Partial differential equations (PDEs) are fundamental for theoretically describing numerous physical processes that are based on some input fields in spatial configurations. Understanding the physical process, in general, requires computational modeling of the PDE in bounded/unbounded regions. Uncertainty in the computational model manifests through lack of precise knowledge of the input field or configuration. Uncertainty quantification (UQ) in the output physical process is typically carried out by modeling the uncertainty using a random field, governed by an appropriate covariance function. This leads to solving high-dimensional stochastic counterparts of the PDE computational models. Such UQ-PDE models require a large number of simulations of the PDE in conjunction with samples in the high-dimensional probability space, with probability distribution associated with the covariance function. Those UQ computational models having explicit knowledge of the covariance function are known as aleatoric UQ (AUQ) models. The lack of such explicit knowledge leads to epistemic UQ (EUQ) models, which typically require solution of a large number of AUQ models. In this article, using a surrogate, post-processing, and domain decomposition framework with coarse stochastic solution adaptation, we develop an offline/online algorithm for efficiently simulating a class of EUQ-PDE models. We demonstrate the algorithm for celebrated bounded and unbounded spatial region models, with high-dimensional uncertainties.

中文翻译：

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一种用于认知不确定性量化的随机域分解和后处理算法

偏微分方程 (PDE) 是从理论上描述基于空间配置中某些输入场的众多物理过程的基础。通常，了解物理过程需要在有界/无界区域中对 PDE 进行计算建模。计算模型中的不确定性表现为缺乏对输入字段或配置的精确了解。输出物理过程中的不确定性量化 (UQ) 通常是通过使用由适当的协方差函数控制的随机场对不确定性进行建模来执行的。这导致求解 PDE 计算模型的高维随机对应物。这样的UQ-PDE模型需要结合高维概率空间中的样本对PDE进行大量模拟，具有与协方差函数关联的概率分布。那些具有协方差函数明确知识的 UQ 计算模型被称为任意 UQ (AUQ) 模型。缺乏这种明确的知识会导致认知 UQ (EUQ) 模型，这通常需要解决大量的 AUQ 模型。在本文中，我们使用具有粗略随机解自适应的代理、后处理和域分解框架，开发了一种离线/在线算法，用于高效地模拟一类 EUQ-PDE 模型。我们演示了具有高维不确定性的著名有界和无界空间区域模型的算法。缺乏这种明确的知识会导致认知 UQ (EUQ) 模型，这通常需要解决大量的 AUQ 模型。在本文中，我们使用具有粗略随机解自适应的代理、后处理和域分解框架，开发了一种离线/在线算法，用于高效地模拟一类 EUQ-PDE 模型。我们演示了具有高维不确定性的著名有界和无界空间区域模型的算法。缺乏这种明确的知识会导致认知 UQ (EUQ) 模型，这通常需要解决大量的 AUQ 模型。在本文中，我们使用具有粗略随机解自适应的代理、后处理和域分解框架，开发了一种离线/在线算法，用于高效地模拟一类 EUQ-PDE 模型。我们演示了具有高维不确定性的著名有界和无界空间区域模型的算法。