Solving Large Nonlinear Systems of First-Order Ordinary Differential Equations With Hierarchical Structure Using Multi-GPGPUs and an Adaptive Runge Kutta ODE Solver

Abstract

The adaptive Runge–Kutta (ARK) method on multi-general-purpose graphical processing units (GPUs) is used for solving large nonlinear systems of first-order ordinary differential equations (ODEs) with over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\sim}{10\, 000}$</tex></formula> variables describing a large genetic network in systems biology for the biological clock. To carry out the computation of the trajectory of the system, a hierarchical structure of the ODEs is exploited, and an ARK solver is implemented in compute unified device architecture/C++ (CUDA/C++) on GPUs. The result is a 75-fold speedup for calculations of 2436 independent modules within the genetic network describing clock function relative to a comparable CPU architecture. These 2436 modules span one-quarter of the entire genome of a model fungal system, Neurospora crassa. The power of a GPU can in principle be harnessed by using warp-level parallelism, instruction level parallelism or both of them. Since the ARK ODE solver is entirely sequential, we propose a new parallel processing algorithm using warp-level parallelism for solving <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\sim}{10\,000}$</tex></formula> ODEs that belong to a large genetic network describing clock genome-level dynamics. A video is attached illustrating the general idea of the method on GPUs that can be used to provide new insights into the biological clock through single cell measurements on the clock.