基于网格的直接求解器的快速稀疏矩阵置换
Fast Sparse Matrix Permutation for Mesh-Based Direct Solvers
AI总结：
本文提出了一种针对三角网格产生的线性系统优化的快速稀疏矩阵置换算法，通过放松传统分区和分离优化要求，提高置换效率和消除树构建效率，从而在CPU和GPU上实现更高效的稀疏Cholesky求解器。
摘要：
我们提出了一种针对三角网格产生的线性系统优化的快速稀疏矩阵置换算法。我们的方法生成类似嵌套分解的置换，同时显著减少置换运行时间开销。与强制严格平衡和分离最优性不同，该算法有意放松这些设计决策，以优先考虑快速分区和高效的消除树构建。我们的方法将置换分解为基于补丁的局部顺序和紧凑的商图顺序分离，保留稀疏Cholesky因子化所需的基本结构，同时避免其最昂贵的部分。我们将该算法集成到供应商维护的稀疏Cholesky求解器中，适用于CPU和GPU。在一系列图形应用中，包括单次因子化和重复因子化，我们的方法将置换时间减少并提高了稀疏Cholesky求解性能，最高提升达6.27倍。我们的代码可在https://github.com/BehroozZare/fast-permute上获得。
英文摘要：
We present a fast sparse matrix permutation algorithm tailored to linear systems arising from triangle meshes. Our approach produces nested-dissection-style permutations while significantly reducing permutation runtime overhead. Rather than enforcing strict balance and separator optimality, the algorithm deliberately relaxes these design decisions to favor fast partitioning and efficient elimination-tree construction. Our method decomposes permutation into patch-level local orderings and a compact quotient-graph ordering of separators, preserving the essential structure required by sparse Cholesky factorization while avoiding its most expensive components. We integrate our algorithm into vendor-maintained sparse Cholesky solvers on both CPUs and GPUs. Across a range of graphics applications, including single factorizations and repeated factorizations, our method reduces permutation time and improves the sparse Cholesky solve performance by up to 6.27x. Our code is available at https://github.com/BehroozZare/fast-permute.
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