We build upon a recently developed approach for solving stochastic inverse problems based on a combination of measure-theoretic principles and Bayes' rule. We propose a multi-fidelity method to reduce the computational burden of performing uncertainty quantification using high-fidelity models. This approach is based on a Monte Carlo framework for uncertainty quantification that combines information from solvers of various fidelities to obtain statistics on the quantities of interest of the problem. In particular, our goal is to generate samples from a high-fidelity push-forward density at a fraction of the costs of standard Monte Carlo methods, while maintaining flexibility in the number of random model input parameters. Key to this methodology is the construction of a regression model to represent the stochastic mapping between the low- and high-fidelity models, such that most of the computations can be leveraged to the low-fidelity model. To that end, we employ Gaussian process regression and present extensions to multi-level-type hierarchies as well as to the case of multiple quantities of interest. Finally, we demonstrate the feasibility of the framework in several numerical examples.

中文翻译：

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基于条件密度的概率多保真方法的随机逆问题的数据一致性解决方案

我们基于测量理论原理和贝叶斯定律的组合，基于最近开发的解决随机逆问题的方法。我们提出了一种多保真度方法，以减少使用高保真度模型执行不确定性量化的计算负担。此方法基于用于不确定性量化的蒙特卡洛框架，该框架结合了来自各种保真度求解器的信息，以获得有关问题关注数量的统计信息。特别是，我们的目标是以高保真推​​算密度生成样本，而成本仅为标准蒙特卡洛方法的一小部分，同时保持随机模型输入参数数量的灵活性。该方法的关键是构建回归模型，以表示低保真模型和高保真模型之间的随机映射，以便可以将大多数计算用于低保真模型。为此，我们采用了高斯过程回归，并提出了对多层次类型层次结构以及多个关注量情况的扩展。最后，我们在几个数值示例中证明了该框架的可行性。