推进测度间的Wasserstein距离及其在不确定性量化中的应用
The Wasserstein Distances Between Pushed-Forward Measures with Applications to Uncertainty Quantification
摘要：
在动力学和物理系统的研究中，输入参数往往是不确定的或按测度$\varrho$随机分布的。系统的响应$f$将$\varrho$推进到我们想要研究的新度量$f\circ \varrho$。但是，我们可能无法访问$f$，而只能访问其近似值$g$。因此，我们得出了一个基本问题--如果$f$和$g$在$L^q$中很接近，那么$g\circ \varrho$是否很好地近似于$f\circ \varrho$，以及在什么意义上？以前，我们证明了这个问题的答案可能是否定的概率密度函数（PDF）之间的$L^p$距离。在这里，我们表明，Wasserstein度量是这个问题的适当框架。对于任何$p\geq 1$，我们从上面通过$\|f-g\|_{q}$限制Wasserstein距离$W_p (f\circ \varrho , g\circ \varrho) $。此外，我们提供的情况下，$p=1,2$的下限。最后，我们将我们的理论应用于分析计算不确定性量化领域的常用数值方法。
英文摘要：
In the study of dynamical and physical systems, the input parameters are often uncertain or randomly distributed according to a measure $\varrho$. The system's response $f$ pushes forward $\varrho$ to a new measure $f\circ \varrho$ which we would like to study. However, we might not have access to $f$ but only to its approximation $g$. We thus arrive at a fundamental question -- if $f$ and $g$ are close in $L^q$, does $g\circ \varrho$ approximate $f\circ \varrho$ well, and in what sense? Previously, we demonstrated that the answer to this question might be negative in terms of the $L^p$ distance between probability density functions (PDF). Here we show that the Wasserstein metric is the proper framework for this question. For any $p\geq 1$, we bound the Wasserstein distance $W_p (f\circ \varrho , g\circ \varrho) $ from above by $\|f-g\|_{q}$. Furthermore, we provide lower bounds for the cases of $p=1,2$. Finally, we apply our theory to the analysis of common numerical methods in the field of computational uncertainty quantification.
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