Efficient subsampling-rate selection for practical Multilevel Markov chain Monte Carlo

2026-10-06

AI Summary

This study addresses the excessive computational overhead in multilevel Markov chain Monte Carlo (ML-MCMC) arising from overly conservative subsampling rate selection. To mitigate this, the proposed method treats finite subsampling as an error budget and derives a cost-error criterion that balances errors across levels to optimize parameters. Departing from traditional independence assumptions, it establishes an empirical diagnostic mechanism grounded in perturbed bias bounds and state-averaged forgetting. The theoretical analysis integrates sample variance, integrated autocorrelation time, and the Poisson equation. Evaluated on elastic beam and Darcy flow benchmarks, this approach significantly reduces computational costs compared with existing methods.

Abstract

The Multilevel Markov chain Monte Carlo algorithm accelerates Bayesian inversion by exploiting correlations between discretized quantities of interest across a hierarchy of model resolutions. A key practical parameter is the subsampling rate used to generate coarse-level proposals for the finer-level chains. Existing practice sets this rate equal to an integrated autocorrelation time as a proxy for proposal decorrelation, but this choice can be overly conservative and lead to substantial computational overhead. In this work we treat finite subsampling as an error-budget question. We derive a cost-error criterion for selecting a computationally efficient subsampling rate using estimates of sample variances, integrated autocorrelation times, and per-level costs. The criterion balances multilevel sampling errors across levels rather than enforcing approximate independence of the coarse proposals. We clarify the theoretical role of independence through an ideal reset-reference kernel, for which independent coarse proposals may be replaced by proposals generated from a reversible coarse-level kernel. We derive a Poisson-equation representation of the perturbation bias resulting from finite subsampling. Under a state-averaged lower-level forgetting assumption, this yields a conditional geometric perturbation-bias bound; in computation, the corresponding projected diagnostic is used as an empirical error-budget check. Numerical experiments on a highly autocorrelated elasticity beam benchmark show substantial reductions in computational cost compared with IAT-based subsampling. A second Darcy-flow inverse problem gives supporting evidence of qualitatively similar finite-subsampling behaviour in another elliptic benchmark.

Problem

• Multilevel Markov chain Monte Carlo
• subsampling rate
• Bayesian inversion
• computational cost
• error budget

Innovation

• Multilevel Markov chain Monte Carlo
• Subsampling-rate selection
• Error-budget criterion
• Perturbation bias bound
• Bayesian inversion