Section: New Results

High performance solvers for large linear algebra problems

Parallel sparse direct solver on runtime systems

The ongoing hardware evolution exhibits an escalation in the number, as well as in the heterogeneity, of the computing resources. The pressure to maintain reasonable levels of performance and portability, forces the application developers to leave the traditional programming paradigms and explore alternative solutions. Algorithms, especially those in critical domains such as linear algebra, need to undergo invasive structural changes and be adapted to new programming paradigms to be in agreement with the latest hardware advances. PaStiX is a parallel sparse direct solver, based on a dynamic scheduler for modern hierarchical architectures. In this paper, we study the replacement of the highly specialized internal scheduler in PaStiX by two generic runtime frameworks: PaRSEC and StarPU . The tasks graph of the factorization step is made available to the two runtimes, providing them with the opportunity to optimize it in order to maximize the algorithm efficiency for a predefined execution environment. A comparative study of the performance of the PaStiX solver with the three schedulers on different execution contexts is performed. The analysis highlights the similarities from a performance point of view between the different execution supports. These results demonstrate that these generic DAG-based runtimes provide a uniform and portable programming interface across heterogeneous environments, and are, therefore, a sustainable solution for hybrid environments.

This work is developed in the framework of Xavier Lacoste's PhD funder by the ANR ANEMOS . These contributions have been presented at the international workshop Sparse Days [37] in Toulouse. More details and results can be found in report RR-8446 [46] .

Hybrid parallel implementation of hybrid solvers

In the framework of the hybrid direct/iterative MaPHyS solver, we have designed and implemented an hybrid MPI-thread variant. More precisely, the implementation rely on the multi-threaded MKL library for all the dense linear algebra calculations and the multi-threaded version of PaStiX . Among the technical difficulties, one was to make sure that the two multi-threaded libraries do not interfere with each other. The resulting software prototype is currently experimented to study its new capability to get flexibility and trade-off between the parallel and numerical efficiency. Parallel experiments have been conducted on the Plafrim plateform as well as on a large scale machine located at the USA DOE NERSC, which has a large number of CPU cores per socket.

This work is developed in the framework of the PhD thesis of Stojce Nakov funded by TOTAL. These contributions have been presented at the NVIDIA GPU Technology Conference [25] in San Jose.

Designing LU-QR hybrid solvers for performance and stability

New hybrid LU-QR algorithms for solving dense linear systems of the form $Ax=b$ have been introduced. Throughout a matrix factorization, these algorithms dynamically alternate LU with local pivoting and QR elimination steps, based upon some robustness criterion. LU elimination steps can be very efficiently parallelized, and are twice as cheap in terms of flops, as QR steps. However, LU steps are not necessarily stable, while QR steps are always stable. The hybrid algorithms execute a QR step when a robustness criterion detects some risk for instability, and they execute an LU step otherwise. Ideally, the choice between LU and QR steps must have a small computational overhead and must provide a satisfactory level of stability with as few QR steps as possible. In this paper, we introduce several robustness criteria and we establish upper bounds on the growth factor of the norm of the updated matrix incurred by each of these criteria. In addition, we describe the implementation of the hybrid algorithms through an extension of the PaRSEC software to allow for dynamic choices during execution. Finally, we analyze both stability and performance results compared to state-of-the-art linear solvers on parallel distributed multicore platforms. These contributions have been presented at the international conference IPDPS [18] in Phoenix.