Synchronized step multilevel Markov chain Monte Carlo

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.

💡 Research Summary

This paper introduces SYNCE (Synchronized Step Correlation Enhancement), a novel coupling strategy for multilevel Markov chain Monte Carlo (ML‑MCMC) applied to Bayesian inverse problems with multiple model fidelities. Traditional multilevel approaches rely on sharing the same proposal point between a low‑fidelity and a high‑fidelity Markov chain. Their effectiveness hinges on substantial overlap between the two posterior distributions; when the models are dissimilar, the shared proposal is often rejected by the high‑fidelity chain, causing the chains to decouple and dramatically reducing the correlation needed for variance reduction.

SYNCE departs from this “state‑sharing” paradigm by synchronizing the random steps of the two chains rather than their states. Both chains use the same random seed and the same proposal distribution to generate an identical candidate θ*. After computing the Metropolis–Hastings acceptance probabilities αₗ(θ,θ*) and α_h(θ,θ*) for the low‑ and high‑fidelity posteriors, a single uniform random number u∈U(0,1) determines the joint move: if u is smaller than the minimum of the two acceptance probabilities, both chains move; if u lies between the two probabilities, only the chain with the larger acceptance moves; otherwise both stay. This mechanism guarantees that the marginal kernels remain correct while the two chains are driven by a common source of randomness, creating strong dependence even when the posterior supports barely intersect.

The authors formalize the joint kernel K̂ on the product space, prove that it preserves each marginal distribution, and establish in Theorem 4.1 that K̂ possesses a unique invariant measure μ̂ and is V‑geometrically ergodic. The convergence rate ρ̂ is strictly better than that of delayed‑acceptance or maximal‑coupling schemes, and remains high (≈0.85) even when posterior overlap is as low as 0.05.

An adaptive extension, SYNCE‑AR, monitors acceptance statistics and dynamically rescales the proposal distribution to maintain high synchronization without compromising the theoretical guarantees.

Extensive numerical experiments are presented on three representative inverse problems: (a) a 2‑D PDE parameter estimation (θ∈ℝ⁴), (b) a high‑dimensional Bayesian regression (θ∈ℝ⁵⁰), and (c) a coupled fluid‑structure model (θ∈ℝ¹⁰). For each case, low‑fidelity models (coarse meshes or simplified physics) are paired with fine‑fidelity counterparts. Compared with state‑of‑the‑art coupling methods—delayed acceptance, maximal coupling, and independent proposals—SYNCE consistently achieves a variance reduction factor of 2–3 and a wall‑clock speed‑up of 1.8–2.5. Notably, the correlation coefficient between levels stays above 0.85 across all experiments, ensuring that the multilevel telescoping sum variance V