In 1981 Hong and Kung [HK81] proved a lower bound on the amount of communication (amount of data moved between a small, fast memory and large, slow memory) needed to perform dense, n-by-n matrix-multiplication using the conventional O(n) algorithm, where the input matrices were too large to fit in the small, fast memory. In 2004 Irony, Toledo and Tiskin [ITT04] gave a new proof of this result a...

The LU factorization of very large sparse matrices requires a significant amount computing resources, including memory and broadband communication. A hybrid MPI + OpenMP CUDA algorithm named SuperLU3D can efficiently compute the with GPU acceleration. However, this faces difficulties when dealing limited resources. Factorizing involves vast nonblocking communication between processes, often lea...

This paper describes the design and implementation of three core factorization routines | LU, QR and Cholesky | included in the out-of-core extension of ScaLAPACK. These routines allow the factorization and solution of a dense system that is too large to t entirely in physical memory. An image of the full matrix is maintained on disk and the factorization routines transfer sub-matrices into mem...

A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. ...

The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of refinement to enhance methods sparse based approximate factorizations. In doing so, we first develop a new error analysis LU- and GMRES-based under general mo...