Accelerating explicit ODE methods on GPUs by kernel fusion. (12th March 2018)

• Record Type: Journal Article
• Authors: Korch, Matthias; Werner, Tim
• Is Part Of: Concurrency and computation. Volume 30:Number 18(2018)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2018-03-12
• Subjects: CUDA -- explicit methods -- GPU -- initial value problems -- kernel fusion -- locality -- numerical integration -- OpenCL -- ordinary differential equations -- parallel -- peer methods -- PIRK methods -- Runge–Kutta methods -- scalability; Parallel processing (Electronic computers) -- Periodicals; Parallel computers -- Periodicals; 004.35
• DOI: 10.1002/cpe.4470
• Languages: English
• ISSNs: 1532-0626

Abstract

Summary: Graphics processing units (GPUs) have a promising architecture for implementing highly parallel solution methods for systems of ordinary differential equations (ODEs). However, their high performance comes at the price of caveats such as small caches or wide SIMD. For ODE methods, optimizing the memory access pattern is often crucial. In this article, instead of considering only one specific method, we generalize the description of explicit ODE methods by using data flow graphs consisting of basic operations that are suitable to cover the types of computations occurring in all common explicit methods. After showing that the straightforward approach for processing the data flow graph by calling one kernel per basic operation is memory bound, we explain how the number of memory accesses can be reduced by the kernel fusion technique, which fuses several basic operations into one kernel. Moreover, we will present enabling transformations that allow additional fusions and thus can reduce the number of memory accesses even further. We apply these optimizations to three different classes of explicit ODE methods: embedded Runge–Kutta (RK) methods, parallel iterated RK (PIRK) methods, and peer methods. A detailed experimental evaluation on three modern GPUs showed speedups between 1.86 and 3.51 compared to unfused implementations.