Implementation of parallel tridiagonal solvers for a heterogeneous computing environment

Macintosh, Hamish; Warne, David J.; Kelson, Neil A.; Banks, Jasmine; Farrell, Troy

doi:10.21914/anziamj.v56i0.9371

Cited by 6 publications

(8 citation statements)

References 10 publications

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“…With the introduction of High-Level synthesis (HLS) tools, a number of more recent works [14], [15], [16], [29] implemented the Thomas, PCR, and Spike algorithms on FPGA using HLS tools. Many of these works did not demonstrate the solver working on full applications, with the exception of Lászl ó et al in 2015 [14] which compared a one factor Black-Scholes option pricing equation using explicit and implicit methods on different architectures such as multi core CPUs, GPUs, and FPGAs.…”

Section: Related Workmentioning

confidence: 99%

“…Macintosh, et al in 2014 [16] uses an OpenCL based implementation targeting an Altera Stratix V FPGA using PCR and Spike algorithms. The performance on the FPGA is compared to a GPU implementation on an Nvidia Quadro 4000 GPU.…”

Section: Related Workmentioning

confidence: 99%

“…As such, our underlying goal is to understand the criteria for a given system solver to be amenable to FPGA implementation and uncover the limitations and profitability of such accelerators. Previous work on tridiagonal system solvers for FPGAs utilized both low-level hardware description languages [18], [28], [31] as well as high-level synthesis tools [3], [14], [15], [16], [29]. They demonstrated implementation of standard tridiagonal system solver algorithms (Thomas, PCR, and Spike), evaluating how to best utilize FPGA resources to maximize performance.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

High Throughput Multidimensional Tridiagonal Systems Solvers on FPGAs

Kamalakkannan¹,

Reguly²,

Fahmy³

et al. 2022

Preprint

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This paper presents a design space exploration for synthesizing optimized, high-throughput implementations of multiple multi-dimensional tridiagonal system solvers on FPGAs. Re-evaluating the characteristics of algorithms for the direct solution of tridiagonal systems, we develop a new tridiagonal solver library aimed at implementing high-performance computing applications on Xilinx FPGA hardware. Key new features of the library are (1) the unification of standard state-of-the-art techniques for implementing implicit numerical solvers with a number of novel high-gain optimizations such as vectorization and batching, motivated by multiple multi-dimensional systems common in real-world applications, (2) data-flow techniques that provide application specific optimizations for both 2D and 3D problems, including integration of explicit loops commonplace in real workloads, and (3) the development of a predictive analytic model to explore the design space, and obtain rapid resource and performance estimates. The new library provide an order of magnitude better performance when solving large batches of systems compared to Xilinx's current tridiagonal solver library. Two representative applications are implemented using the new solver on a Xilinx Alveo U280 FPGA, demonstrating over 85% predictive model accuracy. These are compared with a current state-of-the-art GPU library for solving multi-dimensional tridiagonal systems on an Nvidia V100 GPU, analyzing time to solution, bandwidth, and energy consumption. Results show the FPGAs achieving competitive or better runtime performance for a range of multi-dimensional mesh problems compared to the V100 GPU. Additionally, the significant energy savings offered by FPGA implementations, over 30% for the most complex application, are quantified. We discuss the algorithmic trade-offs required to obtain good performance on FPGAs, giving insights into the feasibility and profitability of FPGA implementations.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High Throughput Multidimensional Tridiagonal Systems Solvers on FPGAs

Kamalakkannan¹,

Reguly²,

Fahmy³

et al. 2022

Preprint

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“…They introduced a nonstandard hybrid Thomas/PCR algorithm for solving the tridiagonal systems for the implicit solver. In 2016 Macintosh et al [12] compared the capability of PCR algorithm implemented on GPUs using OpenCL (Open Computing Language) for solving tridiagonal linear systems. They evaluated these designs in the context of the solver performance, resource efficiency and numerical accuracy.…”

