Local surrogate models with reduced dimensionality via overlapping domain decomposition and proper generalized decomposition

TL;DR

This work presents a non-intrusive offline-online framework that combines overlapping domain decomposition with proper generalized decomposition (PGD) to build local surrogate models for linear elliptic parametric PDEs. By exploiting interface traces and linearity, it reduces the parametric dimensionality of offline local problems and uses PGD-based surrogates in a Schwarz iteration during the online phase, requiring no local solves at runtime. The approach achieves substantial offline-cost reductions and online speedups while maintaining accuracy, outperforming clustering-based variants. Demonstrations on bidomain and multi-domain diffusion problems illustrate the method's efficiency, robustness, and potential for scalable digital twins and multi-physics contexts.

Abstract

We propose an efficient algorithm that combines overlapping domain decomposition and proper generalized decomposition (PGD) to construct surrogate models of linear elliptic parametric problems. The technique is composed of an offline and an online phase that can be implemented in a fully non-intrusive way. The online phase relies on a substructured algebraic formulation of the alternating Schwarz method, while the offline phase exploits the linearity of the boundary value problem to characterize a PGD basis and generate local surrogate models, with minimal parametric dimensionality, in each subdomain. Numerical results show the efficiency of the proposed methodology.

Figures (2)

• Figure 1: Partition of the interface nodes as a collection of single independent boundary parameters.
• Figure 2: Computational domain (left) and scaled error $|u^{\text{\tiny{PGD}}}(\boldsymbol{\mu})-u^h(\mu)| / \max_\Omega | u^h(\boldsymbol{\mu})|$ for the first set of parameters reported in Table \ref{['tab:testPatera']}.