Energy-aware solution of linear systems with many right hand sides

Abstract

The solution of linear systems of equations with many right hand sides is mostly seen as a trivial extension of solving a linear system and the algorithmic developments mostly focus on the efficient computation of the LU decomposition. This is, however, not regarding the case where many right hand sides increase the runtime influence of the forward/backward substitution. In this contribution we present a GPU accelerated Gauss-Jordan-elimination based all-at-once solution scheme which focuses on minimizing the runtime and the energy consumption by switching the forward/backward substitution in favor of a more suitable operation. We obtain a multi-GPU aware algorithm which is up to 2.5 times faster than the current state-of-the-art LU decomposition based solution process of MAGMA and saves 48 % required energy.

Frequency scaling and energy efficiency regarding the Gauss‐Jordan elimination scheme with application to the matrix‐sign‐function on OpenPOWER 8

“…Without loss of generality, we neglect the pivoting matrices P i for deriving the level‐3 formulation of the algorithm. The integration of the pivoting is straightforward, as described in the works of Quintana‐Ortí et al and Köhler et al Note that the application of a Gauss‐Transform G i from the left to a matrix A can be reformulated into a rank‐1 update with the subsequent row scaling The partitioning of the matrix A into…”

“…Using a classic hybrid CPU‐GPU approach, we compute the matrix H on the CPU and perform the update on the remaining parts of the matrix A on the GPUs. Due to the fact that we only have a small number of GPUs inside one system, we chose the Column Block Cyclic (CBC) distribution of A across the available GPUs . The structure of the rank‐ N B updates then allows an easy parallel execution by only distributing the matrix H to all GPUs in each iteration.…”

“…Thus, we have to change the matrix storage scheme on the GPUs to row‐major in contrast to the column‐major scheme used by LAPACK on the host. The previous studies on the Kepler architecture already show that the overhead of combining the CPU to GPU transfers (as well as the GPU to CPU transfers) with a transpose operation can be neglected.…”

“…The straightforward parallelization and data distribution of the general matrix‐matrix product across many computational devices makes the algorithm preferable on (massively) parallel architectures, like multi‐core or multi‐accelerator based systems. Furthermore, one can show that the Gauss‐Jordan Elimination reduces the number of memory accesses and by using general matrix‐matrix multiplies the data locality for the single operations of the algorithm is improved. Moreover, when an architecture appears on the market, the matrix‐matrix is usually the first well optimized routine available.…”

“…In this way, data transfers between the CPU and the GPU are cheaper (with respect to runtime) than on older systems. The ratio of the peak performances between both CPUs and GPUs is a factor of 20. This constitutes an increase by a factor of 5 if we compare it to an older system, like the 16 core Intel Xeon Haswell with two nVidia K20 accelerators, which we used in our related work While keeping the energy consumption for the GPUs in the same order of magnitude as for the old K20 GPUs, the energy consumption of the POWER 8 CPUs is 480 W in idle state and 920 W under full CPU load.…”