论文标题
风险和不确定性传播的敏感性
Wasserstein sensitivity of Risk and Uncertainty Propagation
论文作者
论文摘要
当传播微分方程数据中的不确定性时，描述不确定性的概率定律通常会出现不确定性。我们对具有随机输入的微分方程的不确定性传播提出了敏感性分析，以对输入度量的扰动。我们关注具有随机系数和源项的椭圆扩散方程，在该方程式中，在总变化和Wasserstein距离上，溶液随机场的概率度量被证明是Lipschitz-contiun的。该结果概括为任何微分方程的解决方案图，均具有局部Hölder对输入参数的依赖性。此外，这些结果扩展到Lipschitz的连续数量的解决方案以及这些结果的连贯风险功能，用于评估其不确定性的影响。我们的分析基于风险功能的敏感性和针对局部hölder映射的推动力措施，相对于扰动输入分布的Wasserstein距离。已建立的结果尤其适用于对数正态扩散的情况和输入随机场的串联表示的截断。
When propagating uncertainty in the data of differential equations, the probability laws describing the uncertainty are typically themselves subject to uncertainty. We present a sensitivity analysis of uncertainty propagation for differential equations with random inputs to perturbations of the input measures. We focus on the elliptic diffusion equation with random coefficient and source term, for which the probability measure of the solution random field is shown to be Lipschitz-continuous in both total variation and Wasserstein distance. The result generalizes to the solution map of any differential equation with locally Hölder dependence on input parameters. In addition, these results extend to Lipschitz continuous quantities of interest of the solution as well as to coherent risk functionals of these applied to evaluate the impact of their uncertainty. Our analysis is based on the sensitivity of risk functionals and pushforward measures for locally Hölder mappings with respect to the Wasserstein distance of perturbed input distributions. The established results are applied, in particular, to the case of lognormal diffusion and the truncation of series representations of input random fields.
