Implicit sampling methods and their applications

数据同化将观测数据（可以是实时数据）纳入实际系统的计算模型中。该过程的输出是基于计算模型和观测结果的系统调整状态。这些调整后的状态比仅使用数据或模型获得的状态要好。许多领域都需要数据同化，如统计信号处理、海洋学、气象学、水文学、地球科学、计量经济学和金融学。由于这些应用中模型的大规模和非线性特性，常用的方法依赖于不切实际的假设。PI和她的合作者开发了一种有效的数据同化方法，没有这些不切实际的假设。实例表明，该方法在油藏历史拟合中的成功应用将为油藏管理带来极大的好处。这个项目涉及本科生和研究生。PI为未被充分代表的群体成功参与stem相关学科提供了外联服务。数据驱动的计算，如数据同化，需要从一个不确定的模型中识别系统的状态和/或系统中的未知参数，该模型由噪声和不完整的数据流补充。贝叶斯框架是这类问题的标准方法，它涉及到根据给定的先验分布和数据来描述状态和/或参数的后验分布。常用的方法，如集合卡尔曼滤波和变分方法，依赖于高斯性或近高斯性的假设。相比之下，隐式抽样方法通过具有良好建议密度的重要性抽样获得后验密度的高合格样本，可以应用于更一般的非高斯情况。这些样本是独立的，集中在高概率区域。隐式抽样方法的第一步通常需要求解一个优化问题，这是该方法中最耗时的部分。本文提出的研究是利用领域分解方法（一种广泛应用于并行计算的范式）和高效的非线性求解方法来开发和分析前置条件，以加速该过程并使其适用于高性能计算。PI和她的合作者将这些新开发的隐式采样方法应用于地下流动应用中的数据同化和不确定性量化，包括油藏历史匹配。

Data assimilation incorporates the observations, which can be real-time data, into a computational model of a real system. The output of this process is the adjusted states of the system based on both computational model and the observations. These adjusted states are better than those that could be obtained using just the data or model alone. Data assimilation is required in many fields such as statistical signal processing, oceanography, meteorology, hydrology, geosciences, econometrics, and finance. Due to the large-scale and nonlinear properties of the models in those applications, commonly used methods rely on unrealistic assumptions. The PI and her collaborators develop an efficient data assimilation method without those unrealistic assumptions. As one example, successful application of this method to reservoir history matching will greatly benefit the reservoir management. This project involves undergraduate and graduate students. The PI has outreach for successful participation of underrepresented group in STEM-related disciplines.Data-driven computations, such as data assimilation, need to identify the state of a system and/or unknown parameters in the system from an uncertain model supplemented by a stream of noisy and incomplete data. The Bayesian framework is a standard approach for such problems and it involves characterizing the posterior distribution of the state and/or parameters in terms of given prior distribution and data. Commonly used methods, like ensemble Kalman filter-type and variational methods, rely on assumptions of Gaussianity or near Gaussianity. By contrast, the implicit sampling methods obtain high qualify samples of the posterior density by using importance sampling with good proposal density and can be applied to more general non-Gaussian situations. These samples are independent and focus on the high probability regions. The first step in the implicit sampling methods usually requires solving an optimization problem, which is the most time-consuming part of the methods. The proposed research is to develop and analyze preconditioners using domain decomposition methods, a widely-used paradigm for parallel computation, combined with efficient nonlinear solvers to accelerate this procedure and make it suitable for high performance computation. The PI and her collaborators apply these newly developed implicit sampling methods to data assimilation and uncertainty quantification in subsurface flow applications including reservoir history matching.