## Section: New Results

### Innovative linear system solvers for hybrid multicore/GPU architectures

Participant : Marc Baboulin.

The advent of new processor architectures (e.g. multicore, GPUs) requires the rethinking of most of the scientific applications and innovative methods must be proposed in order to take full advantage of current supercomputers [14] .

To accelerate linear algebra solvers on current parallel machines, we introduced in public domain libraries a class of solvers based on statistical techniques. A first application concerns the solution of a square linear systems $Ax=b$. We study a random transformation of $A$ that enables us to avoid pivoting and then to reduce the amount of communication [16] . Numerical experiments show that this randomization can be performed at a very affordable computational price while providing us with a satisfying accuracy when compared to partial pivoting. This random transformation called Partial Random Butterfly Transformation (PRBT) is optimized in terms of data storage and flops count. In the solver that we developed, PRBT combined with LU factorization with no pivoting take advantage of the latest generation of hybrid multicore/GPU machines and outperform existing factorization routines from current parallel library MAGMA.

A second application is related to solving symmetric indefinite systems via $LD{L}^{T}$ factorization for which there was no existing parallel implementation in the dense library ScaLAPACK. We developed an efficient and innovative parallel tiled algorithm for solving symmetric indefinite systems on multicore architectures [54] & [25] . This solver avoids pivoting by using a multiplicative preconditioning based on symmetric randomization. This randomization prevents the communication overhead due to pivoting, is computationally inexpensive and requires very little storage. Following randomization, a tiled LDLT factorization is used that reduces synchronization by using static or dynamic scheduling. We compare Gflop/s performance of our solver with other types of factorizations on a current multicore machine and we provide tests on accuracy using LAPACK test cases.