Efficient Markov chain Monte Carlo sampling for Bayesian inverse problems with covariance matrix adaptation

Bayesian inference provides a principled framework for solving inverse problems, yet efficient sampling of high-dimensional non-Gaussian posterior distributions remains challenging for conventional Markov chain Monte Carlo (MCMC) methods. This study introduces the covariance matrix adaptation Metropolis (CMAM) algorithm, an efficient MCMC approach that synergistically integrates the population-based covariance matrix adaptation evolution strategy (CMA-ES) optimization algorithm with Metropolis sampling. The proposed CMAM employs multiple parallel chains to enhance exploration in Bayesian inversion and leverages the adaptive mechanisms of CMA-ES to dynamically adapt both the direction and scale of proposal distributions. A decoupled Metropolis acceptance mechanism ensures proper Markov chain construction throughout the adaptation process. The method further incorporates augmentation and dual adaptation strategies, utilizing information from rejected samples and dual CMA-ES optimizations, to improve robustness in multimodal and high-dimensional scenarios. Theoretical analysis confirms ergodicity of the proposed method. Numerical benchmarks demonstrate that CMAM is capable of sampling with sub-dimensional chains, while still achieving comparable performance in inversion accuracy and convergence efficiency to state-of-the-art multi-chain adaptive MCMC methods. Hydrogeological case studies further show that CMAM accurately reconstructs spatially heterogeneous conductivity fields and simultaneously identifies contaminant source locations and release histories. These results highlight CMAM as a practical tool for Bayesian inference in real-world hydrological and environmental applications.