Breidt J, Butler T, Estep D
Department of Statistics, Colorado State University, Fort Collins, CO 80523.
SIAM J Numer Anal. 2011 Sep 1;49(5):1836-1859. doi: 10.1137/100785946.
We consider the inverse sensitivity analysis problem of quantifying the uncertainty of inputs to a deterministic map given specified uncertainty in a linear functional of the output of the map. This is a version of the model calibration or parameter estimation problem for a deterministic map. We assume that the uncertainty in the quantity of interest is represented by a random variable with a given distribution, and we use the law of total probability to express the inverse problem for the corresponding probability measure on the input space. Assuming that the map from the input space to the quantity of interest is smooth, we solve the generally ill-posed inverse problem by using the implicit function theorem to derive a method for approximating the set-valued inverse that provides an approximate quotient space representation of the input space. We then derive an efficient computational approach to compute a measure theoretic approximation of the probability measure on the input space imparted by the approximate set-valued inverse that solves the inverse problem.
给定确定性映射输出的线性泛函中的特定不确定性，我们考虑量化该映射输入不确定性的逆灵敏度分析问题。这是确定性映射的模型校准或参数估计问题的一个版本。我们假设感兴趣量的不确定性由具有给定分布的随机变量表示，并使用全概率定律来表达输入空间上相应概率测度的逆问题。假设从输入空间到感兴趣量的映射是光滑的，我们通过使用隐函数定理来求解通常不适定的逆问题，以推导一种近似集值逆的方法，该方法提供输入空间的近似商空间表示。然后，我们推导一种有效的计算方法，以计算由解决逆问题的近似集值逆赋予输入空间的概率测度的测度理论近似。
