In this work, we propose a new algorithm for coupling Markov chains in a Markovian multilevel Monte Carlo estimator. We apply this approach for solving Bayesian inverse problems that consist of multiple model fidelities. The coupling methodology, termed as synchronized step correlation enhancement (SYNCE), is inspired by the concept of using common random numbers in Markov chain Monte Carlo sampling. This methodology is shown to be more efficient and cost-effective than existing couplings in the literature. This improvement is achieved because SYNCE leads to higher correlation of samples obtained from level dependent posteriors than previous works. SYNCE is especially effective at coarse levels of the hierarchy where posteriors differ significantly from each other, resulting in orders of magnitude improvement in variance reduction. We first demonstrate the effectiveness of our proposed methodology by comparing it with existing algorithms on two simple examples taken from the literature. We then apply our methodology to a more complex example in the context of uncertainty quantification in subsurface flow simulations.

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