# An Efficient Multilevel Preconditioned Nonlinear Conjugate Gradient Method for Incremental Potential Contact

arXiv: https://arxiv.org/abs/2604.19892 · HTML: https://mlsanity.com/paper/2604.19892/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 30 references that are in the map.
- First posted on arXiv 21 Apr 2026.

## Builds on (most cited first, 7 of 7)

- [Differentiable solver for time-dependent deformation problems with contact](https://mlsanity.com/paper/2205.13643/) (2022)
- [GIPC: Fast and stable Gauss-Newton optimization of IPC barrier energy](https://mlsanity.com/paper/2308.09400/) (2023)
- [StiffGIPC: Advancing GPU IPC for stiff affine-deformable simulation](https://mlsanity.com/paper/2411.06224/) (2024)
- [A Large Scale Benchmark and an Inclusion-Based Algorithm for Continuous Collision Detection](https://mlsanity.com/paper/2009.13349/) (2020)
- [A Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling](https://mlsanity.com/paper/2301.01976/) (2023)
- [High-Order Incremental Potential Contact for Elastodynamic Simulation on Curved Meshes](https://mlsanity.com/paper/2205.13727/) (2022)
- [Preconditioned Nonlinear Conjugate Gradient Method for Real-time Interior-point Hyperelasticity](https://mlsanity.com/paper/2405.08001/) (2024)

## Abstract (arXiv)

Incremental Potential Contact (IPC) guarantees intersection-free simulation but suffers from high computational costs due to the expensive Hessian assembly and linear solves required by Newton's method. While Preconditioned Nonlinear Conjugate Gradient (PNCG) avoids Hessian assembly, it has historically struggled with poor convergence in stiff, contact-rich scenarios due to the lack of effective preconditioners; simple Jacobi preconditioners fail to capture the global coupling, while advanced hierarchy-based preconditioners like Multilevel Additive Schwarz (MAS) are computationally prohibitive to rebuild at every nonlinear iteration. We present MAS-PNCG, a method that unlocks the power of hierarchical preconditioning for nonlinear optimization. Our key technical innovation is a Sparse-Input Woodbury update algorithm that incrementally adapts the fine-level MAS components to rapidly evolving contact sets. This bypasses the need for full preconditioner rebuilds, reducing maintenance cost to near-zero while capturing the complex spectral properties of the contact system. Furthermore, we replace heuristic PNCG search directions with a Hessian-aware 2D subspace minimization that optimally combines the preconditioned gradient and previous direction. We also apply a fast per-subdomain conservative CCD method that ensures penetration-free trajectories while avoiding overly restrictive global step sizes. Experiments demonstrate that our MAS-PNCG outperforms state-of-the-art Newton-PCG solvers, GIPC and StiffGIPC, both preconditioned with MAS up to 5.66× and 2.07× respectively.