Inverse problems are prevalent in both scientific research and engineering applications. In the context of Bayesian inverse problems, sampling from the posterior distribution can be particularly challenging when the forward models are computationally expensive. This challenge is further compounded when the posterior distribution is multimodal. To address this issue, we propose a Gaussian process (GP)-based method to indirectly build surrogates for the forward model. Specifically, the unnormalized posterior density is expressed as a product of an auxiliary density and an exponential GP surrogate. Iteratively, the auxiliary density converges to the posterior distribution, starting from an arbitrary initial density. However, the efficiency of GP regression is highly influenced by the quality of the training data. Therefore, we utilize the iterative local updating ensemble smoother (ILUES) to generate high-quality samples that are concentrated in regions with high posterior probability. Subsequently, based on the surrogate model and mode information extracted using a clustering method, Markov chain Monte Carlo (MCMC) with a Gaussian mixed (GM) proposal is used to draw samples from the auxiliary density. Through numerical examples, we demonstrate that the proposed method can accurately and efficiently represent the posterior with a limited number of forward simulations.

翻译：反问题在科学研究和工程应用中普遍存在。在贝叶斯反问题的背景下，当正演模型计算成本高昂时，从后验分布中采样可能尤为困难。当后验分布是多模态时，这一挑战会进一步加剧。为解决此问题，我们提出一种基于高斯过程的方法，间接地为正演模型构建代理模型。具体而言，未归一化的后验密度被表示为一个辅助密度与一个指数型高斯过程代理模型的乘积。迭代过程中，辅助密度从任意初始密度开始，收敛于后验分布。然而，高斯过程回归的效率很大程度上受训练数据质量的影响。因此，我们利用迭代局部更新集合平滑器来生成高质量样本，这些样本集中于后验概率较高的区域。随后，基于代理模型以及通过聚类方法提取的模态信息，采用具有高斯混合提议分布的马尔可夫链蒙特卡洛方法从辅助密度中抽取样本。通过数值算例，我们证明了所提方法能够以有限次数的正演模拟，准确且高效地表征后验分布。