  This work presents an efficient approach for accelerating multilevel Markov
Chain Monte Carlo (MCMC) sampling for large-scale problems using low-fidelity
machine learning models. While conventional techniques for large-scale Bayesian
inference often substitute computationally expensive high-fidelity models with
machine learning models, thereby introducing approximation errors, our approach
offers a computationally efficient alternative by augmenting high-fidelity
models with low-fidelity ones within a hierarchical framework. The multilevel
approach utilizes the low-fidelity machine learning model (MLM) for inexpensive
evaluation of proposed samples thereby improving the acceptance of samples by
the high-fidelity model. The hierarchy in our multilevel algorithm is derived
from geometric multigrid hierarchy. We utilize an MLM to acclerate the coarse
level sampling. Training machine learning model for the coarsest level
significantly reduces the computational cost associated with generating
training data and training the model. We present an MCMC algorithm to
accelerate the coarsest level sampling using MLM and account for the
approximation error introduced. We provide theoretical proofs of detailed
balance and demonstrate that our multilevel approach constitutes a consistent
MCMC algorithm. Additionally, we derive conditions on the accuracy of the
machine learning model to facilitate more efficient hierarchical sampling. Our
technique is demonstrated on a standard benchmark inference problem in
groundwater flow, where we estimate the probability density of a quantity of
interest using a four-level MCMC algorithm. Our proposed algorithm accelerates
multilevel sampling by a factor of two while achieving similar accuracy
compared to sampling using the standard multilevel algorithm.
