We present Bayesian techniques for solving inverse problems which involve mean-square convergent random approximations of the forward map. Noisy approximations of the forward map arise in several fields, such as multiscale problems and probabilistic numerical methods. In these fields, a random approximation can enhance the quality or the efficiency of the inference procedure, but entails additional theoretical and computational difficulties due to the randomness of the forward map. A standard technique to address this issue is to combine Monte Carlo averaging with Markov chain Monte Carlo samplers, as for example in the pseudo-marginal Metropolis--Hastings methods. In this paper, we consider mean-square convergent random approximations, and quantify how Monte Carlo errors propagate from the forward map to the solution of the inverse problems. Moreover, we review and describe simple techniques to solve such inverse problems, and compare performances with a series of numerical experiments.
翻译:我们提出巴伊西亚解决反问题的技巧,这些技巧涉及前方地图的平均平方集合随机近似。前方地图的噪音近近似出现在多个领域,例如多尺度问题和概率数字方法。在这些方面,随机近近近可能提高推论程序的质量或效率,但由于前方地图的随机性而引起额外的理论和计算困难。解决这一问题的标准技巧是把蒙特卡洛平均平均和马尔科夫链的蒙特卡洛采样器结合起来,例如假边边际大都会-哈斯廷方法。在本文中,我们考虑暗方趋近,并量化蒙特卡洛错误如何从前方地图传播到解决反面问题的方法。此外,我们审查并描述解决此类反面问题的简单技术,并将性能与一系列数字实验进行比较。