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.
This paper presents a workflow for synthesizing near-optimal FPGA implementations for structured-mesh based stencil applications for explicit solvers. It leverages key characteristics of the application class, its computation-communication pattern, and the architectural capabilities of the FPGA to accelerate solvers from the high-performance computing domain. Key new features of the workflow are (1) the unification of standard state-of-the-art techniques with a number of highgain optimizations such as batching and spatial blocking/tiling, motivated by increasing throughput for real-world work loads and (2) the development and use of a predictive analytic model for exploring the design space, resource estimates and performance. Three representative applications are implemented using the design workflow on a Xilinx Alveo U280 FPGA, demonstrating near-optimal performance and over 85% predictive model accuracy. These are compared with equivalent highly-optimized implementations of the same applications on modern HPC-grade GPUs (Nvidia V100) analyzing time to solution, bandwidth and energy consumption. Performance results indicate equivalent runtime performance of the FPGA implementations to the V100 GPU, with over 2× energy savings, for the largest non-trivial application synthesized on the FPGA compared to the best performing GPU-based solution. Our investigation shows the considerable challenges in gaining high performance on current generation FPGAs compared to traditional architectures. We discuss determinants for a given stencil code to be amenable to FPGA implementation, providing insights into the feasibility and profitability of a design and its resulting performance.
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