We are concerned with a novel Bayesian statistical framework for the characterization of natural subsurface formations, a very challenging task. Because of the large dimension of the stochastic space of the prior distribution in the framework, typically a dimensional reduction method, such as a Karhunen-Leove expansion (KLE), needs to be applied to the prior distribution to make the characterization computationally tractable. Due to the large variability of properties of subsurface formations (such as permeability and porosity) it may be of value to localize the sampling strategy so that it can better adapt to large local variability of rock properties. In this paper, we introduce the concept of multiscale sampling to localize the search in the stochastic space. We combine the simplicity of a preconditioned Markov Chain Monte Carlo method with a new algorithm to decompose the stochastic space into orthogonal subspaces, through a one-to-one mapping of the subspaces to subdomains of a non-overlapping domain decomposition of the region of interest. The localization of the search is performed by a multiscale blocking strategy within Gibbs sampling: we apply a KL expansion locally, at the subdomain level. Within each subdomain, blocking is applied again, for the sampling of the KLE random coefficients. The effectiveness of the proposed framework is tested in the solution of inverse problems related to elliptic partial differential equations arising in porous media flows. We use multi-chain studies in a multi-GPU cluster to show that the new algorithm clearly improves the convergence rate of the preconditioned MCMC method. Moreover, we illustrate the importance of a few conditioning points to further improve the convergence of the proposed method.

翻译：本文关注一种新颖的贝叶斯统计框架，用于表征极其复杂的天然地下地层结构。由于该框架中先验分布的随机空间维数庞大，通常需要对先验分布应用维数约简方法（如Karhunen-Loève展开）以使其在计算上可行。考虑到地下地层属性（如渗透率和孔隙度）存在显著变异性，引入局部化采样策略以更好地适应岩石属性的强局部变化具有重要价值。本文提出多尺度采样概念，用于实现随机空间中的局部化搜索。我们结合预条件马尔可夫链蒙特卡洛方法的简洁性，提出一种新算法，通过将随机空间与感兴趣区域的无重叠区域分解子域建立一一映射，将随机空间分解为正交子空间。局部化搜索通过吉布斯采样中的多尺度分块策略实现：在子域层面局部应用KL展开，并在每个子域内对KL随机系数再次实施分块采样。该框架的有效性在与多孔介质流动问题相关的椭圆型偏微分方程反问题求解中得到了验证。我们在多GPU集群上开展多链研究，结果表明新算法显著提升了预条件MCMC方法的收敛速度。此外，我们论证了在若干条件点约束下可进一步改进所提方法的收敛性能。