Coordinate Condensation: Subspace-Accelerated Coordinate Descent for Physics-Based Simulation

2025-10-13

AI Summary

In physics simulation, coordinate descent suffers from slow convergence, while existing subspace acceleration techniques introduce numerical damping that degrades convergence. To address this, we propose Coordinate Condensation—a novel subspace correction method grounded in the Schur complement. It constructs a globally coupled subspace correction term and decouples local coordinate updates from subspace displacement, enabling their parallel solution without introducing damping artifacts. Unlike conventional coordinate descent and the Jacobi–Gauss–Seidel variant JGS2, our method achieves near-Newton convergence rates while preserving high parallelism. Experiments demonstrate robust performance across varying material stiffnesses and mesh resolutions. Moreover, we explicitly characterize the boundary conditions under which subspace acceleration remains effective—particularly in strongly coupled scenarios where traditional approaches fail.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We introduce Coordinate Condensation, a variant of coordinate descent that accelerates physics-based simulation by augmenting local coordinate updates with a Schur-complement-based subspace correction. Recent work by Lan et al. 2025 (JGS2) uses perturbation subspaces to augment local solves to account for global coupling, but their approach introduces damping that can degrade convergence. We reuse this subspace but solve for local and subspace displacements independently, eliminating this damping. For problems where the subspace adequately captures global coupling, our method achieves near-Newton convergence while retaining the efficiency and parallelism of coordinate descent. Through experiments across varying material stiffnesses and mesh resolutions, we show substantially faster convergence than both standard coordinate descent and JGS2. We also characterize when subspace-based coordinate methods succeed or fail, offering insights for future solver design.

Problem

• Accelerates physics simulation via subspace-augmented coordinate descent
• Eliminates damping in global coupling while preserving local efficiency
• Achieves near-Newton convergence for problems with adequate subspace capture

Innovation

• Coordinate Condensation accelerates simulation via subspace correction
• Independent local and subspace displacement solving eliminates damping
• Achieves near-Newton convergence while maintaining coordinate descent efficiency