Stochastic Domain Decomposition Based on Variable-Separation Method

TL;DR

This work addresses efficient solution of steady-state stochastic PDEs with random inputs by introducing SDD-VS, a hybrid of non-overlapping Schur-complement domain decomposition and the Variable-Separation (VS) method. The offline-online workflow constructs affine, low-rank representations of the stochastic interface operators $S(\bm{\xi})$ and right-hand sides $F(\bm{\xi})$, and a reduced model for the stochastic interface $\mathbf{u}_{\Gamma}(\bm{\xi})$, enabling rapid evaluation for many samples. Key contributions include: (i) low-rank affine representations for $S(\bm{\xi})$ and $F(\bm{\xi})$, (ii) a separated representation for the interface unknowns, and (iii) surrogate models for subproblems that substantially reduce online cost while preserving accuracy; demonstrated on 1D and 2D diffusion and diffusion-convection tests with high-dimensional randomness. The online phase yields cost that is largely independent of spatial discretization, making the method suitable for many-query tasks such as optimization and inverse analysis in engineering and physics.

Abstract

In this work, we propose a new stochastic domain decomposition method for solving steady-state partial differential equations (PDEs) with random inputs. Based on the efficiency of the Variable-separation (VS) method in simulating stochastic partial differential equations (SPDEs), we extend it to stochastic algebraic systems and apply it to stochastic domain decomposition. The resulting Stochastic Domain Decomposition based on the Variable-separation method (SDD-VS) effectively addresses the ``curse of dimensionality" by leveraging the explicit representation of stochastic functions derived from physical systems. The SDD-VS method aims to obtain a separated representation of the solution for the stochastic interface problem. To enhance efficiency, an offline-online computational decomposition is introduced. In the offline phase, the affine representation of stochastic algebraic systems is obtained through the successive application of the VS method. This serves as a crucial foundation for the SDD-VS method. In the online phase, the interface unknowns of SPDEs are estimated using a quasi-optimal separated representation, enabling the construction of efficient surrogate models for subproblems. The effectiveness of the proposed method is demonstrated via the numerical results of three concrete examples.

Figures (13)

• Figure 1: Illustration of a domain partitioned into two subdomains.
• Figure 2: Flowchart of the SDD-VS method.
• Figure 3: Comparison of the relative mean error corresponding to the different numbers of the separated terms $N_{S_1}$.
• Figure 4: The relative error for 100 random samples by the SDD-VS method in two conditions: (1) $N_{S_1}=4, N_{S_2}=1, N_{F_1}=4, N_{F_2}=1$; (2) $N_{S_1}=6, N_{S_2}=1, N_{F_1}=4, N_{F_2}=1$.
• Figure 5: Comparison of the mean solution for reference and the SDD-VS method with $N_{S_1}=4, N_{S_2}=1, N_{F_1}=4, N_{F_2}=1$ in subdomains $D_1$ and $D_2$.

Theorems & Definitions (1)

• Remark 3.1