Tackling Parallelization Challenges of Randomized Preconditioners With Dependency Tracking

Abstract

Incomplete Cholesky (IC) preconditioners are the standard black-box, memory-efficient first resort for large graph Laplacian systems, but classical IC can break down on irregular graphs. Approximate Cholesky (AC) addresses this with randomized dropping that is provably breakdown-free. Its stochastic fill-in pattern, however, resists static parallelization, creating a "setup bottleneck." We present ParAC, which parallelizes the sequential rchol algorithm on CPUs and GPUs using dynamic dependency tracking and architecture-aware memory management, without altering its numerics. Total time-to-solution speedups over deterministic IC reach $57.3\times$ on CPU and $52.2\times$ on GPU. Algebraic multigrid (AMG) remains preferable on structured PDE discretizations; on irregular graph Laplacians where AMG struggles, ParAC attains up to $7.2\times$ and $6.0\times$. On 200 adversarial SDD systems, ParAC converges on every instance, whereas the AMG solvers fail on 15 and 19, on CPU and GPU, respectively.