High Performance Portable Solver for Tridiagonal Toeplitz Systems of Linear Equations

Dmitruk, Beata; Stpiczynski, Przemyslaw

doi:10.1007/978-3-030-71593-9_14

Cited by 1 publication

(10 citation statements)

References 12 publications

(14 reference statements)

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“…Our OpenMP implementation of the algorithm achieved satisfying speedup on Intel Xeon CPUs and Intel Xeon Phi. In our next paper, 13 we showed further improvements in the implementation of the algorithm using more sophisticated vectorization techniques. We also used OpenACC, a standard for accelerated computing, 14,15 which introduces compiler directives for offloading selected computations from host to attached accelerator devices.…”

Section: Introductionmentioning

confidence: 90%

“…Moreover, it cannot be automatically vectorized and parallelized. To obtain an efficient vectorizable parallel algorithm for solving () let us consider the following divide and conquer method 7,13 . First, we choose two integers

r, s > 1

r s = n

, and rewrite

L

in the following block form:

L = [\begin{array}{cccc} L_{s} \\ B & L_{s} \\ ⋱ & ⋱ \\ B & L_{s} \end{array}], L_{s} = [\begin{array}{cccc} 1 \\ α & 1 \\ ⋱ & ⋱ \\ α & 1 \end{array}], B = [\begin{array}{cccc} 0 & \dots & 0 & α \\ ⋮ & 0 & 0 \\ ⋮ & . . . & ⋮ \\ 0 & \dots & \dots & 0 \end{array}] .

Let us define

e_{k} = {(\underset{k}{\underset{⏟}{0, \dots, 0}}, 1, 0, \dots, 0)}^{T} \in ℝ^{s}, k = 0, \dots, s - 1,

and split

d

z

into vectors

…”

Section: Parallel Algorithmsmentioning

confidence: 99%

“…OpenACC programs can be compiled for a variety of accelerators and parallel computers based on multicore CPUs. It is also possible to use OpenACC and OpenMP together to utilize CPU and GPU cores at the same time 13 …”

Section: Openacc‐based Implementationsmentioning

confidence: 99%

“…We also used OpenACC, a standard for accelerated computing, 14,15 which introduces compiler directives for offloading selected computations from host to attached accelerator devices. Our new portable implementation that uses OpenACC can be executed on both CPU‐based and GPU‐accelerated systems 13 . Moreover, more sophisticated variants of the implementation are suitable for systems with multiple GPUs or ready to use CPU and GPU cores simultaneously 13 .…”

Section: Introductionmentioning

confidence: 99%

“…Our new portable implementation that uses OpenACC can be executed on both CPU‐based and GPU‐accelerated systems 13 . Moreover, more sophisticated variants of the implementation are suitable for systems with multiple GPUs or ready to use CPU and GPU cores simultaneously 13 . We also considered the use of both column‐wise and row‐wise storage formats for two‐dimensional arrays and showed how to efficiently convert between these two formats using cache memory to ensure coalesced access to device's global memory.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Solving tridiagonal Toeplitz systems of linear equations on GPU‐accelerated computers

Dmitruk

Stpiczynski

2021

Concurrency and Computation

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The aim of this article is to show that solvers for tridiagonal Toeplitz systems of linear equations can be efficiently implemented for a variety of modern GPU-accelerated and multicore architectures using OpenACC. We consider two parallel algorithms for solving such systems with special assumptions about coefficient matrices. As the first algorithm, we propose a new, faster implementation of the divide and conquer method.The next algorithm is a new, vectorizable algorithm based on a recently introduced sequential method. We consider the use of both column-wise and row-wise storage formats for two-dimensional arrays and show how to efficiently convert between these two formats using cache memory and improve the overall performance of our implementations. We also show how to tune the performance by predicting the best values of the methods' parameters. Numerical experiments performed on Intel CPUs and Nvidia GPUs show that our new implementations achieve relatively good performance and accuracy.

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Section: Introductionmentioning

confidence: 90%