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

为解决物理和生物系统随机建模中的基本开放问题，如随机系统的维度灾变、缺乏正则性和长时间积分问题，提出了新的理论和相应的数值算法。这类问题存在于相对关联长度较小或随机参数较多的过程，以及具有不确定性的时变非线性系统。根据系统响应和随机激励之间的联合概率密度函数(PDF)的时间演化关系，建立了新的方程。特别地，利用泛函积分方法确定了与非线性随机常、偏微分方程组的随机解相关的联合响应激励函数所满足的新型线性确定性偏微分方程组。到目前为止，对于非线性和具有随机边界条件、随机初始条件或随机强迫项的一阶拟线性随机偏微分方程解的理论是完备的。对于高阶方程，如随机波动方程或Oberbeck-Boussinesq热对流方程，提出了一种新的基于微分约束的PDF方法。随机建模和不确定性量化是计算数学中重要的新方向，在气候、能源和新产品设计等关键应用中，它将使物理和生物现象的准确预测成为可能。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的类型，而模拟和实验之间的交互将变得更有意义和更动态。这项工作还将有助于在计算数学和概率论之间的这一元学科中培养一批新的模拟科学家。

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.