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首页> 外文期刊>Journal of Computational Physics >New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs
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New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs

机译：一阶非线性PDE随机解的联合响应-激发概率密度函数的新演化方程

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摘要

By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed for the classical linear and nonlinear advection equations and for the advection-reaction equation. By using a Fourier-Galerkin spectral method we obtain numerical solutions of the proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.
机译：通过使用函数积分方法，我们确定了与一阶非线性偏微分方程（PDE）的随机解相关的联合响应-激发概率密度函数（PDF）所满足的新演化方程。针对完全非线性和准线性标量PDE提出了该理论，其服从于随机边界条件，随机初始条件或随机强迫项。讨论了经典线性和非线性对流方程以及对流反应方程的特殊应用。通过使用傅里叶-加勒金谱方法，我们获得了所提出的响应-激励PDF方程的数值解。将这些数值解与使用更常规的统计方法（例如概率配置和多元素概率配置方法）获得的数值解进行比较。发现响应激励方法可以准确预测系统的统计属性。此外，它可以直接确定概率分布的尾巴，从而有助于评估罕见事件和相关风险。如果PDE受到高维随机边界或初始条件的影响，则响应激励方法的计算成本比传统的统计方法之一要小几个数量级。还解决了涉及多维联合响应-激励PDF的演化方程的高维性问题。

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