A reduced order modeling algorithm for the estimation of space varying parameter patterns in numerical models is proposed. In this approach domain decomposition is applied to construct separate approximations to the numerical model in every subdomain. We introduce a new local parameterization that decouples the computational cost of the algorithm from the number of global principal components and therefore provides attractive scaling for models with a very large number of uncertain parameter patterns. By defining uncertain parameter patterns only in the various subdomains the number of full order simulation required for the derivation of the reduced order models can be reduced drastically. To avoid non-smoothness at the boundaries of the subdomains, the optimal local parameters patterns are projected onto global parameter patterns. The computational effort of the new methodology hardly increases when the number of parameter patterns increases. The number of training models depends primarily on the maximum number of local parameters in a subdomain, which can be decreased by refining the domain decomposition. We apply the new algorithm to a large-scale reservoir model parameter estimation problem. In this application 282 parameters could be estimated using only 90 full order model runs.

提出了一种用于估计数值模型中空间变化参数模式的降阶建模算法。在这种方法中，将域分解应用于在每个子域中构造对数值模型的单独近似。我们引入了一种新的局部参数化方法，该方法将算法的计算成本与全局主成分的数量脱钩，从而为具有大量不确定参数模式的模型提供了有吸引力的缩放比例。通过仅在各个子域中定义不确定的参数模式，可以大大减少派生降阶模型所需的全阶模拟次数。为了避免子域边界处的不平滑，将最佳局部参数模式投影到全局参数模式上。当参数模式的数量增加时，新方法的计算量几乎不会增加。训练模型的数量主要取决于子域中局部参数的最大数量，可以通过优化域分解来减少数量。我们将新算法应用于大型油藏模型参数估计问题。在此应用中，仅使用90个完整订单模型运行即可估算282个参数。