## Section: New Results

### High Performance Linear Algebra on the Grid

Participants : Thomas Hérault, Camille Coti.

Previous studies have reported that common dense linear algebra operations do not achieve speed up by using multiple geographical sites of a computational grid. Because such operations are the building blocks of most scientific applications, conventional supercomputers are still strongly predominant in high-performance computing and the use of grids for speeding up large-scale scientific problems is limited to applications exhibiting parallelism at a higher level.

In this work, we have identified two performance bottlenecks in the distributed memory algorithms implemented in ScaLAPACK, a state-of-the-art dense linear algebra library. First, because ScaLAPACK assumes a homogeneous communication network, the implementations of ScaLAPACK algorithms lack locality in their communication pattern. Second, the number of messages sent in the ScaLAPACK algorithms is significantly greater than other algorithms that trade flops for communication.

This year, we presented a new approach for computing a QR factorization one of the main dense linear algebra kernels of tall and skinny matrices in a grid computing environment that overcomes these two bottlenecks. Our contribution is to articulate a recently proposed algorithm (Communication Avoiding QR) with a topology-aware middleware (QCG-OMPI) in order to confine intensive communications (ScaLAPACK calls) within the different geographical sites.

An experimental study conducted on the Grid5000 platform shows that the resulting performance increases linearly with the number of geographical sites on large-scale problems (and is in particular consistently higher than ScaLAPACKs).