1998 MCS Divisional Seminars & Colloquia

Computing Rigorous Bounds on the Solution of  an Initial Value Problem for an Ordinary Differential Equation

Ned Nedialkov
University of Toronto
  Hosted by Paul Hovland

10:30 AM, December 21, 1998
Building 221,  Room A-216

y'(t)  = f(y),  y(t₀) = y₀,                                   (0.1)

where t Î [t₀, T] for some T > t₀. Our goal is to compute a set of points {t_j : j = 1,...,N} in (t₀, T] and an associated set of interval vectors {[y_j]: j=1,...,N} such that y(t_j), the true solution of (0.1) at t_j, is guaranteed to be contained in [y_j] for all j=1,...,N. In most validated methods for IVPs for ordinary differential equations (ODEs), each step consists of two phases:

1. Compute an interval that is guaranteed to contain a unique solution and an a priori enclosure of the solution on this interval.
2. Using this a priori enclosure, compute a tighter enclosure of the solution.

To date, the only effective approach for computing tight enclosures of the solution has been interval methods based on Taylor series. In this talk, we overview interval Taylor series methods (ITS) and discuss a new approach, an interval Hermite-Obreschkoff (IHO) method, for computing such enclosures. Compared to ITS methods, for the same order and step size, our IHO scheme has a smaller truncation error and better stability. As a result, the IHO method allows larger step sizes than the corresponding ITS methods, thus saving computation time. In addition, since fewer Taylor coefficients and their Jacobians are required by IHO than ITS methods, the IHO method performs better than the ITS methods when the function for computing the right side contains many terms. We also show that the stability of an interval method is determined not only by the stability function of the underlying formula, as in a standard method for an IVP for an ODE, but also by the associated formula for the truncation error.