SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 4, Page 1493-1521, January 2020.
The generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accuracy, PDE solutions need to have high regularity in the random space, but this is what hyperbolic type problems cannot provide. We provide a counterargument in this paper and show that even though the solution profile develops singularities in the random space, which destroys the spectral accuracy of gPC, the physical quantities (such as the shock emergence time, the shock location, and the shock strength) are all smooth functions of the uncertainties coming from both initial data and the wave speed. With proper shifting, the solution's polynomial interpolation approximates the real solution accurately, and the error decays as the order of the polynomial increases. Therefore this work provides a new perspective to “quantify uncertainties" and significantly improves the accuracy of the gPC method with a slight reformulation. We use the Burgers' equation as an example for thorough analysis, and the analysis could be extended to general conservation laws with convex fluxes.

中文翻译：

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具有冲击的Burgers方程的灵敏度分析

SIAM / ASA不确定性量化期刊，第8卷，第4期，第1493-1521页，2020年1月。
广义多项式混沌（gPC）方法已广泛用于方程包含随机变量的不确定性量化问题中。为了使gPC达到高精度，PDE解决方案需要在随机空间中具有较高的规则性，但这是双曲线型问题无法提供的。我们在本文中提供了一个反对意见，并表明，即使解决方案配置文件在随机空间中发展出奇异性，也破坏了gPC的光谱准确性，物理量（例如，电击出现时间，电击位置和电击强度）都是来自初始数据和波速的不确定性的平滑函数。通过适当的移位，解决方案的多项式插值可以准确地逼近实际解决方案，并且误差随着多项式阶数的增加而衰减。