Modeling the effect of application-specific program transformations on energy and performance improvements of parallel ODE solvers

Abstract

Ordinary differential equations (ODEs) are important for modelling many problems from science and engineering and efficient ODE solvers are required, for example when solving time-dependent partial differential equations (PDEs) with the method of lines. Since an ODE solver may perform a large number of iteration steps, the execution time for solving an ODE problem might be quite large. Thus, a reduction of the execution time is desirable and should affect each iteration step of the simulation. Programming techniques to reduce the execution time of ODE solver are parallelism and modification of the memory access structure such that the memory access time decreases. In this article, we investigate multithreaded solution methods for ODEs with different memory access behavior and their influence on the performance. Additionally the energy consumption is considered. The parallelism is implemented as shared memory program for multicore processors. The memory access behavior is investigated using different program variants which result from application-specific program transformations changing the memory access order while guaranteeing the numerical correctness. For the investigation of the performance, experimental data have been gathered on five different recent multicore processors. Additionally, an analytical power and energy model for modeling the performance and energy consumption is introduced. As ODE solver, the popular embedded Runge-Kutta methods with error correction is used. The simulation problems are two different ODEs resulting from discretized PDEs. The experimental data give insight into the quite diverse performance behavior of the ODE solver variants solving the same problem on different platforms.