Accelerating explicit ODE methods on GPUs by kernel fusion

Abstract

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.

An in‐depth introduction of multi‐workgroup tiling for improving the locality of explicit one‐step methods for ODE systems with limited access distance on GPUs

“…The application of kernel fusion to ODE methods on GPUs for general ODE systems was also considered 2 . For those systems it is only allowed to fuse RHS → LC, RHS → RED, LC → LC and LC → RED dependencies, while a global barrier is required for each LC → RHS dependency.…”

“…We have added the ability to generate multi‐workgroup tilings for explicit one‐step methods along a user defined dependency chain to our automatic prototype framework 2,3 . This framework allows a user to solve an arbitrary IVP by an arbitrary explicit ODE method of several supported classes (RK methods, PIRK methods, peer methods, Adams–Bashforth methods).…”

“…To reduce the time‐to‐solution on GPUs, we use the well‐known technique of kernel fusion , which exploits the on‐die memories of the GPU (caches, scratchpad, registers) to increase data reuse. In our previous work, 2 we focused on kernel fusion for problems with an arbitrary coupling of the RHS. Since this allowed the fusion of different stages of the method only if they were independent, we lifted this restriction in a subsequent work 3 by a specialization in problems with a limited access distance, which commonly arise, for example, from stencil‐based problems (e.g., partial differential equations [PDEs] discretized by the method of lines) or block‐structured electrical circuits.…”

“…Remarks . Please note that for some methods A , b , and are not fully populated, thus thinning out the dependencies between the computations, which also can be exploited for improving the runtime 2 …”

FusionStitching: Boosting Execution Efficiency of Memory Intensive Computations for DL Workloads

“…Another known approach is kernel fusion, a technique to fuse multiple memory intensive ops with data dependencies into a single kernel to reduce off chip memory accesses. Prior works have explored this idea extensively in database [32], image processing [5,16,23], HPC applications [18,30], and AI workloads [7,19]. However, there are two notable limitations when targeting memory intensive DL models.…”