In this work we are interested in the (ill-posed) inverse problem for absolute permeability characterization that arises in predictive modeling of porous media flows. We consider a Bayesian statistical framework with a preconditioned Markov Chain Monte Carlo (MCMC) algorithm for the solution of the inverse problem. Reduction of uncertainty can be accomplished by incorporating measurements at sparse locations (static data) in the prior distribution. We present a new method to condition Gaussian fields (the log of permeability fields) to available sparse measurements. A truncated Karhunen-Lo\`eve expansion (KLE) is used for dimension reduction. In the proposed method the imposition of static data is made through the projection of a sample (expressed as a vector of independent, identically distributed normal random variables) onto the nullspace of a data matrix, that is defined in terms of the KLE. The numerical implementation of the proposed method is straightforward. Through numerical experiments for a model of second-order elliptic equation, we show that the proposed method in multi-chain studies converges much faster than the MCMC method without conditioning. These studies indicate the importance of conditioning in accelerating the MCMC convergence.

翻译：在这项工作中,我们感兴趣的是,在预测多孔媒体流的模型中出现的绝对渗透性定性(ill-posed)逆向问题。我们考虑的是巴伊西亚统计框架,其先决条件是Markov Chain Monte Carlo(MCMC)算法,用于解决反向问题。可以通过在先前分布中将稀疏地点的测量(静态数据)纳入先前分布中来减少不确定性。我们提出了一种新的方法,使高斯域(渗透性域的日志)处于可用稀薄的测量状态。在尺寸缩小时,使用了快速的Karhunen-Lo ⁇ eve扩展(KLEE)法。在拟议方法中,采用静态数据的方式是通过在数据矩阵的空格上投一个样本(作为独立、同样分布的正常随机变量的矢量表示)来进行(以 KLEE 定义的数据矩阵的空格),从而减少不确定性。我们提出了一种新的方法,将高斯域域(渗透性域域域域域图)的数值实验,我们表明,拟议的多链研究方法比MC方法的结合速度要快得多,而没有调节。这些研究显示加速调节的重要性。