Quantised State Simulation (QSS): Advances in the Numerical Solution of Ordinary Differential Equations
Vassiliadis, Vassilios S.
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Vassiliadis, V. S., & Fiorelli, F. (2014). Quantised State Simulation (QSS): Advances in the Numerical Solution of Ordinary Differential Equations [Presentation file]. https://www.repository.cam.ac.uk/handle/1810/245285
This presentation handout presents the idea of discretising the state variables (quantising them) instead of time, to effect the numerical integration/simulation of ordinary differential equation systems. The aim is to provide the computational technology to address huge scale systems at extremely high efficiency, at unprecedented speeds over existing methods. The methodology results in a matrix free algorithm, which can also be very readily be used for sensitivity evaluations and is highly parallelisable (in fact it is completely scalable). A multitude of potential uses is outlined, i.e. replacing totally the need for stochastic simulation algorithms for dynamical systems as the method is completely rigorous and robust, and nonrandom. In other words it comprises an innovative standard type numerical integration scheme. The potential for applications in Molecular Dynamic Simulation, Polymerisation reaction simulations, population dynamic balances, and of course in combustion reactions is tremendous. The importance also and its particular incomparable strength over stochastic simulation methods is that it can produce rigorous high precision values for sensitivity equations. As such it can be integrated within a parameter estimation scheme robustly, obviating the need for nonrigorous stochastic gradient methods. Furthermore it can be used in conjunction with optimisation software to provide stable and reliable gradients (of tunable precision at limited extra cost) so as to assist in the computer aided design of huge scale dynamical systems (e.g. as in the simulation of combustion in a distributed way within an internal combustion engine in the automobile industry). With a proper integration within an optimisation framework, furthermore constraints can be added so as to design engines and their operation such that any emissions be minimised and completely controlled. The future for these methods, and in particular to the way we approach this research project is totally new, innovative and guaranteed to lead to potential the future method of choice to deal with systems dynamics. Coupled also with the natural ability to handle discrete events in a most efficient manner (solving one equation in one variable only to lock onto discrete event locations) it seems clearly this is the richest area to investigate in terms of the numerical analysis and solution of ordinary differential equation systems.
Ordinary Differential Equations, State discretisation, Effective continuous time solution of ODE's, First order sensitivity equations, Discrete events, Stiffness resolution, one-variable-at-a-time scheme, High-performance computing, discrete event location, Sensitivity Equations, ODE, philosophy of continuous time systems
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