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.

海洋的动力学表现出几种大规模的持续电流，包括墨西哥湾流和中纬度中的黑鲁西奥，作为突出的例子。与其他低纬度地区的其他电流一起，它们将大量的热量和动量从热带地区传递到极地地区，从而影响了局部和全球气候。宜人的海水喷气式飞机还具有产生清洁无碳能源的巨大潜力。因此，了解这种水流的空间和时间变化对我们的社会至关重要。在该项目中，研究人员将分析内在的非线性与外在随机强迫之间的相互作用，以塑造观察到的变异性。为了消除这种相互作用，并分析噪声对管理系统动态和统计行为的影响仍然是许多实际应用的巨大挑战。为了解决这些问题，研究人员将建立一个新的范式，以进行参数化和随机迫使非线性耗散方程的有效减少，例如那些管理大规模海洋流的范围。所提出的方法至关重要的是研究人员及其同事最近开发的降低方法。预计该项目中获得的知识将为大规模气候模式的基本机制带来新的了解。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低方法基于一种新的随机参数化技术，用于基础非线性非线性随机偏微分方程的未解决的小规模动力学。研究者将根据大规模动力学和噪声路径的历史近似小规模动力学的明确公式，从而导致仅涉及大规模变量的低维随机方程。此类还原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访问。 在拟议的理论框架中，研究了噪声对经典雷利 - 贝纳德对流的模式形成的影响以及双gyre风驱动的海洋循环的影响。在随机不变的歧管的背景下，未解决的小规模动力学的参数化公式是严格合理的。这些公式将在此项目中扩展，以处理使用变分框架从第一个不稳定性开始的参数制度。本质上，参数化是路径上的，它非常适合当人们不仅对统计量感兴趣，而且还可以通过轨迹的动态行为感兴趣。公式涉及噪声的历史，该噪声将记忆引入相应的还原方程中。这种记忆效应起了简化的方程式的基本作用，可以在定性和定量上捕获原始系统的动态和统计特征，并且已经被说明是为了实现良好的建模性能，即使在已知在其他传统方法中都充满挑战的情况，也可以实现良好的建模性能。这些减少的系统将有助于我们更好地了解噪声对所研究系统的影响，否则这些系统在计算上太昂贵了。通过研究这些减少模型受到各种噪声的影响，提出的方法还将为进一步改善基本随机模型的可能方式提供见解。该奖项反映了NSF的法定任务，并认为值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来获得支持。

项目成果

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
• 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

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

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

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
• 期刊: Chaos
• 影响因子: 2.9
• 作者: Chen, N.;Liu, H.;Lu, F.
• 通讯作者: Lu, F.

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
• 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

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
• 期刊: Chaos
• 影响因子: 2.9
• 作者: Chen, N.;Li, Y.;Liu, H.
• 通讯作者: Liu, H.