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. 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 - native PaStiX , StarPU and PaRSEC 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 has been developed in the framework of Xavier Lacoste's PhD funded by the ANR ANEMOS . These contributions have been presented at the Heterogeneous Computing Workshop held jointly with the international conference IPDPS 2014 [32] . Xavier Lacoste will defend his PhD in February 2015.

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 relies 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.

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 work, 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 2014 [30] in Phoenix. An extended version has been submitted to JPDC journal.

Divide and conquer symmetric tridiagonal eigensolver for multicore architectures

Computing eigenpairs of a symmetric matrix is a problem arising in many industrial applications, including quantum physics and finite-elements computation for automobiles. A classical approach is to reduce the matrix to tridiagonal form before computing eigenpairs of the tridiagonal matrix. Then, a back-transformation allows one to obtain the final solution. Parallelism issues of the reduction stage have already been tackled in different shared-memory libraries. In this work, we focus on solving the tridiagonal eigenproblem, and we describe a novel implementation of the Divide and Conquer algorithm. The algorithm is expressed as a sequential task-flow, scheduled in an out-of-order fashion by a dynamic runtime which allows the programmer to play with tasks granularity. The resulting implementation is between two and five times faster than the equivalent routine from the INTEL MKL library, and outperforms the best MRRR implementation for many matrices.

These contributions will be presented at the international conference IPDPS 2015 [34] in Hyderabad.