基于ILUES的自适应高斯过程方法用于多模态贝叶斯反问题
An ILUES-based adaptive Gaussian process method for multimodal Bayesian inverse problems
AI总结：
本文提出基于ILUES的自适应高斯过程方法，用于解决多模态贝叶斯反问题中的高效采样问题。
摘要：
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，当正向模型计算成本较高时，从后验分布中采样尤其具有挑战性。当后验分布是多模态时，这一挑战进一步加剧。为了解决这个问题，我们提出了一种基于高斯过程（GP）的方法，间接构建正向模型的替代模型。具体来说，未归一化的后验密度被表示为辅助密度和指数高斯过程替代模型的乘积。迭代过程中，辅助密度会收敛到后验分布，从任意初始密度开始。然而，高斯过程回归的效率高度依赖于训练数据的质量。因此，我们利用迭代局部更新集合平滑器（ILUES）来生成高质量的样本，这些样本集中在具有高后验概率的区域。随后，基于替代模型和通过聚类方法提取的模态信息，使用具有高斯混合（GM）提议的马尔可夫链蒙特卡罗（MCMC）方法从辅助密度中采样。通过数值示例，我们证明所提出的方法能够用有限的正向模拟准确而高效地表示后验分布。
英文摘要：
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.
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