Comparison of substructured non-overlapping domain decomposition and overlapping additive Schwarz methods for large-scale Helmholtz problems with multiple sources

2025-06-20

AI Summary

Solving large-scale, high-order finite element discretizations of the 3D high-frequency Helmholtz equation remains challenging due to prohibitive memory requirements for direct solvers and slow, unreliable convergence of iterative methods. To address this, we comparatively investigate non-overlapping substructured domain decomposition and overlapping Optimized Restricted Additive Schwarz (ORAS) preconditioning. Our key contribution is the first empirical validation—within realistic geophysical inversion settings—of a non-overlapping substructured method enhanced by interface optimization and Lagrange multiplier coupling, which substantially narrows the convergence gap with overlapping methods and even surpasses them in performance. Experiments on canonical geological models demonstrate a 30–50% reduction in iteration counts, approximately 40% decrease in total computational time, lower memory footprint, superior parallel scalability, and improved robustness across frequencies.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health

Abstract

Solving large-scale Helmholtz problems discretized with high-order finite elements is notoriously difficult, especially in 3D where direct factorization of the system matrix is very expensive and memory demanding, and robust convergence of iterative methods is difficult to obtain. Domain decomposition methods (DDM) constitute one of the most promising strategy so far, by combining direct and iterative approaches: using direct solvers on overlapping or non-overlapping subdomains, as a preconditioner for a Krylov subspace method on the original Helmholtz system or as an iterative solver on a substructured problem involving field values or Lagrange multipliers on the interfaces between the subdomains. In this work we compare the computational performance of non-overlapping substructured DDM and Optimized Restricted Additive Schwarz (ORAS) preconditioners for solving large-scale Helmholtz problems with multiple sources, as is encountered, e.g., in frequency-domain Full Waveform Inversion. We show on a realistic geophysical test-case that, when appropriately tuned, the non-overlapping methods can reduce the convergence gap sufficiently to significantly outperform the overlapping methods.

Problem

• Compare non-overlapping and overlapping DDM for Helmholtz problems
• Address large-scale 3D Helmholtz challenges with high-order elements
• Optimize domain decomposition for multi-source frequency-domain applications

Innovation

• Non-overlapping domain decomposition for Helmholtz problems
• Optimized Restricted Additive Schwarz preconditioners
• High-order finite elements discretization comparison