Section: Introductionmentioning

confidence: 99%

Parallel Thomas approach development for solving tridiagonal systems in GPU programming − steady and unsteady flow simulation

Souri

Akbarzadeh

Darian

2020

Mechanics & Industry

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The solution of tridiagonal system of equations using graphic processing units (GPU) is assessed. The parallel-Thomas-algorithm (PTA) is developed and the solution of PTA is compared to two known parallel algorithms, i.e. cyclic-reduction (CR) and parallel-cyclic-reduction (PCR). Lid-driven cavity problem is considered to assess these parallel approaches. This problem is also simulated using the classic Thomas algorithm that runs on a central processing unit (CPU). Runtimes and physical parameters of the mentioned GPU and CPU algorithms are compared. The results show that the speedup of CR, PCR and PTA against the CPU runtime is 4.4x,5.2x and 38.5x, respectively. Furthermore, the effect of coalesced and uncoalesced memory access to GPU global memory is examined for PTA, and a 2x-speedup is achieved for the coalesced memory access. Additionally, the PTA performance in a time dependent problem, the unsteady flow over a square, is assessed and a 9x-speedup is obtained against the CPU.

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“…Previous efforts in developing FPGA-based routines to solve tridiagonal systems have been limited to solving small systems with the serial omas algorithm [11][12][13]. We have previously investigated the feasibility of FPGA implementations of parallel algorithms including the parallel cyclic reduction and SPIKE [14] for solving small tridiagonal linear systems.…”

Section: Introductionmentioning

confidence: 99%

Implementing and Evaluating an Heterogeneous, Scalable, Tridiagonal Linear System Solver with OpenCL to Target FPGAs, GPUs, and CPUs

Macintosh

Banks

Kelson

2019

International Journal of Reconfigurable Computing

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Solving diagonally dominant tridiagonal linear systems is a common problem in scientific high-performance computing (HPC). Furthermore, it is becoming more commonplace for HPC platforms to utilise a heterogeneous combination of computing devices. Whilst it is desirable to design faster implementations of parallel linear system solvers, power consumption concerns are increasing in priority. This work presents the oclspkt routine. The oclspkt routine is a heterogeneous OpenCL implementation of the truncated SPIKE algorithm that can use FPGAs, GPUs, and CPUs to concurrently accelerate the solving of diagonally dominant tridiagonal linear systems. The routine is designed to solve tridiagonal systems of any size and can dynamically allocate optimised workloads to each accelerator in a heterogeneous environment depending on the accelerator’s compute performance. The truncated SPIKE FPGA solver is developed first for optimising OpenCL device kernel performance, global memory bandwidth, and interleaved host to device memory transactions. The FPGA OpenCL kernel code is then refactored and optimised to best exploit the underlying architecture of the CPU and GPU. An optimised TDMA OpenCL kernel is also developed to act as a serial baseline performance comparison for the parallel truncated SPIKE kernel since no FPGA tridiagonal solver capable of solving large tridiagonal systems was available at the time of development. The individual GPU, CPU, and FPGA solvers of the oclspkt routine are 110%, 150%, and 170% faster, respectively, than comparable device-optimised third-party solvers and applicable baselines. Assessing heterogeneous combinations of compute devices, the GPU + FPGA combination is found to have the best compute performance and the FPGA-only configuration is found to have the best overall estimated energy efficiency.

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Implementation of parallel tridiagonal solvers for a heterogeneous computing environment

Cited by 6 publications

References 10 publications

High Throughput Multidimensional Tridiagonal Systems Solvers on FPGAs

High Throughput Multidimensional Tridiagonal Systems Solvers on FPGAs

Parallel Thomas approach development for solving tridiagonal systems in GPU programming − steady and unsteady flow simulation

Implementing and Evaluating an Heterogeneous, Scalable, Tridiagonal Linear System Solver with OpenCL to Target FPGAs, GPUs, and CPUs

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