Analysis and Optimization of an Adaptive Interpolation Algorithm for the Numerical Solution of a System of Ordinary Differential Equations with Interval Parameters

Abstract

We consider an adaptive interpolation algorithm for numerical integration of systems of ordinary differential equations (ODEs) with interval parameters and initial conditions. At each time moment, a piecewise polynomial function of a prescribed degree is constructed in the course of algorithm operation that interpolates the dependence of the solution on particular values of interval uncertainties with a controlled accuracy. We study the question of computational costs of the algorithm. An analytic estimate is derived for the number of operations; it depends on the algorithm parameters, in particular, the degree of interpolation and the specific features of the ODE system being integrated. Using a number of representative examples of various dimension and containing a varying number of interval uncertainties, we show that there exists an optimum value of interpolation degree from the viewpoint of computational costs.