Accelerating numerical solution of stochastic differential equations with CUDA

Numerical integration of stochastic differential equations is commonly used in many branches of science. In this paper we present how to accelerate this kind of numerical calculations with popular NVIDIA Graphics Processing Units using the CUDA programming environment. We address general aspects of numerical programming on stream processors and illustrate them by two examples: the noisy phase dynamics in a Josephson junction and the noisy Kuramoto model. In presented cases the measured speedup can be as high as 675× compared to a typical CPU, which corresponds to several billion integration steps per second. This means that calculations which took weeks can now be completed in less than one hour. This brings stochastic simulation to a completely new level, opening for research a whole new range of problems which can now be solved interactively.Program summaryProgram title: SDECatalogue identifier: AEFG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFG_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Gnu GPL v3No. of lines in distributed program, including test data, etc.: 978No. of bytes in distributed program, including test data, etc.: 5905Distribution format: tar.gzProgramming language: CUDA CComputer: any system with a CUDA-compatible GPUOperating system: LinuxRAM: 64 MB of GPU memoryClassification: 4.3External routines: The program requires the NVIDIA CUDA Toolkit Version 2.0 or newer and the GNU Scientific Library v1.0 or newer. Optionally gnuplot is recommended for quick visualization of the results.Nature of problem: Direct numerical integration of stochastic differential equations is a computationally intensive problem, due to the necessity of calculating multiple independent realizations of the system. We exploit the inherent parallelism of this problem and perform the calculations on GPUs using the CUDA programming environment. The GPU's ability to execute hundreds of threads simultaneously makes it possible to speed up the computation by over two orders of magnitude, compared to a typical modern CPU.Solution method: The stochastic Runge–Kutta method of the second order is applied to integrate the equation of motion. Ensemble-averaged quantities of interest are obtained through averaging over multiple independent realizations of the system.Unusual features: The numerical solution of the stochastic differential equations in question is performed on a GPU using the CUDA environment.Running time: < 1 minute