Parallel algorithms for parabolic problems on graphs in neuroscience

Abstract

Three parallel numerical algorithms have been developed for solution of parabolic problems on graphs in neuroscience, suitable for structures containing closed loops. This chapter investigates parallel versions of implicit schemes, based on the superposition principle, and predictor-corrector schemes, which are more stable than explicit methods and suitable for structures with closed loops. It focuses mainly on linear problems. The chapter also investigates the worst case scenario of the given parallel calculations. It describes formulation of the mathematical model and the three algorithms which includes the ?-method and two predictor-corrector type algorithms. The parallel versions of the algorithms are also given, where the complexity and scalability analysis is performed. It also presents some results of computation experiments which illustrate the results of theoretical analysis. Finally, the chapter concludes that the theoretical investigation and the results of numerical experiments showed good scalability of all algorithms.