A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow
This work addresses the computational bottleneck of MCMC for high-dimensional parameter spaces in subsurface flow uncertainty quantification, offering significant cost savings.
The paper tackles the high computational cost of Markov chain Monte Carlo methods for large-scale uncertainty quantification in subsurface flow. It proposes a multilevel Metropolis-Hastings algorithm that achieves over an order of magnitude cost reduction compared to standard methods for tolerances ε < 10^{-2}.
In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large--scale applications with high dimensional parameter spaces, e.g. in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis-Hastings algorithm, and give an abstract, problem dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard Metropolis-Hastings estimator. Numerical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of more than an order of magnitude in the cost of the multilevel estimator over the standard Metropolis-Hastings algorithm for tolerances $\varepsilon < 10^{-2}$.