We formulate a novel approach to solve a class of stochastic problems, referred to as data-consistent inverse (DCI) problems, which involve the characterization of a probability measure on the parameters of a computational model whose subsequent push-forward matches an observed probability measure on specified quantities of interest (QoI) typically associated with the outputs from the computational model. Whereas prior DCI solution methodologies focused on either constructing non-parametric estimates of the densities or the probabilities of events associated with the pre-image of the QoI map, we develop and analyze a constrained quadratic optimization approach based on estimating push-forward measures using weighted empirical distribution functions. The method proposed here is more suitable for low-data regimes or high-dimensional problems than the density-based method, as well as for problems where the probability measure does not admit a density. Numerical examples are included to demonstrate the performance of the method and to compare with the density-based approach where applicable.

翻译：我们提出了一种新颖的方法来解决一类随机问题，即数据一致反演问题。这类问题涉及对计算模型参数上的概率测度进行刻画，使其后续的推前测度与观测到的、关于特定感兴趣量的概率测度相匹配（这些感兴趣量通常与计算模型的输出相关）。以往的数据一致反演求解方法主要聚焦于构建与感兴趣量映射原像相关联的密度非参数估计或事件概率估计，而我们则发展并分析了一种基于加权经验分布函数估计推前测度的约束二次优化方法。与基于密度的方法相比，本文提出的方法更适用于低数据量场景或高维问题，以及概率测度不存在密度函数的问题。文中还包含数值算例以展示该方法的性能，并在适用情况下与基于密度的方法进行对比。