Preconditioned Crank–Nicolson Differential Evolution Markov Chain Method

Bayesian inverse problems governed by partial differential equations (PDEs) are often characterized by high-dimensional parameter spaces and computationally expensive forward models, making efficient posterior exploration challenging. Function-space Markov chain Monte Carlo (McMC) methods such as the preconditioned Crank–Nicolson (pCN) algorithm provide dimension-robust proposals by preserving the prior distribution, but their exploration efficiency may deteriorate in anisotropic or strongly likelihood-informed regimes. Population-based Differential Evolution (DE) proposals can improve global exploration by exploiting ensemble information, yet they typically sacrifice prior invariance and lack guarantees in function-space settings. In this work, we introduce the Preconditioned Crank–Nicolson Differential Evolution (pCNDE) method, a new McMC proposal that integrates differential evolution directions into the pCN framework while preserving exact prior invariance and detailed balance. The resulting proposal retains the dimension-robust properties of pCN while automatically learning jump scale and orientation from an ensemble of chains, enabling efficient sampling with relatively small populations. We establish reversibility and posterior invariance of the resulting Markov chain and develop an asynchronous parallel implementation that reduces communication overhead in large-scale simulations. Numerical experiments on a nonlinear PDE-constrained Bayesian inverse problem in porous media demonstrate substantial efficiency gains over classical pCN, including higher acceptance rates and significantly fewer iterations to reach convergence. Performance is assessed using multichain stabilization diagnostics and wall-clock-normalized efficiency metrics. The proposed pCNDE method provides a scalable, theoretically grounded sampling strategy for high-dimensional Bayesian inverse problems governed by PDEs.