Preconditioning Markov chain Monte Carlo simulations using coarse-scale models
Preconditioning Markov chain Monte Carlo simulations using coarse-scale models
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DOI：
10.1137/050628568
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发表时间：
2006-01-01
影响因子：
3.1
通讯作者：
Luo, W.
中科院分区：
文献类型：
--
作者：
Efendiev, Y.;Hou, T.;Luo, W.
我们利用粗尺度模型研究了马尔可夫链蒙特卡罗（MCMC）方法的预处理，并将其应用于地下表征。预处理的目的是减少。降低了MCMC采样的计算成本，提高了采样的接受率。这一目标是通过基于两阶段计算生成马尔可夫链来实现的。在第一阶段，首先用基于多尺度有限体积法的粗尺度模型对新方案进行验证。完整的。只有提案通过了粗尺度筛选，才会进行新尺度的计算。为了更有效的模拟，也可以使用预先计算的多尺度基函数来近似全精细尺度计算。与常规MCMC方法相比，预条件MCMC方法通过结合问题的粗尺度信息生成修正的马尔可夫链。本文给出了修正后的马尔可夫链收敛于正确后验分布的条件。这些假设对我们的应用的有效性以及保证高接受率的条件也进行了讨论。我们想指出的是，模拟中使用的粗尺度模型需要便宜，但不一定非常准确，正如我们的分析和数值模拟所证明的那样。我们给出了用两点地质统计方法对渗透率场进行采样的数值例子。Karhunen-Loeve展开式用于表示受动态数据（如生产数据）以及一些静态数据约束的渗透率场的实现。我们的数值算例表明，如果使用粗尺度模型对MCMC模拟进行预处理，则接受率可以提高10倍以上。
We study the preconditioning of Markov chain Monte Carlo (MCMC) methods using coarse-scale models with applications to subsurface characterization. The purpose of preconditioning is to reduce the. ne-scale computational cost and increase the acceptance rate in the MCMC sampling. This goal is achieved by generating Markov chains based on two-stage computations. In the first stage, a new proposal is first tested by the coarse-scale model based on multiscale finite volume methods. The full. ne-scale computation will be conducted only if the proposal passes the coarse-scale screening. For more efficient simulations, an approximation of the full fine-scale computation using precomputed multiscale basis functions can also be used. Comparing with the regular MCMC method, the preconditioned MCMC method generates a modifed Markov chain by incorporating the coarse-scale information of the problem. The conditions under which the modifed Markov chain will converge to the correct posterior distribution are stated in the paper. The validity of these assumptions for our application and the conditions which would guarantee a high acceptance rate are also discussed. We would like to note that coarse-scale models used in the simulations need to be inexpensive but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. The Karhunen-Loeve expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as production data, as well as some static data. Our numerical examples show that the acceptance rate can be increased by more than 10 times if MCMC simulations are preconditioned using coarse-scale models.