8 research outputs found
An efficient algorithm for the parallel solution of high-dimensional differential equations
The study of high-dimensional differential equations is challenging and
difficult due to the analytical and computational intractability. Here, we
improve the speed of waveform relaxation (WR), a method to simulate
high-dimensional differential-algebraic equations. This new method termed
adaptive waveform relaxation (AWR) is tested on a communication network
example. Further we propose different heuristics for computing graph partitions
tailored to adaptive waveform relaxation. We find that AWR coupled with
appropriate graph partitioning methods provides a speedup by a factor between 3
and 16
Waveform relaxation as a dynamical system
In this paper the properties of waveform relaxation are studied when applied to the dynamical system generated by an autonomous ordinary differential equation. In particular, the effect of the waveform relaxation on the invariant sets of the flow is analysed. Windowed waveform relaxation is studied, whereby the iterative technique is applied on successive time intervals of length T and a fixed, finite, number of iterations taken on each window. This process does not generate a dynamical system on R+ since two different applications of the waveform algorithm over different time intervals do not, in general, commute. In order to generate a dynamical system it is necessary to consider the time T map generated by the relaxation process. This is done, and C^1-closeness of the resulting map to the time T map of the underlying ordinary differential equation is established. Using this, various results from the theory of dynamical systems are applied, and the results discussed
Waveform relaxation as a dynamical system
Abstract. In this paper the properties of waveform relaxation are studied when applied to the dynamical system generated by an autonomous ordinary differential equation. In particular, the effect of the waveform relaxation on the invariant sets of the flow is analysed. Windowed waveform relaxation is studied, whereby the iterative technique is applied on successive time intervals of length T and a fixed, finite, number of iterations taken on each window. This process does not generate a dynamical system on R + since two different applications of the waveform algorithm over different time intervals do not, in general, commute. In order to generate a dynamical system it is necessary to consider the time T map generated by the relaxation process. This is done, and C 1-closeness of the resulting map to the time T map of the underlying ordinary differential equation is established. Using this, various results from the theory of dynamical systems are applied, and the results discussed. 1
Waveform Relaxation as a Dynamical System
In this paper the properties of waveform relaxation are studied when applied to the dynamical system generated by an autonomous ordinary differential equation. In particular, the effect of the waveform relaxation on the invariant sets of the flow is analysed. Windowed waveform relaxation is studied, whereby the iterative technique is applied on succesive time intervals of length T and a fixed, finite, number of iterations taken on each window. This process does not generate a dynamical system on R + since two different applications of the waveform algorithm over different time intervals do not, in general, commute. In order to generate a dynamical system it is necessary to consider the time T map generated by the relaxation process. This is done, and C 1 -closeness of the resulting map to the time T map of the underlying ordinary differential equation is established. Using this, various results from the theory of dynamical systems are applied, and the results discussed. 1 Introduct..