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

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性问题，如：维数缺失、缺乏规律性和随机系统的长时间集成。这类问题出现在涉及相对相关长度较小或随机参数较多的过程的应用中，以及受不确定性影响的时变非线性系统中。用系统响应与随机激励之间的联合概率密度函数（PDF）的时间演化形式来表示新方程。特别地，利用泛函积分方法确定了一类由非线性随机常微分方程和偏微分方程的随机解联合响应激励PDF所满足的新型线性确定性偏微分方程。到目前为止，该理论对于具有随机边界条件、随机初始条件或随机强迫项的非线性和拟线性一阶随机偏微分方程是完整的。对于随机波动方程或Oberbeck-Boussinesq热对流方程等高阶方程，提出了一种基于微分约束的PDF求解方法。研究了该方法在高维随机系统中的理论和数值有效性，例如在有界域中受高维随机边界或初始条件约束的随机流动。随机建模和不确定性量化是计算数学中重要的新方向，它将使物理和生物现象的准确预测成为可能，在气候、能源和新产品设计等关键应用中。这项工作将产生重大而广泛的影响，因为它将为许多物理和生物系统的不确定性量化、数据同化和敏感性分析奠定新的严格基础。它将从根本上影响我们设计新实验的方式和我们可以解决的问题类型，而模拟和实验之间的互动将变得更有意义，更有活力。这项工作也将有助于在计算数学和概率论的界面上，在这门元学科中培养新的模拟科学家骨干。

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.