**Magma and Batched Small Dense Matrix Computation on the GPU**
Tingxing Dong

*26 August 2014, 10h30 - 26 August 2014, 11h30*
Salle/Bat : 465/PCRI-N

Contact :

**Activités de recherche : **Calcul à haute performance

**Résumé :**
The Recent Progress of MAGMA (less than 10min)

The MAGMA (Matrix Algebra on GPU and Multicore Architectures) project aims to develop a dense linear algebra library similar to LAPACK but for heterogeneous/hybrid architectures, like "Multicore+GPU", "Multcore+MIC" systems.

MAGMA uses a hybrid methodology where algorithms of interest are slit into tasks of varying

granularity and their execution scheduled over the available hardware component. Small non-parallelizable tasks often on critical path are schedule on the CPU, and large parallelizable tasks are schedule on accelerators. We talk about the recent features of MAGMA for CUDA 1.5, MAGMA MIC 1.2, clMAGMA 1.1.

Batched Small Dense Matrix Computation on the GPU (20min)

Ones-sided factorizations (Cholesky, LU and QR) are commonly used to solve

dense linear systems in scientific models. In a large number of

applications, a need arises to solve many small size problems,

instead of few large linear systems. The size of each of these

small linear systems depends, for example, on the number of

the ordinary differential equations (ODEs) used in the model,

and can be on the order of hundreds of unknowns. To efficiently

exploit the computing power of modern accelerator hardware,

these linear systems are processed in batches. To improve the

numerical stability of the Gaussian Elimination（LU), at least partial

pivoting is required, most often accomplished with row pivoting.

However, row pivoting can result in a severe performance penalty

on GPUs because it brings in thread divergence and non-coalesced

memory accesses. In this paper, we propose a batched LU

factorization for GPUs by using a multi-level blocked right

looking algorithm that preserves the data layout but minimizes

the penalty of partial pivoting. We extend this algorithm to Cholesky and LU.

Our batched LU achieves up to 2.5-fold speedup when compared to the alternative CUBLAS

solution on a K40c GPU. Our batched Cholesky, batched QR achieves 1.8 speedup

compared to the optimized parallel implementation in the MKL

library on two sockets of Intel Sandy Bridge CPUs.

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