities of such currents is thus of vital importance for our society. In this project, the investigator will analyze the interplay between intrinsic nonlinearity and extrinsic stochastic forcing in shaping the observed variabilities. To disentangle such interactions and to analyze the impact of noise on dynamical and statistical behaviors of the governing systems are still grand challenges for many practical applications. To address these questions, the investigator will establish a new paradigm for the parameterization and the effective reduction of stochastically forced nonlinear dissipative equations, such as those governing large-scale oceanic flows. The proposed approach relies crucially on a dimension reduction methodology developed recently by the investigator and his colleagues. The knowledge gained in this project is expected to bring new understanding of the fundamental mechanisms of large-scale climate patterns. The award will also provide opportunities for the involvement of graduate students in this research.The dimension reduction methodology adopted and further developed in this project is based on a new stochastic parameterization technique for the unresolved small-scale dynamics of the underlying nonlinear stochastic partial differential equations. The investigator will derive explicit formulas that approximate the small-scale dynamics in terms of both the large-scale dynamics and the history of the noise path, leading thus to low-dimensional stochastic equations involving only large-scale variables. Such reduced equations are able to capture key dynamical features of the original stochastic systems and are much more accessible both theoretically and numerically. The impact of noise on both pattern formation in the classical Rayleigh-Benard convection and time-variability of the double-gyre wind-driven ocean circulation will be studied within the proposed theoretic framework. The parameterization formulas of unresolved small-scale dynamics are rigorously justified in the context of stochastic invariant manifolds. These formulas will be extended in this project to handle parameter regimes that are away from the onset of the first instability using a variational framework. The parameterization is pathwise in nature, which is very well suited for cases when one is not only interested in statistical quantities but also trajectory-wise dynamical behaviors. The formulas involve the history of the noise, which introduces memory into the corresponding reduced equations. This memory effect plays a fundamental role for the reduced equations to capture both qualitatively and quantitatively the dynamical and statistical features of the original system, and it has already been illustrated to be responsible for achieving good modeling performance even in situations that are known to be challenging for other traditional methods to operate. These reduced systems will help us understand better the impact of noise on the studied systems, which are otherwise computationally too expensive to obtain. By studying these reduced models subject to various types of noise, the proposed approach will also bring insights into possible ways of further improving the underlying stochastic models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

海洋动力学表现出几种大规模的持续洋流，包括墨西哥湾流和中纬度地区的黑潮就是突出的例子。它们与低纬度和高纬度地区的其他洋流一起，将大量的热量和动量从热带转移到极地地区，影响当地和全球气候。温和的海水喷射还具有生产清洁的近海无碳能源的巨大潜力。因此，了解此类电流的空间和时间变化对于我们的社会至关重要。在这个项目中，研究人员将分析内在非线性和外在随机强迫在形成观测到的变异性方面的相互作用。对于许多实际应用来说，解开这种相互作用并分析噪声对控制系统的动力学和统计行为的影响仍然是巨大的挑战。为了解决这些问题，研究人员将建立一个新的范式，用于参数化和有效减少随机强迫非线性耗散方程，例如控制大规模海洋流动的方程。所提出的方法主要依赖于研究者和他的同事最近开发的降维方法。该项目获得的知识预计将为大规模气候模式的基本机制带来新的理解。该奖项还将为研究生参与这项研究提供机会。该项目采用并进一步开发的降维方法基于一种新的随机参数化技术，用于解决基础非线性随机偏微分方程的未解决的小规模动力学问题。研究人员将推导出根据大尺度动力学和噪声路径历史来近似小尺度动力学的显式公式，从而得出仅涉及大尺度变量的低维随机方程。这种简化的方程能够捕获原始随机系统的关键动力学特征，并且在理论上和数值上都更容易理解。 将在所提出的理论框架内研究噪声对经典瑞利-贝纳德对流模式形成和双旋风驱动海洋环流时变性的影响。未解决的小尺度动力学的参数化公式在随机不变流形的背景下得到了严格的证明。这些公式将在本项目中扩展，以使用变分框架处理远离第一次不稳定开始的参数状态。参数化本质上是路径化的，这非常适合人们不仅对统计量感兴趣而且对轨迹动态行为感兴趣的情况。这些公式涉及噪声的历史，这将记忆引入相应的简化方程中。这种记忆效应对于简化方程定性和定量地捕获原始系统的动态和统计特征起着基础作用，并且已经被证明即使在已知具有挑战性的情况下也能实现良好的建模性能以便其他传统方法进行操作。这些简化的系统将帮助我们更好地理解噪声对所研究系统的影响，否则获得这些系统的计算成本太高。通过研究这些受到各种类型噪声影响的简化模型，所提出的方法还将深入了解进一步改进底层随机模型的可能方法。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持以及更广泛的影响审查标准。

项目成果

Least-squares finite element method for ordinary differential equations

常微分方程的最小二乘有限元法

• DOI: 10.1016/j.cam.2022.114660
• 发表时间: 2023-01
• 期刊: Journal of Computational and Applied Mathematics
• 影响因子: 2.4
• 作者: Chung, Matthias;Krueger, Justin;Liu, Honghu
• 通讯作者: Liu, Honghu

Verifiability of the Data-Driven Variational Multiscale Reduced Order Model

数据驱动的变分多尺度降阶模型的可验证性

• DOI: 10.1007/s10915-022-02019-y
• 发表时间: 2022-11
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Koc, Birgul;Mou, Changhong;Liu, Honghu;Wang, Zhu;Rozza, Gianluigi;Iliescu, Traian
• 通讯作者: Iliescu, Traian

Conditional Gaussian nonlinear system: A fast preconditioner and a cheap surrogate model for complex nonlinear systems

条件高斯非线性系统：复杂非线性系统的快速预处理器和廉价代理模型

• DOI: 10.1063/5.0081668
• 发表时间: 2022-05
• 期刊: Chaos
• 影响因子: 2.9
• 作者: Chen, N.;Li, Y.;Liu, H.
• 通讯作者: Liu, H.

Shock trace prediction by reduced models for a viscous stochastic Burgers equation

通过粘性随机 Burgers 方程的简化模型进行冲击轨迹预测

• DOI: 10.1063/5.0084955
• 发表时间: 2022-04
• 期刊: Chaos
• 影响因子: 2.9
• 作者: Chen, N.;Liu, H.;Lu, F.
• 通讯作者: Lu, F.

Transitions in stochastic non-equilibrium systems: Efficient reduction and analysis

随机非平衡系统中的转变：有效还原和分析

• DOI: 10.1016/j.jde.2022.11.025
• 发表时间: 2022-02-14
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: M. Chekroun;Honghu Liu;J. McWilliams;Shouhong Wang
• 通讯作者: Shouhong Wang