Performance Analysis of Accelerator Architectures and Programming Models for Parareal Algorithm Solutions of Ordinary Differential Equations

Increasing needs for the study of complex dynamical systems require computing solutions of a large number of ordinary and partial differential time-dependent equations in near real-time. Numerical integration algorithms, which are computationally expensive and inherently sequential, are typically used to compute solutions of ordinary and partial differential time-dependent equations. This presents challenges to study complex dynamical systems in near real-time. This paper examines the challenges of computing solutions of ordinary differential time-dependent equations using the Parareal algorithm belonging to the class of parallel-in-time algorithms on various high-performance computing accelerator-based architectures and associated programming models. The paper presents the code refactoring steps and performance analysis of the Parareal algorithm on two accelerator computing architectures: the Intel Xeon Phi CPU and Graphics Processing Unit many-core architectures, and with OpenMP, OpenACC, and CUDA programming models. The speedup and scaling performance analysis are used to demonstrate the suitability of the Parareal to compute the solutions of a single ordinary differential time-dependent equation and a family of interdependent ordinary differential time-dependent. The speedup, weak and strong scaling results demonstrate the suitability of Graphical Processing Units with the CUDA programming model as the most efficient accelerator for computing solutions of ordinary differential time-dependent equations using parallel-in-time algorithms. Considering the time and effort required to refactor the code for execution on the accelerator architectures, the Graphical Processing Units with the OpenACC programming model is the most efficient accelerator for computing solutions of ordinary differential time-dependent equations using parallel-in-time algorithms.

Graphical Processing Unit Based Time-Parallel Numerical Method for Ordinary Differential Equations

On-line transient stability analysis of a power grid is crucial in determining whether the power grid will traverse to a steady state stable operating point after a disturbance. The transient stability analysis involves computing the solutions of the algebraic equations modeling the grid network and the ordinary differential equations modeling the dynamics of the electrical components like synchronous generators, exciters, governors, etc., of the grid in near real-time. In this research, we investigate the use of time-parallel approach in particular the Parareal algorithm implementation on Graphical Processing Unit using Compute Unified Device Architecture to compute solutions of ordinary differential equations. The numerical solution accuracy and computation time of the Parareal algorithm executing on the GPU are demonstrated on the single machine infinite bus test system. Two types of dynamic model of the single synchronous generator namely the classical and detailed models are studied. The numerical solutions of the ordinary differential equations computed by the Parareal algorithm are compared to that computed using the modified Euler’s method demonstrating the accuracy of the Parareal algorithm executing on GPU. Simulations are performed with varying numerical integration time steps, and the suitability of Parareal algorithm in computing near real-time solutions of ordinary different equations is presented. A speedup of 25× and 31× is achieved with the Parareal algorithm for classical and detailed dynamic models of the synchronous generator respectively compared to the sequential modified Euler’s method. The weak scaling efficiency of the Parareal algorithm when required to solve a large number of ordinary differential equations at each time step due to the increase in sequential computations and associated memory transfer latency between the CPU and GPU is discussed.