We present a non-trivial integration of dimension-independent likelihood-informed (DILI) MCMC (Cui, Law, Marzouk, 2016) and the multilevel MCMC (Dodwell et al., 2015) to explore the hierarchy of posterior distributions. This integration offers several advantages: First, DILI-MCMC employs an intrinsic likelihood-informed subspace (LIS) (Cui et al., 2014) -- which involves a number of forward and adjoint model simulations -- to design accelerated operator-weighted proposals. By exploiting the multilevel structure of the discretised parameters and discretised forward models, we design a Rayleigh-Ritz procedure to significantly reduce the computational effort in building the LIS and operating with DILI proposals. Second, the resulting DILI-MCMC can drastically improve the sampling efficiency of MCMC at each level, and hence reduce the integration error of the multilevel algorithm for fixed CPU time. Numerical results confirm the improved computational efficiency of the multilevel DILI approach.

翻译：本文提出了维度无关似然信息（DILI）MCMC方法（Cui, Law, Marzouk, 2016）与多层级MCMC方法（Dodwell et al., 2015）的非平凡整合，用于探索后验分布的层次结构。该整合具有多项优势：首先，DILI-MCMC利用内在的似然信息子空间（LIS）（Cui et al., 2014）——该子空间涉及多次正演和伴随模型模拟——设计加速算子加权提案。通过利用离散化参数与离散化正演模型的多层级结构，我们设计了一种Rayleigh-Ritz过程，显著降低了构建LIS和使用DILI提案的计算代价。其次，所得到的DILI-MCMC能够大幅提升各层级MCMC的采样效率，从而在固定计算时间内减少多层级算法的积分误差。数值结果证实了多层级DILI方法在计算效率上的提升。