Stochastic Galerkin methods can be used to approximate the solution to a differential equation in the presence of uncertainties represented as stochastic inputs or parameters. The strategy is to express the resulting stochastic solution using $M+1$ terms of a polynomial chaos expansion and then derive and solve a deterministic, coupled system of PDEs with standard numerical techniques. Some challenges with conventional numeric techniques applied in this context are as follows: (1) the solution to a polynomial chaos $M$ system cannot easily reuse an already-existing computer solution to an $M_0$ system for some $M_0<M,$ and (2) there is no flexibility around choosing which variables in an $M$ system are more important or advantageous to estimate accurately. This latter point is especially relevant when, rather than focusing on the PDE solution itself, the objective is to approximate some function of the PDE solution that weights the solution variables with relative levels of importance. In Lyman and Iaccarino (2020) we first present a promising iterative strategy (bluff-and-fix) to address challenge (1); we find that numerical estimates of the accuracy and efficiency demonstrate that bluff-and-fix can be more effective than using monolithic methods to solve a whole $M$ system directly. This paper is an extended version of Lyman and Iaccarino (2020) that showcases how bluff-and-fix successfully addresses challenge (2) as well by allowing for choice in which variables are approximated more accurately, in particular when estimating statistical properties such as the mean and variance of an $M$ system solution.

在存在表示为随机输入或参数的不确定性的情况下，可以使用随机Galerkin方法来逼近微分方程的解。该策略是使用以下方法表达最终的随机解决方案$\mathrm{中号}+\mathrm{1个}$多项式混沌扩展的项，然后用标准数值技术推导和求解确定性的PDE耦合系统。在这种情况下应用常规数值技术面临的挑战如下：（1）多项式混沌的解决方案$\mathrm{中号}$ 系统无法轻松地将现有的计算机解决方案重用于 ${\mathrm{中号}}_0$ 一些系统 ${\mathrm{中号}}_0<\mathrm{中号}，$ （2）选择一个变量中的哪些变量没有灵活性 $\mathrm{中号}$系统对于准确估算更为重要或有利。当目标不是近似于PDE解决方案本身，而是要以相对重要程度对PDE解决方案的某些功能进行加权时，后一点尤为重要。在Lyman和Iaccarino（2020）中，我们首先提出了一种有前途的迭代策略（虚张声势并修复）来应对挑战（1）；我们发现，对准确性和效率的数值估计表明，与使用整体方法求解整体相比，虚张声势和修复效果更好$\mathrm{中号}$系统直接。本文是Lyman and Iaccarino（2020）的扩展版本，展示了虚张声势并修复如何成功地应对挑战（2），以及允许选择更精确地近似变量的方法，特别是在估计统计属性时，例如均值和方差$\mathrm{中号}$ 系统解决方案。