Introduction to scientific computing technologies for global analysis of multidimensional nonlinear dynamical systems

Abstract

To determine global behaviour of a dynamical system, one must find invariant sets (attractors) and their respective basins of attraction. Since this cannot be made extensively with analytical methods, the numerical global analysis is currently the subject of intensive research, especially for strongly nonlinear, multidimensional dynamical systems. Numerical analysis in dimensions higher than four present a challenge, since it requires significant computing resources. Numerical methods used in global analysis that can benefit from high-power computing are those that can parallelize either data or task elaboration on a large scale. Mass parallelization comes with large number of difficulties, restrictions and programming hazards. When not implemented in compliance with hardware organization, data and instruction management can lead to severe loss of parallel algorithm performance. Systematic and methodical approach to design parallel programs is, therefore, critical to get the most from expensive high-power computing systems and to avoid unrealistic speed-up expectation. Considering these difficulties, the goal of this chapter is to introduce readers to the world of high-power computing systems for science and global analysis of strongly nonlinear, multidimensional dynamical systems. Topic covered are classification and performance of hardware and software, classes of computing problems and methodical design of programs. Two major hardware platforms used for scientific computing, clusters and systems with computational GPU are considered. Functionality of widely utilized software solutions (OpenMP, MPI, CUDA and OpenCL) for high-power computing systems is described. Performance of individual computer components are addressed so that the reader can understand advantages, disadvantages, efficiency and limits of each hardware platform. With this knowledge users can judge if their computation problem is suitable for mass parallelization. If this is the case, which hardware and software platforms to use. To avoid many traps of parallel programming, one of the methodical design approaches is covered. Topic is closed with example applications in science and global analysis.