So far the theory and numerical practice of stochastic partial differential equations (SPDEs) have dealt almost exclusively with Gaussian noise or Lévy noise. Recently, Mikulevicius and Rozovskii (Stoch Partial Differ Equ Anal Comput 4:319–360, 2016) proposed a distribution-free Skorokhod–Malliavin calculus framework that is based on generalized stochastic polynomial chaos expansion, and is compatible with arbitrary driving noise. In this paper, we conduct systematic investigation on numerical results of these newly developed distribution-free SPDEs, exhibiting the efficiency of truncated polynomial chaos solutions in approximating moments and distributions. We obtain an estimate for the mean square truncation error in the linear case. The theoretical convergence rate, also verified by numerical experiments, is exponential with respect to polynomial order and cubic with respect to number of random variables included.

中文翻译：

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任意噪声驱动随机PDE的数值解

到目前为止，随机偏微分方程（SPDE）的理论和数值实践几乎只涉及高斯噪声或Lévy噪声。最近，Mikulevicius和Rozovskii（Stoch Partial Differ Equ Anal Comput 4：319–360，2016）提出了一种无分布的Skorokhod-Malliavin演算框架，该框架基于广义随机多项式混沌扩展，并且与任意行驶噪声兼容。在本文中，我们对这些新开发的无分布SPDE的数值结果进行了系统的研究，展示了截断多项式混沌解在逼近矩和分布方面的效率。我们获得了线性情况下均方截断误差的估计。理论收敛速度，也通过数值实验验证，