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. Additionally, we show that assumptions of exponential covariance and log-normality of random fields, widely held in multilevel Monte Carlo literature, can be extended to a wide range of covariance structures and random fields. 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：针对利用高分辨率数据进行偏微分方程随机场参数贝叶斯推断的问题，我们提出一种多层马尔可夫链蒙特卡洛（MCMC）方法。与现有多层MCMC方法相比，我们额外考虑了随层变化的数据分辨率，并引入合适的似然尺度化操作以实现跨层一致性比较。理论分析表明，该方法在显著降低计算成本的同时，能够达到与数据层间无关处理方法相同的收敛速率。此外，我们证明多层蒙特卡洛文献中广泛假设的指数协方差与随机场对数正态性可推广至更广泛的协方差结构与随机场类型。通过二维平面应力问题的数值实验验证上述结论，在该问题中杨氏模量由位移场的离散化观测数据进行估计。