Performance modeling of serial and parallel implementations of the fractional Adams-Bashforth-Moulton method

Numerical schemes for solving fractional differential equations are computationally heavy, due to the floating-point operations needed and, more importantly, the data flow within the entire memory system of a computer. We choose the fractional Adams-Bashforth-Moulton method as a representative numerical scheme, and review various code optimizations that can be applied to its serial and parallel implementations. As the most important contribution of this paper, we propose a simple methodology to analyze the achievable serial and parallel performance, based on quantifying the amount of data flow through various stages of the entire memory system, together with a small set of easily obtainable hardware parameters. This quantitative approach to performance modeling can in most cases pinpoint the real performance bottleneck, while also verifying the actual performance improvements due to various code optimizations. Moreover, the optimization techniques and performance modeling approach can both be applied to other convolution-intensive numerical methods for solving fractional differential equations.