New evolution equations of the joint response-excitation PDF for stochastic modeling: Theory and numerical methods

用于随机建模的联合响应激励 PDF 的新演化方程：理论和数值方法

• 批准号: 1216437
• 负责人: George Karniadakis
• 金额: $ 35.06万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1216437&HistoricalAwards=false
• 关键词: New; evolution; equations; joint; response

New theory and corresponding numerical algorithms are proposed for addressing fundamental open questions in stochastic modeling of physical and biological systems, e.g., the curse-of-dimensionality, the lack of regularity and the long-time integration of stochastic systems. Such problems arise in applications involving processes with small relative correlation length or large number of random parameters, and for time-dependent nonlinear systems subject to uncertainty. The new equations are formulated in terms of the time-evolution of the joint probability density function (PDF) between the system's response and the stochastic excitation. In particular, functional integral methods are employed to determine new types of linear deterministic partial differential equations satisfied by the joint response-excitation PDF associated with the stochastic solution of nonlinear stochastic ordinary and partial differential equations. So far the theory is complete for nonlinear and for quasilinear first-order stochastic PDEs subject to random boundary conditions, random initial conditions or random forcing terms. For higher-order equations, such the stochastic wave equation or the Oberbeck-Boussinesq thermal convection equations, it is proposed to develop a new PDF method based on differential constraints for the PDF of the solution. It is proposed to investigate the theoretical and numerical effectiveness of this new approach for high-dimensional random systems, such as random flows subject to high-dimensional random boundary or initial conditions in bounded domains.Stochastic modeling and uncertainty quantification are important new directions in computational mathematics that will enable accurate predictions of physical and biological phenomena,in critical applications such as climate, energy and the design of new products. The proposed work will have significant and broad impact as it will set new rigorous foundations in uncertainty quantification, data assimilation and sensitivity analysis for many physical and biological systems. It will affect fundamentally the way we design new experiments and the type of questions that we can address, while the interaction between simulation and experiment will become more meaningful and more dynamic. This work will also aid in educating a new cadre of simulation scientists in this metadiscipline at the interface of computational mathematics and probability theory.

提出了新的理论和相应的数值算法，以解决物理和生物系统随机建模中的基本开放问题，例如，差异性，缺乏规律性和随机系统的长期整合。此类问题出现在涉及具有较小相关长度或大量随机参数的过程中，并且对于时间依赖性的非线性系统而产生。新方程是根据系统响应和随机激发之间的关节概率密度函数（PDF）的时间进化来制定的。特别是，使用功能积分方法来确定与非线性随机普通和部分微分方程的随机解决方案相关的联合响应激发PDF满足的新型线性确定性偏微分方程。到目前为止，对于非线性和准线性一阶随机PDE，该理论是完整的，受随机边界条件，随机初始条件或随机强迫项的影响。对于高阶方程，例如随机波方程或Oberbeck-Boussinesq热对流方程，建议开发一种基于溶液PDF差异约束的新PDF方法。有人建议研究这种新方法对高维随机系统的理论和数值有效性，例如受到高维随机边界或有限域中的初始条件的随机流。构建和不确定性量化是计算数学中的重要新方向，这些方向将可以准确地预测物理和生物学现象的准确预测，例如批判性现象，以及批判性的现象。 拟议的工作将产生重大和广泛的影响，因为它将为许多物理和生物系统的不确定性量化，数据同化和灵敏度分析树立新的严格基础。它将从根本上影响我们设计新实验的方式以及可以解决的问题的类型，而模拟与实验之间的相互作用将变得更有意义，更具动态性。这项工作还将有助于在计算数学和概率理论的界面上，在该元学学中教育新的模拟科学家干部。