Abstract
Solving ordinary differential equations (ODE) on heterogenous or multi-core/parallel embedded systems does significantly increase the operational capacity of many sensing systems in view of processing tasks such as self-calibration, model-based measurement and self-diagnostics. The main challenge is usually related to the complexity of the processing task at hand which costs/requires too much processing power, which may not be available, to ensure a real-time processing. Therefore, a distributed solving involving multiple cores or nodes is a good/precious option. Also, speeding-up the processing does also result in significant energy consumption or sensor nodes involved. There exist several methods for solving differential equations on single processors. But most of them are not suitable for an implementation on parallel (i.e., multi-core) systems due to the increasing communication related network delays between computing nodes, which become a main and serious bottleneck to solve such problems in a parallel computing context. Most of the problems faced relate to the very nature of differential equations. Normally, one should first complete calculations of a previous step in order to use it in the next/following step. Hereby, it appears also that increasing performance (e.g., through increasing step sizes) may possibly result in decreasing the accuracy of calculations on parallel/multi-core systems like GPUs. In this paper, we do create a new adaptive algorithm based on the Adams–Moulton and Parareal method (we call it PAMCL) and we do compare this novel method with other most relevant implementations/schemes such as the so-called DOPRI5, PAM, etc. Our algorithm (PAMCL) is showing very good performance (i.e., speed-up) while compared to related competing algorithms, while thereby ensuring a reasonable accuracy. For a better usage of computing units/resources, the OpenCL platform is selected and ODE solver algorithms are optimized to work on both GPUs and CPUs. This platform does ensure/enable a high flexibility in the use of heterogeneous computing resources and does result in a very efficient utilization of available resources when compared to other comparable/competing algorithm/schemes implementations.
Highlights
The history of using differential equations has traces in calculus from the old Newton’s times.Since it has evolved so much, and it is extensively used in many different branches of science and engineering
For a better usage of computing units/resources, the OpenCL platform is selected and ordinary differential equations (ODE) solver algorithms are optimized to work on both GPUs and CPUs
Quorum sensing (QS), which is based on bacterial communication, can be modeled with differential equations
Summary
The history of using differential equations has traces in calculus from the old Newton’s times. A lower value of ∆t provides a higher accuracy for solving a given problem but it does thereby increase the resulting calculation time This above-named further approach, i.e., a “parallel integration method”, is only possible while using multi-steps methods such as Runge-Kutta or Adams-Bashforth integrators, which are classified as larger group of Generalized Linear Model (GLM) solvers. At the end of each step, the different computing nodes should send their results to one node to sum-up or combine them appropriately and calculate a new value This last part of the lastly described scheme does visibly create a bottleneck w.r.t. to the potential speeding-up of the solving of differential equations while using a multi-step method.
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