[Paper] A Structure-Aware Irregular Blocking Method for Sparse LU Factorization

Source: arXiv - 2512.04389v1

Overview

Sparse LU factorization is a work‑horse in scientific computing, graph analytics, and many machine‑learning pipelines that need to solve large linear systems. The paper “A Structure‑Aware Irregular Blocking Method for Sparse LU Factorization” introduces a new way to partition sparse matrices that respects their intrinsic non‑uniform pattern, delivering up to 3.8× speed‑up on modern NVIDIA A100 GPUs compared with state‑of‑the‑art libraries.

Key Contributions

• Diagonal‑block feature: A lightweight metric that captures the local density of nonzeros around each diagonal block, enabling the algorithm to “see” where the matrix is dense or sparse.
• Irregular blocking scheme: Dynamically adjusts block sizes—fine‑grained in dense zones, coarse‑grained in sparse zones—so that each GPU thread block processes roughly the same amount of work.
• Dependency‑tree balancing: Extends the balancing across multiple levels of the LU elimination tree, preventing bottlenecks that arise from traditional uniform 2‑D blocking.
• GPU‑focused implementation: Integrated into a CUDA kernel stack and evaluated on single‑ and multi‑GPU (4 × A100) configurations.
• Performance gains: Shows average speedups of 1.50× vs. PanguLU and 3.32× vs. SuperLU_DIST on one GPU, and 1.40× / 3.84× respectively on four GPUs.

Methodology

1. Symbolic analysis & feature extraction – After the symbolic phase (which determines the fill‑in pattern), the authors slide a diagonal window across the matrix and compute the diagonal‑block feature: the count of nonzeros inside each window normalized by its area. This yields a 1‑D profile of density along the diagonal.
2. Irregular block generation – Using the density profile, the algorithm splits the matrix into blocks of varying sizes:
   • Dense regions → small blocks → more parallelism.
   • Sparse regions → larger blocks → fewer blocks but each still contains enough work to keep GPU cores busy.
3. Tree‑aware balancing – The LU factorization follows a dependency tree (the elimination tree). The method ensures that blocks belonging to the same tree level have comparable nonzero counts, and it also balances across levels to avoid a single “heavy” level that stalls the whole pipeline.
4. GPU kernel design – Each irregular block is mapped to a CUDA thread block. The kernels use shared memory for intra‑block operations and warp‑level primitives for reduction, preserving high occupancy despite the irregular shapes.
5. Multi‑GPU scaling – Blocks are distributed across GPUs using a simple round‑robin scheme, with peer‑to‑peer transfers only when a dependency crosses GPU boundaries.

Results & Findings

• Load balance: Per‑block nonzero counts vary within a ±10 % band, compared to ±45 % for uniform 2‑D blocking.
• Memory footprint: Irregular blocking adds < 5 % overhead for storing block metadata, negligible relative to the overall matrix size.
• Scalability: The method scales almost linearly up to 4 GPUs; beyond that, communication overhead starts to dominate (not explored in the paper).

Practical Implications

• High‑performance libraries: Developers of sparse linear‑algebra packages (e.g., cuSOLVER, PETSc) can adopt the diagonal‑block feature as a cheap pre‑processing step to improve GPU utilization without redesigning the entire factorization pipeline.
• Domain‑specific solvers: Applications such as circuit simulation, CFD, and graph‑based ML (e.g., PageRank on massive graphs) often encounter matrices with a “dense diagonal strip” pattern. The irregular blocking strategy directly targets this pattern, promising faster turnaround times.
• Multi‑GPU workloads: The simple block‑distribution scheme means existing multi‑GPU orchestration frameworks (e.g., NCCL‑based pipelines) can integrate the method with minimal code changes.
• Energy efficiency: By keeping more cores busy and reducing idle time, the approach can lower the energy per solve—a tangible benefit for large‑scale data‑center deployments.

Limitations & Future Work

• Pattern dependence: The diagonal‑block feature assumes a pronounced diagonal‑dominant fill pattern; matrices with completely unstructured sparsity may not benefit as much.
• Static blocking: Block sizes are decided once after symbolic analysis; dynamic re‑blocking during factorization (e.g., to adapt to runtime load imbalance) is not explored.
• Communication cost at larger scales: Experiments stop at 4 GPUs; scaling to dozens of GPUs will likely require more sophisticated data‑movement strategies and possibly hierarchical blocking.
• Integration effort: While the method is conceptually simple, retrofitting existing CPU‑centric libraries could involve non‑trivial engineering to expose the required metadata to the GPU kernels.

Overall, the paper offers a pragmatic, performance‑driven tweak to sparse LU factorization that can be adopted by developers looking to squeeze extra speed out of modern GPU clusters, especially when dealing with matrices that exhibit the typical diagonal‑dense, off‑diagonal‑sparse structure.

Authors

• Zhen Hu
• Dongliang Xiong
• Kai Huang
• Changjun Wu
• Xiaowen Jiang

Paper Information

• arXiv ID: 2512.04389v1
• Categories: cs.DC
• Published: December 4, 2025
• PDF: Download PDF