Acceptance Rate-Adaptive pCN MCMC for Moderate-Dimensional Bayesian Inverse Problems

The preconditioned Crank–Nicolson (pCN) Markov chain Monte Carlo method is a standard tool for high-dimensional Bayesian inverse problems because its proposals preserve the prior and its performance degrades gracefully under mesh refinement. Yet its efficiency depends heavily on the step-size parameter β, which practitioners must tune manually for each problem. To remove this burden, we propose an acceptance rate-adaptive pCN algorithm that calibrates β automatically during a finite burn-in phase via a Robbins–Monro stochastic approximation driven by cumulative acceptance rate feedback. We also studied a dual-signal extension that supplements acceptance rate adaptation with an auxiliary effective sample size (ESS) feedback term, activated gradually through a smooth logistic switch. To contextualize the proposed methods, we conducted a systematic comparison with the adaptive Metropolis (AM) and the Metropolis-adjusted Langevin algorithm (MALA). In both adaptive pCN variants, adaptation is confined to the burn-in and the step size is then frozen, so that the post-burn-in samples are produced by a time-homogeneous Metropolis–Hastings kernel targeting the correct posterior. Numerical experiments on five linear-Gaussian benchmarks with a controlled spectral structure (dimensions: 10 to 100, condition numbers: 2 to 200, SNR from −9 dB to +8 dB) and a nonlinear PDE-constrained inverse problem governed by the cubic–quintic nonlinear Schrödinger equation showed that the acceptance rate-adaptive pCN yields consistent improvements over vanilla pCN, reaching approximately an 1.8× higher ESS in ill-conditioned regimes with low inter-seed variance. The AM achieved a superior ESS on well-conditioned and low-dimensional problems, while the MALA excelled in low-SNR regimes, but degraded under sharp posteriors. A sensitivity analysis confirmed the robustness to hyperparameter choices, and multi-seed experiments validated the reproducibility of all findings. These results indicate that automatic step-size adaptation can substantially improve the practical efficiency of pCN for challenging moderate-dimensional Bayesian inverse problems without manual tuning.