PGD-based local surrogate models via overlapping domain decomposition: a computational comparison

TL;DR

The paper addresses efficient, non-intrusive surrogate modeling for parametric elliptic PDEs using overlapping domain decomposition. It introduces a reduced-dimensional PGD-based local surrogate framework that constructs physics-based surrogates in each subdomain and couples them online via a matrix-free GMRES interface solve, avoiding Lagrange multipliers and interface trace clustering. Across multiple benchmarks, the method shows substantial offline speedups (up to 110x) and real-time online performance (sub-second evaluations) while preserving accuracy on par with full-order solves. The work demonstrates the practical potential of PGD-ROMs in large-scale, parametric multi-domain problems, offering robust, efficient alternatives to previous DD-PGD strategies.

Abstract

An efficient strategy to construct physics-based local surrogate models for parametric linear elliptic problems is presented. The method relies on proper generalized decomposition (PGD) to reduce the dimensionality of the problem and on an overlapping domain decomposition (DD) strategy to decouple the spatial degrees of freedom. In the offline phase, the local surrogate model is computed in a non-intrusive way, exploiting the linearity of the operator and imposing arbitrary Dirichlet conditions, independently at each node of the interface, by means of the traces of the finite element functions employed for the discretization inside the subdomain. This leads to parametric subproblems with reduced dimensionality, significantly decreasing the complexity of the involved computations and achieving speed-ups up to 100 times with respect to a previously proposed DD-PGD algorithm that required clustering the interface nodes. A fully algebraic alternating Schwarz method is then formulated to couple the subdomains in the online phase, leveraging the real-time (less than half a second) evaluation capabilities of the computed local surrogate models, that do not require the solution of any additional low-dimensional problems. A computational comparison of different PGD-based local surrogate models is presented using a set of numerical benchmarks to showcase the superior performance of the proposed methodology, both in the offline and in the online phase.

Figures (10)

• Figure 1: Partition of the domain $\Omega$ into two overlapping subdomains $\Omega_1$ (light blue) and $\Omega_2$ (light red), with overlap $\Omega_{12}$ (dotted purple) and interfaces $\Gamma_1$ (blue) and $\Gamma_2$ (red).
• Figure 2: Partition of the interface nodes as a collection of single independent interface parameters.
• Figure 3: Example of clustering of the interface nodes into two sets of active interface parameters $\mathcal{N}_i^1$ and $\mathcal{N}_i^2$.
• Figure 4: Map of the scaled nodal error $| u_{\texttt{PGD}}(\mu) - u_{\text{ex}}(\mu)|/\max_{\Omega} |u_{\text{ex}}(\mu)|$ for $\mu=3$ using four different approaches to handle the interface parameters.
• Figure 5: Computational domain for the geometrically-parametrized convection-diffusion problem.

Theorems & Definitions (2)

• Remark 1
• Remark 2