Accepted for publication in Applied Numerical Mathematics 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.

Linear systems whose coefficients have large uncertainties arise routinely in finite element calculations for structures with uncertain geometry, material properties, or loads. However, a true worst case analysis of the influence of such uncertainties was previously possible only for very small systems and uncertainties, or in special cases where the coefficients do not exhibit dependence. This paper presents a method for computing rigorous bounds on the solution of such systems, with a computable overestimation factor that is frequently quite small. The merits of the new approach are demonstrated by computing realistic bounds for some large, uncertain truss structures, some leading to linear systems with over 5000 variables and over 10000 interval parameters, with excellent bounds for up to about 10 % input uncertainty. Also discussed are some counterexamples for the performance of traditional approximate methods for worst case uncertainty analysis.

Branch and bound methods for nding all zeros of a nonlinear system of equations in a box frequently have the diculty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box. This has the eect that near each zero, many small boxes are created by repeated splitting, whose processing may dominate the total work spent on the global search. This paper discusses the reasons for the occurrence of this so-called cluster eect, and how to reduce the cluster eect by de ning exclusion regions around each zero found, that are guaranteed to contain no other zero and hence can safely be discarded. Such exclusion regions are traditionally constructed using uniqueness tests based on the Krawczyk operator or the Kantorovich theorem. These results are reviewed; moreover, re nements are proved that signi cantly enlarge the size of the exclusion region. Existence and uniqueness tests are also given. Keywords: zeros, system of equations, validated enclosure, existence test, uniqueness test, inclusion region, exclusion region, branch and bound, cluster eect, Krawczyk operator, Kantorovich theorem, backboxing, ane invariant 2000 MSC Classi cation: primary 65H20, secondary 65G30 1

Interval analysis provides techniques that make it possible to determine all solutions to a nonlinear algebraic equation system and to do so with mathemat-ical and computational certainty. Such methods are based on the processing of granules in the form of intervals and thus can be regarded as one facet of granular computing. We review here some of the key concepts used in these methods and then focus on some specific application areas, namely ecological modeling, transition state analysis, and the modeling of phase equilibrium. 1

This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing.

Abstract. This paper serves as background information for the Vienna proposal for interval standardization, explaining what is needed in practice to make competent use of the interval arithmetic provided by an implementation of the standard to be. Discussed are methods to improve the quality of interval enclosures of the range of a function over a box, considerations of possible hardware support facilitating the implementation of such methods, and the results of a simple interval challenge that I had posed to the reliable computing mailing list on November 26, 2008. Also given is an example of a bound constrained global optimization problem in 4 variables that has a 2-dimensional continuum of global minimizers. This makes standard branch and bound codes extremely slow, and therefore may serve as a useful degenerate test problem.

Scope of the talk. The problem is in a rather general form: (P) minimize 0 ( ) x ϕ

Epidemiological models can be used to study the impact of an infection within a population. These models often involve parameters that are not known with certainty. Using a method for the verified solution of nonlinear dynamic models we can bound the disease trajectories that are possible for given bounds on the uncertain parameters. The method used is based on the use of an interval Taylor series to represent dependence on time and the use of Taylor models to represent dependence on uncertain parameters and/or initial conditions. The use of this method in epidemiology is demonstrated using the SIRS model, and other variations of Kermack-McKendrick models, including the case of time-dependent transmission.