Parallel GPU-Accelerated Randomized Construction of Approximate Cholesky Preconditioners

TL;DR

This paper presents a novel parallel GPU-accelerated algorithm for constructing approximate Cholesky preconditioners for large sparse Laplacian matrices, utilizing randomization to improve efficiency and scalability.

Contribution

It introduces a new randomized, parallel approach for incomplete factorization of Laplacian matrices, optimized for GPU architectures, with minimal pre-processing.

Findings

Efficient parallel implementation on GPUs and CPUs.

Competitive performance against existing methods.

Low pre-processing time required.

Abstract

We introduce a parallel algorithm to construct a preconditioner for solving a large, sparse linear system where the coefficient matrix is a Laplacian matrix (a.k.a., graph Laplacian). Such a linear system arises from applications such as discretization of a partial differential equation, spectral graph partitioning, and learning problems on graphs. The preconditioner belongs to the family of incomplete factorizations and is purely algebraic. Unlike traditional incomplete factorizations, the new method employs randomization to determine whether or not to keep fill-ins, i.e., newly generated nonzero elements during Gaussian elimination. Since the sparsity pattern of the randomized factorization is unknown, computing such a factorization in parallel is extremely challenging, especially on many-core architectures such as GPUs. Our parallel algorithm dynamically computes the dependency among row/column indices of the Laplacian matrix to be factorized and processes the independent indices in parallel. Furthermore, unlike previous approaches, our method requires little pre-processing time. We implemented the parallel algorithm for multi-core CPUs and GPUs, and we compare their performance to other state-of-the-art methods.