We propose SYNCE (synchronized step correlation enhancement), a new algorithm for coupling Markov chains within multilevel Markov chain Monte Carlo (ML-MCMC) estimators. We apply this algorithm to solve Bayesian inverse problems using multiple model fidelities. SYNCE is inspired by the concept of common random number coupling in Markov chain Monte Carlo sampling. Unlike state-of-the-art methods that rely on the overlap of level-wise posteriors, our approach enables effective coupling even when posteriors differ substantially. This overlap-independence generates significantly higher correlation between samples at different fidelity levels, improving variance reduction and computational efficiency in the ML-MCMC estimator. We prove that SYNCE admits a unique invariant probability measure and demonstrate that the coupled chains converge to this measure faster than existing overlap-dependent methods, particularly when models are dissimilar. Numerical experiments validate that SYNCE consistently outperforms current coupling strategies in terms of computational efficiency and scalability across varying model fidelities and problem dimensions.

翻译：我们提出SYNCE（同步步长相关性增强），一种用于在多级马尔可夫链蒙特卡洛（ML-MCMC）估计器中耦合马尔可夫链的新算法。我们将该算法应用于利用多模型保真度求解贝叶斯反问题。SYNCE的灵感来源于马尔可夫链蒙特卡洛采样中的公共随机数耦合概念。与依赖层级后验分布重叠的最先进方法不同，即使在后验分布差异显著时，我们的方法也能实现有效耦合。这种对重叠的独立性在不同保真度层级的样本间产生了显著更高的相关性，从而改善了ML-MCMC估计器的方差缩减和计算效率。我们证明了SYNCE具有唯一的平稳概率测度，并证明相较于现有的依赖重叠的方法，耦合链能更快地收敛到该测度，尤其是在模型差异较大时。数值实验验证了在不同模型保真度和问题维度下，SYNCE在计算效率和可扩展性方面始终优于当前的耦合策略。