Parallelization and scalability analysis of inverse factorization using the chunks and tasks programming model

Abstract

We present three methods for distributed memory parallel inverse factorization of block-sparse Hermitian positive definite matrices. The three methods are a recursive variant of the AINV inverse Cholesky algorithm, iterative refinement, and localized inverse factorization, respectively. All three methods are implemented using the Chunks and Tasks programming model, building on the distributed sparse quad-tree matrix representation and parallel matrix-matrix multiplication in the publicly available Chunks and Tasks Matrix Library (CHTML). Although the algorithms are generally applicable, this work was mainly motivated by the need for efficient and scalable inverse factorization of the basis set overlap matrix in large scale electronic structure calculations. We perform various computational tests on overlap matrices for quasi-linear Glutamic Acid-Alanine molecules and three-dimensional water clusters discretized using the standard Gaussian basis set STO-3G with up to more than 10 million basis functions. We show that for such matrices the computational cost increases only linearly with system size for all the three methods. We show both theoretically and in numerical experiments that the methods based on iterative refinement and localized inverse factorization outperform previous parallel implementations in weak scaling tests where the system size is increased in direct proportion to the number of processes. We show also that compared to the method based on pure iterative refinement the localized inverse factorization requires much less communication.

The Chunks and Tasks Matrix Library 2.0

“…The Chunks and Tasks Matrix Library has been indispensable in the development, implementation, and analysis of a number of novel parallel sparse matrix algorithms for distributed memory systems, including a general purpose parallel sparse matrixmatrix multiply efficiently exploiting locality of nonzero matrix entries [18], a sparse approximate matrix-matrix multiply for matrices with decay [2,3], localized inverse factorization [19,4], a new communication-avoiding divide and conquer method for inverse factorization of symmetric positive definite matrices, and density matrix purification [15].…”