RCHOL: Randomized Cholesky Factorization for Solving SDD Linear Systems

TL;DR

This paper presents RCHOL, a randomized, scalable algorithm for approximate Cholesky factorization of Laplacian matrices, enabling efficient solutions to large sparse SDD linear systems with demonstrated robustness and parallel performance.

Contribution

RCHOL introduces a novel randomized approach for sparse Cholesky factorization of Laplacian matrices, improving scalability and robustness for large SDD systems.

Findings

Successfully solved a billion-unknowns problem using parallel RCHOL.

RCHOL is breakdown-free and maintains sparsity through random sampling.

Parallel implementation scales well up to 64 threads.

Abstract

We introduce a randomized algorithm, namely RCHOL, to construct an approximate Cholesky factorization for a given Laplacian matrix (a.k.a., graph Laplacian). From a graph perspective, the exact Cholesky factorization introduces a clique in the underlying graph after eliminating a row/column. By randomization, RCHOL only retains a sparse subset of the edges in the clique using a random sampling developed by Spielman and Kyng. We prove RCHOL is breakdown-free and apply it to solving large sparse linear systems with symmetric diagonally dominant matrices. In addition, we parallelize RCHOL based on the nested dissection ordering for shared-memory machines. We report numerical experiments that demonstrate the robustness and the scalability of RCHOL. For example, our parallel code scaled up to 64 threads on a single node for solving the 3D Poisson equation, discretized with the 7-point stencil on a $1024\times 1024 \times 1024$ grid, a problem that has one billion unknowns.