Scheduling of parallel matrix computations and data layout conversion for HPC and Multi-Core Architectures

Abstract: Dense linear algebra represents fundamental building blocks in many computational science and engineering applications. The dense linear algebra algorithms must be numerically stable, robust, and reliable in order to be usable as black-box solvers by expert as well as non-expert users. The algorithms also need to scale and run efficiently on massively parallel computers with multi-core nodes. Developing high-performance algorithms for dense matrix computations is a challenging task, especially since the widespread adoption of multi-core architectures. Cache reuse is an even more critical issue on multi-core processors than on uni-core processors due to their larger computational power and more complex memory hierarchies. Blocked matrix storage formats, in which blocks of the matrix are stored contiguously, and blocked algorithms, in which the algorithms exhibit large amounts of cache reuse, remain key techniques in the effort to approach the theoretical peak performance.In Paper I, we present a packed and distributed Cholesky factorization algorithm based on a new blocked and packed matrix storage format. High performance node computations are obtained as a result of the blocked storage format, and the use of look-ahead leads to improved parallel efficiency. In Paper II and Paper III, we study the problem of in-place matrix transposition in general and in-place matrix storage format conversion in particular. We present and evaluate new high-performance parallel algorithms for in-place conversion between the standard column-major and row-major formats and the four standard blocked matrix storage formats. Another critical issue, besides cache reuse, is that of efficient scheduling of computational tasks. Many weakly scalable parallel algorithms are efficient only when the problem size per processor is relatively large. A current research trend focuses on developing parallel algorithms which are more strongly scalable and hence more efficient also for smaller problems.In Paper IV, we present a framework for dynamic node-scheduling of two-sided matrix computations and demonstrate that by using priority-based scheduling one can obtain an efficient scheduling of a QR sweep. In Paper V and Paper VI, we present a blocked implementation of two-stage Hessenberg reduction targeting multi-core architectures. The main contributions of Paper V are in the blocking and scheduling of the second stage. Specifically, we show that the concept of look-ahead can be applied also to this two-sided factorization, and we propose an adaptive load-balancing technique that allow us to schedule the operations effectively.