Accelerating explicit ODE methods on GPUs by kernel fusion

摘要

Graphics processing units (GPUs) have a promising architecture for implementing highly parallelrnsolution methods for systems of ordinary differential equations (ODEs). However, their highrnperformance comes at the price of caveats such as small caches or wide SIMD. For ODE methods,rnoptimizing the memory access pattern is often crucial. In this article, instead of consideringrnonly one specific method, we generalize the description of explicit ODE methods by using datarnflow graphs consisting of basic operations that are suitable to cover the types of computationsrnoccurring in all common explicit methods. After showing that the straightforward approach forrnprocessing the data flow graph by calling one kernel per basic operation is memory bound, wernexplain how the number of memory accesses can be reduced by the kernel fusion technique,rnwhich fuses several basic operations into one kernel. Moreover, we will present enabling transformationsrnthat allow additional fusions and thus can reduce the number of memory accessesrneven further. We apply these optimizations to three different classes of explicit ODE methods:rnembedded Runge–Kutta (RK) methods, parallel iterated RK (PIRK) methods, and peer methods.rnA detailed experimental evaluation on three modern GPUs showed speedups between 1.86 andrn3.51 compared to unfused implementations.