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

TL;DR

This work addresses slow convergence of coordinate descent for implicit elastodynamics caused by global coupling by introducing Coordinate Condensation, a Schur-complement-based subspace correction that decouples local updates from subspace corrections. Building on JGS2, it preserves locality and parallelism while aiming for near-Newton convergence when the subspace accurately captures global coupling. Empirical results show substantially faster convergence than standard coordinate descent and JGS2 across varying mesh resolutions and material stiffness, with detailed analysis of subspace quality, large-deformation effects, and unanticipated coupling. Limitations arise from subspace basis quality and potential staleness under strong nonlinearities, motivating adaptive or online basis updates and coupling-aware mechanisms in future work.

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.

Figures (8)

• Figure 1: Slow convergence of coordinate descent on a 1D elastic rod with an applied impulse (right end). (Top) Heat map showing displacement magnitude across vertices (x-axis) and iterations (y-axis). Information propagates slowly from the impulse location, with vertices at the opposite end requiring many more iterations to reach their equilibrium displacement. (Bottom) Rest configuration (gray) and final configuration after one timestep.
• Figure 2: Convergence scaling with material stiffness for the 1D rod with impulse. (Top) Final configurations at three stiffness values showing the transition from mostly rigid translation (1e6 Pa) to localized deformation around the impulse location (1e1 Pa). (Bottom) Coordinate Condensation maintains constant iterations (optimal for this quadratic problem with the rest-shape basis), while Coordinate Descent and JGS2 both degrade significantly as stiffness increases, saturating the 10000 iteration limit. JGS2's damping provides minimal benefit over standard coordinate descent here.
• Figure 3: Perturbation basis $\mathbf{U}_{iC}$ for a single vertex (center, marked in red) in a 2D elastic mesh. (Left) 2D mesh with the perturbed vertex highlighted. (Right) Surface plot showing the x-direction displacement response of the mesh, where height and color indicate displacement magnitude.
• Figure 4: Solver convergence as mesh resolution increases on the 2D elastic stretch test. (Top) Converged configurations for the coarsest mesh (5$\times$5, 25 vertices) and finest mesh (29$\times$29, 841 vertices) tested, showing the stretched deformation. (Bottom) Iterations to convergence as we increase the number of vertices. Standard Coordinate Descent quickly saturates the 500 iteration limit. JGS2 shows significant improvement but still scales poorly. Coordinate Condensation remains close to Newton's method, demonstrating near-optimal convergence across all resolutions.
• Figure 5: Subspace quality analysis on a cantilever beam quasi-static simulation. (Left) Beam configurations throughout the simulation. (Right) Convergence comparison. Standard Coordinate Condensation with a fixed basis shows degraded convergence as the beam deforms and elements rotate. Restarted Coordinate Condensation (basis reconstructed every 5 steps) recovers the performance intermittently, confirming that convergence breakdown stems from subspace staleness. Corotated Coordinate Condensation, which rotates basis entries with estimated per-vertex rotations at each iteration, maintains good convergence without expensive basis reconstruction.