About the granularity portability of block‐based Krylov methods in heterogeneous computing environments

Abstract

Summary Large‐scale problems in engineering and science often require the solution of sparse linear algebra problems and the Krylov subspace iteration methods (KM) have led to a major change in how users deal with them. But, for these solvers to use extreme‐scale hardware efficiently a lot of work was spent to redesign both the KM algorithms and their implementations to address challenges like extreme concurrency, complex memory hierarchies, costly data movement, and heterogeneous node architectures. All the redesign approaches bases the KM algorithm on block‐based strategies which lead to the Block‐KM (BKM) algorithm which has high granularity (i.e., the ratio of computation time to communication time). The work proposes novel parallel revisitation of the modules used in BKM which are based on the overlapping of communication and computation. Such revisitation is evaluated by a model of their granularity and verified on the basis of a case study related to a classical problem from numerical linear algebra.

Special Issue on High‐end Heterogeneous Architectures, Methodologies, and Algorithms (HHAMA20)

“…To highlight the capability of DD‐KF of exploiting the computing capabilities provided by future designs of microprocessors based on multi/many‐cores CPU/GPU technologies, the authors consider parallelism at the microprocessor level and discuss the scalability of DD‐KF algorithm at a fine‐grained level. To this end, the authors employ the DD‐KF algorithm to constrained least square model underlying variational data assimilation problems. In Reference 5, the authors note that large‐scale problems in engineering and science often require the solution of sparse linear algebra problems, and the Krylov subspace iteration methods (KM) have led to a significant change in how users deal with them. For these solvers to use extreme‐scale hardware efficiently, a lot of work was spent on redesigning both the KM algorithms and their implementations to address challenges like extreme concurrency, complex memory hierarchies, costly data movement, and heterogeneous node architectures.…”

Toward a new linpack‐like benchmark for heterogeneous computing resources

“…Furthermore, we cannot overlook the fact that for the Krylov methods (KM) solvers to efficiently utilize extreme‐scale hardware, a lot of work has been dedicated to redesign both the Krylov methods algorithms and their implementations over the last three decades (e.g., see Yamazaki et al, 7 Bai et al, 12 Hoemmen, 13 Carracciuolo et al, 14 Laccetti at al 15 …”

“…Nevertheless, paraphrasing the authors of the new benchmarks cited above, 9 we state "Presently Linpack-like benchmarks remains tremendously valuable as a measure of historical trends, and as a stress test … Furthermore, it provides the HPC community with a valuable outreach tool, understandable to the outside world". Furthermore, we cannot overlook the fact that for the Krylov methods (KM) solvers to efficiently utilize extreme-scale hardware, a lot of work has been dedicated to redesign both the Krylov methods algorithms and their implementations over the last three decades (e.g., see Yamazaki et al, 7 Bai et al, 12 Hoemmen, 13 Carracciuolo et al, 14 Laccetti at al. 15 ) to address challenges like extreme concurrency, complex memory hierarchies, costly data movement, and heterogeneous node architectures.…”