Matusiak Mariusz
Institute of Applied Computer Science, Lodz University of Technology, 90-924 Lodz, Poland.
Entropy (Basel). 2020 May 18;22(5):566. doi: 10.3390/e22050566.
In this article, some practical software optimization methods for implementations of fractional order backward difference, sum, and differintegral operator based on Grünwald-Letnikov definition are presented. These numerical algorithms are of great interest in the context of the evaluation of fractional-order differential equations in embedded systems, due to their more convenient form compared to Caputo and Riemann-Liouville definitions or Laplace transforms, based on the discrete convolution operation. A well-known difficulty relates to the non-locality of the operator, implying continually increasing numbers of processed samples, which may reach the limits of available memory or lead to exceeding the desired computation time. In the study presented here, several promising software optimization techniques were analyzed and tested in the evaluation of the variable fractional-order backward difference and derivative on two different Arm Cortex-M architectures. Reductions in computation times of up to 75% and 87% were achieved compared to the initial implementation, depending on the type of Arm core.
本文提出了一些基于 Grünwald-Letnikov 定义实现分数阶向后差分、求和及分数阶积分-微分算子的实用软件优化方法。这些数值算法在嵌入式系统中评估分数阶微分方程时具有重要意义，因为与基于离散卷积运算的 Caputo 和 Riemann-Liouville 定义或拉普拉斯变换相比，它们的形式更为便捷。一个众所周知的难题与算子的非局部性有关，这意味着需要处理的样本数量不断增加，可能会达到可用内存的极限或导致超出预期的计算时间。在本文所呈现的研究中，分析并测试了几种有前景的软件优化技术，用于在两种不同的 Arm Cortex-M 架构上评估可变分数阶向后差分和导数。根据 Arm 内核的类型，与初始实现相比，计算时间减少了高达 75%和 87%。
