Validated solutions of initial value problems for parametric ODEs

Abstract

In initial value problems for ODEs with interval-valued parameters and/or initial values, it is desirable in many applications to be able to determine a validated enclosure of all possible solutions to the ODE system. Much work has been done for the case in which initial values are given by intervals, and there are available software packages that deal with this case. However, less work has been done on the case in which parameters are given by intervals. We describe here a new method for obtaining validated solutions of initial value problems for ODEs with interval-valued parameters. The method also accounts for interval-valued initial values. The effectiveness of the method is demonstrated using several numerical examples involving parametric uncertainties.

Verified Solution of Nonlinear Dynamic Models in Epidemiology

“…We have demonstrated here the use of a recently developed interval method (Lin and Stadtherr, 2007) for determining mathematically and computationally guaranteed bounds on the population trajectories that are possible for given bounds on the uncertain quantities. Using Monte Carlo analysis, it was also shown that it is possible for these bounds to be quite tight.…”

“…To control the wrapping effect, the state enclosures are propagated using a new type of Taylor model consisting of a polynomial and a parallelepiped (as opposed to an interval) remainder bound. Complete details of the computation of T yj+1 are given by Lin and Stadtherr (2007). An implementation of this approach, called VSPODE (Verifying Solver for Parametric ODEs), has been developed and tested by Lin and Stadtherr (2007), who compared its performance with results obtained using the popular VNODE package (Nedialkov et al, 1999;Nedialkov et al, 2001).…”

“…Complete details of the computation of T yj+1 are given by Lin and Stadtherr (2007). An implementation of this approach, called VSPODE (Verifying Solver for Parametric ODEs), has been developed and tested by Lin and Stadtherr (2007), who compared its performance with results obtained using the popular VNODE package (Nedialkov et al, 1999;Nedialkov et al, 2001). For the test problems used, VSPODE provided tighter enclosures on the state variables than VNODE, and required significantly less computation time.…”

“…Alternative bounding schemes are mostly focused on the exact bounding of the firstand second-order terms of p (Neumaier, 2003), but exact bounding of a general interval quadratic can also be computationally expensive (in the worst case, exponential in the number of variables). Lin and Stadtherr (2007) use a simple compromise approach in which only the first-order and the diagonal second-order terms are considered for exact bounding, and other terms are evaluated directly using interval arithmetic. This is the approach used on the problems considered here.…”

“…However, relatively little work has been done on the case in which parameters are given by intervals. In the work described here, we will use a recently developed solver for parametric ODEs (Lin and Stadtherr, 2007) called VSPODE (Validating Solver for Parametric ODEs), which is used to produce guaranteed bounds on the solutions of nonlinear dynamic systems with interval-valued initial states and parameters. Both COSY VI and VSPODE use Taylor models (Makino and Berz, 1996;Makino and Berz, 1999;Makino and Berz, 2003), though in different ways, to deal with the uncertain quantities (parameters and initial values).…”