Fast Sparse Matrix Permutation for Mesh-Based Direct Solvers

TL;DR

This paper introduces a fast sparse matrix permutation algorithm designed for mesh-based linear systems, significantly reducing permutation time and enhancing Cholesky solver performance on CPUs and GPUs.

Contribution

The proposed algorithm relaxes traditional permutation constraints to enable faster partitioning and efficient elimination-tree construction, tailored for mesh-based problems.

Findings

Reduces permutation time by up to 6.27x

Improves sparse Cholesky solve performance significantly

Effective on both CPU and GPU implementations

Abstract

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, repeated factorizations, our method reduces permutation time and improves the sparse Cholesky solve performance by up to 6.27x.