We propose a multilevel Markov chain Monte Carlo (MCMC) method for the Bayesian inference of random field parameters in PDEs using high-resolution data. Compared to existing multilevel MCMC methods, we additionally consider level-dependent data resolution and introduce a suitable likelihood scaling to enable consistent cross-level comparisons. We theoretically show that this approach attains the same convergence rates as when using level-independent treatment of data, but at significantly reduced computational cost. The convergence analysis focuses on Lipschitz continuous transformations of Gaussian random fields with Mat\'ern covariance structure. These results are illustrated using numerical experiments for a 2D plane stress problem, where the Young's modulus is estimated from discretisations of the displacement field.

翻译：本文提出了一种多级马尔可夫链蒙特卡洛（MCMC）方法，用于利用高分辨率数据对偏微分方程中的随机场参数进行贝叶斯推断。与现有的多级MCMC方法相比，我们进一步考虑了层级依赖的数据分辨率，并引入了适当的似然缩放机制以实现跨层级的一致性比较。理论分析表明，该方法在保持与数据层级无关处理方法相同收敛速率的同时，显著降低了计算成本。收敛性分析主要针对具有Matérn协方差结构的高斯随机场的Lipschitz连续变换展开。这些结果通过二维平面应力问题的数值实验加以验证，其中杨氏模量是通过位移场的离散化进行估计的。