A Domain Decomposition Approach for Uncertainty Analysis

Abstract

Abstract. This paper proposes a decomposition approach for uncertainty analysis of systems governed by partial differential equations (PDEs). The system is split into local components using domain decomposition. Our domain-decomposed uncertainty quantification (DDUQ) approach performs uncertainty analysis independently on each local component in an "offline" phase, and then assembles global uncertainty analysis results using precomputed local information in an "online" phase. At the heart of the DDUQ approach is importance sampling, which weights the precomputed local PDE solutions appropriately so as to satisfy the domain decomposition coupling conditions. To avoid global PDE solves in the online phase, a proper orthogonal decomposition reduced model provides an efficient approximate representation of the coupling functions. 1. Introduction. Many problems arising in computational science and engineering are described by mathematical models of high complexity-involving multiple disciplines, characterized by a large number of parameters, and impacted by multiple sources of uncertainty. Decomposition of a system into subsystems or component parts has been one strategy to manage this complexity. For example, modern engineered systems are typically designed by multiple groups, usually decomposed along disciplinary or subsystem lines, sometimes spanning different organizations and even different geographical locations. Mathematical strategies have been developed for decomposing a simulation task (e.g., domain decomposition methods [45]), for decomposing an optimization task (e.g., domain decomposition for optimal design or control [28,10,29,5]), and for decomposing a complex design task (e.g., decomposition approaches to multidisciplinary optimization [15,34,55]). In this paper we propose a decomposition approach for uncertainty quantification (UQ). In particular, we focus on the simulation of systems governed by stochastic partial differential equations (PDEs) and a domain decomposition approach to quantification of uncertainty in the corresponding quantities of interest.In the design setting, decomposition approaches are not typically aimed at improving computational efficiency, although in some cases decomposition can admit parallelism that would otherwise be impossible [57]. Rather, decomposition is typically employed in situations for which it is infeasible or impractical to achieve tight coupling among the system subcomponents. This inability to achieve tight coupling becomes a particular problem when the goal is to wrap an outer loop (e.g., optimization) around the simulation. In the field of multidisciplinary design optimization (MDO), decomposition is achieved through mathematical formulations such as col-

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