In inverse problems, the parameters of a model are estimated based on observations of the model response. The Bayesian approach is powerful for solving such problems; one formulates a prior distribution for the parameter state that is updated with the observations to compute the posterior parameter distribution. Solving for the posterior distribution can be challenging when, e.g., prior and posterior significantly differ from one another and/or the parameter space is high-dimensional. We use a sequence of importance sampling measures that arise by tempering the likelihood to approach inverse problems exhibiting a significant distance between prior and posterior. Each importance sampling measure is identified by cross-entropy minimization as proposed in the context of Bayesian inverse problems in Engel et al. (2021). To efficiently address problems with high-dimensional parameter spaces we set up the minimization procedure in a low-dimensional subspace of the original parameter space. The principal idea is to analyse the spectrum of the second-moment matrix of the gradient of the log-likelihood function to identify a suitable subspace. Following Zahm et al. (2021), an upper bound on the Kullback-Leibler-divergence between full-dimensional and subspace posterior is provided, which can be utilized to determine the effective dimension of the inverse problem corresponding to a prescribed approximation error bound. We suggest heuristic criteria for optimally selecting the number of model and model gradient evaluations in each iteration of the importance sampling sequence. We investigate the performance of this approach using examples from engineering mechanics set in various parameter space dimensions.
翻译:模型的参数是根据模型反应的观测结果估计的。Bayesian 方法对于解决这些问题具有很强的作用; 一种方法为参数状态预设了先前的分布, 并附有计算后方参数分布的观测结果( 2021年) 更新了参数状态。 后方参数分布的解决方案具有挑战性, 例如, 前方和后方之间差异很大, 且/ 或者参数空间是高维的。 我们使用一系列重要的抽样措施, 减缓了在前方和后方之间出现显著距离的反向问题的可能性。 每种重要取样措施都是按照Bayesian 反向问题( Engel 等人. (2021年) 中提议的跨倍增殖序列最小化方法确定的。 要有效解决高维参数空间分布的问题, 我们将在原始参数空间空间空间的低维次空间空间空间设置最小化程序。 我们的主要想法是分析日志相似值函数梯度模型的第二移动模型的频谱, 以确定适当的亚空间空间。 在Zahm 等人 之后( 2021年) 之后, 以跨度的跨度测序顺序为一个高空基度标准标准, 提供了对地平面标准的直径度标准的上,, 提供了对地平面精确度的精确度评估,, 的精确度标准, 提供了对地标的精确度的精确度的上对地标的精确度, 。