During the past few decades, uncertainty quantification (UQ) techniques have been developed and applied to many applications. The majority of these techniques have been applied directly to specifically defined problems, that is, problems described by a mathematical operator and a specific source term, both which may be endowed with uncertainty. In this article, we take an alternative approach: applying UQ techniques to probe the operator. The goal is to extract intrinsic structures of the problem, particularly operator structures that reveal the propagation of uncertainty which a UQ analysis of a specific problem may not accurately extract. There are many practical reasons for this. For example, the operator often contains more aspects of uncertainty than the forcing term, and the operator is often computationally expensive to form and hence, wasteful to use only for determining the solution of a sample instantiation. The anticipation is that the detailed information can lead to more accurate and efficient UQ analysis of the general problem. In this article, we consider only linear problems since we want to explore the potential benefits of this approach without the added complexities introduced by nonlinearities. This linearity assumption allows us to explore structures exposed by the spectrum of the operator. We explore sensitivity of the eigenvectors to determine a relationship between the spatial and uncertainty parameter dimensions, and then construct an adaptive parameter procedure based on this relationship. We also explore the propagation of uncertainties in multicomponent systems, like in multiphysics applications, by examining the coupling in the smooth-frequency eigenvectors; and explore componentized UQ processing based on the eigenvalues/vectors of the state-equation operator in goal-oriented multiple-input/multiple-output systems. Analysis and numerical examples demonstrating the applicability of the methods are presented.

中文翻译：

---

不确定性量化的光谱分析

在过去的几十年中，不确定性量化 (UQ) 技术已经开发并应用于许多应用。大多数这些技术已直接应用于特定定义的问题，即由数学运算符和特定源项描述的问题，两者都可能具有不确定性。在本文中，我们采用了另一种方法：应用 UQ 技术来探测算子。目标是提取问题的内在结构，特别是揭示不确定性传播的算子结构，特定问题的 UQ 分析可能无法准确提取。这有很多实际原因。例如，算子通​​常包含比强制项更多的不确定性方面，并且算子的形成通常在计算上很昂贵，因此，仅用于确定样本实例化的解决方案是浪费的。预期是详细信息可以导致对一般问题的更准确和更有效的 UQ 分析。在本文中，我们只考虑线性问题，因为我们想探索这种方法的潜在好处，而不会增加非线性带来的复杂性。这种线性假设使我们能够探索算子频谱暴露的结构。我们探索特征向量的敏感性以确定空间和不确定性参数维度之间的关系，然后基于这种关系构建自适应参数程序。我们还探索了多分量系统中不确定性的传播，例如在多物理场应用中，通过检查平滑频率特征向量中的耦合；并在面向目标的多输入/多输出系统中探索基于状态方程算子的特征值/向量的组件化 UQ 处理。分析和数值例子证明了这些方法的适用性。