We introduce a novel adaptive Gaussian Process Regression (GPR) methodology for efficient construction of surrogate models for Bayesian inverse problems with expensive forward model evaluations. An adaptive design strategy focuses on optimizing both the positioning and simulation accuracy of training data in order to reduce the computational cost of simulating training data without compromising the fidelity of the posterior distributions of parameters. The method interleaves a goal-oriented active learning algorithm selecting evaluation points and tolerances based on the expected impact on the Kullback-Leibler divergence of surrogated and true posterior with a Markov Chain Monte Carlo sampling of the posterior. The performance benefit of the adaptive approach is demonstrated for two simple test problems.

翻译：我们提出了一种新颖的自适应高斯过程回归（GPR）方法，用于高效构建贝叶斯反问题的代理模型，以应对正向模型评估成本高昂的问题。该方法采用自适应设计策略，重点优化训练数据的位置和仿真精度，从而在不降低参数后验分布保真度的前提下，减少训练数据仿真的计算成本。该算法结合了一种目标导向的主动学习策略，基于代理后验与真实后验之间的库尔贝克-莱布勒散度的预期影响，选择评估点和容差，同时结合马尔可夫链蒙特卡洛方法对后验进行采样。通过两个简单测试问题，验证了自适应方法的性能优势。