Parallel integration of ODEs based on convolution algorithms
A method for parallel computing of ordinary differential equations (ODE's) is introduced. The method is based on the potential parallelism of convolution algorithms. Initial value problems for a system of nonlinear ODE's are considered. The solution of the problems is presented in convolution form. The idea of the proposed approach is to discretize the continuous convolution and after that to apply any existing parallel convolution algorithms, e.g. FFT algorithms, circle convolution by segmenting, direct convolution etc. The opportunity of applying convolution algorithms in finite fields which supplies new properties and additional speedup with special parallel signal processors using number theoretic transformations is discussed. The author's approach is considered on the example of the well-known Lorenz model of liquid convection. A connection between properties of shift operators on trajectories, the convolution, and ODE parallelism across time is shown. In this way there is an easy explanation for \textit{C. W. Gear's} observation [Calcolo 25, No. 1/2, 1-20 (1988; Zbl 0675.65068)] about high parallelism across time for equations with ``large'' and ``small'' Jacobians. An estimation for the norm of the shift operator derivative gives some information about a signal transmission from one point of time to another along trajectories. The parallel convolution algorithms: optical digital filtration, direct convolution on systolic arrays, circle convolution by segmenting, FFT algorithms are considered.
scientific article; zbMATH DE number 490617
scientific article