Multithreaded Multifrontal Sparse Cholesky Factorization Using Threading Building Blocks

Abstract

The multifrontal method is a well-established approach to parallel sparse direct solvers of linear algebraic equations systems with sparse symmetric positive-definite matrices. This paper discusses the approaches and challenges of scalable parallel implementation of the numerical phase of the multifrontal method for shared memory systems based on high-end server CPUs with dozens of cores. The commonly used parallelization schemes are often guided by an elimination tree, containing information about dependencies between logical tasks in a computational loop of the method. We consider a dynamic two-level scheme for the organization of parallel computations. This scheme employs the task-based model with dynamic switching between solving relatively small tasks in parallel and using parallel functions of BLAS for relatively large tasks. There are several problems with the implementation of this scheme, including time-consuming synchronizations and the need for smart memory management. We found a way to improve performance and scaling efficiency using the model of parallelism and memory management tools from the Threading Building Blocks library. Experiments on large symmetric matrices from the SuiteSparse Matrix Collection show that our implementation is competitive with the commercial direct sparse solver Intel MKL PARDISO.