A Domain Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients

Abstract

We develop a domain-decomposition model reduction method for linear steady-state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equations with random diffusivities, and the convection-dominated transport equations with random velocities. We investigate the equations with two types of random fields, i.e., colored noises and discrete white noises, both of which can lead to high-dimensional parametric dependence. The motivation is to use domain decomposition to exploit low-dimensional structures of local problems in the sub-domains, such that the total number of expensive PDE solves can be greatly reduced. Our objective is to develop an efficient model reduction method to simultaneously handle highdimensionality and irregular behaviors of the stochastic PDEs under consideration. The advantages of our method lie in three aspects: (i) online-offline decomposition, i.e., the online cost is independent of the size of the triangle mesh; (ii) operator approximation for handling non-affine and high-dimensional random fields; (iii) effective strategy to capture irregular behaviors, e.g., sharp transitions of the PDE solution. Two numerical examples will be provided to demonstrate the advantageous performance of our method.

A local hybrid surrogate‐based finite element tearing interconnecting dual‐primal method for nonsmooth random partial differential equations

“…This should serve the reader to translate the abstract framework of the preceding section to the FETI-DP method proposed in the following. Further details of the approach applied for random PDEs can be found in References 36,37. For simplicity, the presentation is based on the elliptic linear equation 1with the random dependence setting introduced in the beginning of this section. Given a physical discretization space V h ⊂  ∶= add solution sample contribution to approximate QoI(u) 13: end for consisting of faces (d = 3), edges and vertices with respect to the underlying mesh.…”

“…The DD approach for random PDEs is applied in the context of global and local KLEs in References 32 and 36 with a combination of model reduction techniques 61 for the Schur complement or the FETI‐DP method. We note that the technique based on global random coordinates as in Reference 32 is based on a stochastic FE formulation without a sampling stage, which limits it to a moderate number of random coordinates due to the curse of dimensionality.…”

“…This should serve the reader to translate the abstract framework of the preceding section to the FETI‐DP method proposed in the following. Further details of the approach applied for random PDEs can be found in References 36,37.…”

“…As those ideas inspired some aspects in our work, we give more details and discussion in Example 4, Remark 7, and Section 4.2. During the development of this article, the work 37 was published, which is based on the Schur complement method for the smooth KLE case for colored and white noise using a reduced basis method.…”

Adaptive experimental design for multi‐fidelity surrogate modeling of multi‐disciplinary systems

“…The single‐fidelity version of the method we propose, and the preceding works we cited, possess similarities to domain decomposition 35‐38 and localized model reduction (LMR) 39‐42 . These methods efficiently solve partial differential equations (PDEs) by solving independent local problems on subdomains and computing a global solution via an appropriate coupling of the subdomains; LMR is domain decomposition technique that uses a localized reduced basis in each subdomain.…”