Large-scale Multigrid with Adaptive Galerkin Coarsening

2025-11-17

AI Summary

To address the lack of robustness in coarse-grid correction within geometric multigrid methods for strongly variable-coefficient PDEs (e.g., generalized Stokes problems), this paper proposes a matrix-free, adaptive coarse-grid operator construction. In regions with large coefficient gradients, Galerkin projection is applied locally; in smoother regions, lightweight direct approximations are used. Local Galerkin operators are assembled and stored only where necessary, significantly reducing memory overhead. The method integrates uniform geometric coarsening with heterogeneous coarse-grid operators, enabling fully matrix-free implementation and excellent parallel scalability. Numerical experiments demonstrate robust convergence on problems with up to 10 billion degrees of freedom, viscosity jumps of order $10^6$, and efficient strong scaling beyond 100,000 cores—achieving simultaneous robustness, low memory footprint, and high parallel scalability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We propose a robust, adaptive coarse-grid correction scheme for matrix-free geometric multigrid targeting PDEs with strongly varying coefficients. The method combines uniform geometric coarsening of the underlying grid with heterogeneous coarse-grid operators: Galerkin coarse grid approximation is applied locally in regions with large coefficient gradients, while lightweight, direct coarse grid approximation is used elsewhere. This selective application ensures that local Galerkin operators are computed and stored only where necessary, minimizing memory requirements while maintaining robust convergence. We demonstrate the method on a suite of sinker benchmark problems for the generalized Stokes equation, including grid-aligned and unaligned viscosity jumps, smoothly varying viscosity functions with large gradients, and different viscosity evaluation techniques. We analytically quantify the solver's memory consumption and demonstrate its efficiency by solving Stokes problems with $10^{10}$ degrees of freedom, viscosity jumps of $10^{6}$ magnitude, and more than 100{,}000 parallel processes.

Problem

• Developing adaptive coarse-grid correction for matrix-free geometric multigrid methods
• Solving PDEs with strongly varying coefficients using heterogeneous coarse-grid operators
• Addressing memory efficiency while maintaining robust convergence in large-scale simulations

Innovation

• Adaptive coarse-grid correction for multigrid methods
• Combines uniform coarsening with heterogeneous coarse-grid operators
• Selective Galerkin application minimizes memory while maintaining convergence