Spectral analysis for uncertainty quantification

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.