A detailed comparison between Taylor model methods and other tools for validated computations is provided. Basic elements of the Taylor model (TM) methods are reviewed, beginning with the arithmetic for elementary operations and intrinsic functions. We discuss some of the fundamental properties, including high approximation order and the ability to control the dependency problem, and pointers to many of the more advanced TM tools are provided. Aspects of the current imple- mentation, and in particular the issue of floating point error control, are discussed. For the purpose of providing range enclosures, we compare with modern versions of centered forms and mean value forms, as well as the direct computation of remainder bounds by high-order interval auto- matic differentiation and show the advantages of the TM methods. We also compare with the so-called boundary arithmetic (BA) of Lanford, Eckmann, Wittwer, Koch et al., which was developed to prove existence of fixed points in several comparatively small systems, and the ultra-arithmetic (UA) developed by Kaucher, Miranker et al. which

/pdf/taylor-models-and-other-validated-functional-inclusion-1uwobhi3u7.pdf

Taylor models and other validated functional inclusion methods

Global optimization of nonconvex nonlinear programs via interval analysis

In this paper we propose a new method for reachability analysis of the class of discrete-time polynomial dynamical systems Our work is based on the approach combining the use of template polyhedra and optimization [25, 7] These problems are non-convex and are therefore generally difficult to solve exactly Using the Bernstein form of polynomials, we define a set of equivalent problems which can be relaxed to linear programs Unlike using affine lower-bound functions in [7], in this work we use piecewise affine lower-bound functions, which allows us to obtain more accurate approximations In addition, we show that these bounds can be improved by increasing artificially the degree of the polynomials This new method allows us to compute more accurately guaranteed over-approximations of the reachable sets of discrete-time polynomial dynamical systems We also show different ways to choose suitable polyhedral templates Finally, we show the merits of our approach on several examples

/pdf/reachability-analysis-of-polynomial-systems-using-linear-2aqbepeprt.pdf

Reachability analysis of polynomial systems using linear programming relaxations

An overview of interval arithmetical tools and basic techniques is presented that can be used to construct deterministic global optimization algorithms. These tools are applicable to unconstrained and constrained optimization as well as to nonsmooth optimization and to problems over unbounded domains. Since almost all interval based global optimization algorithms use branch-and-bound methods with iterated bisection of the problem domain we also embed our overview in such a setting.

http://www-sbras.nsc.ru/interval/Library/Thematic/GlobOptim/IAGlobOpt.pdf

What can interval analysis do for global optimization

An already classical attempt at solving a circuit design problem leads to a system of 9 nonlinear equations in 9 variables. The sensitivity of the problem to small perturbations is extraordinarily high. Since 1974 several investigations have been made into this problem and they hint at one solution in the restricted domain of the nonnegative reals. The investigations did not give error estimates nor did they present conclusive evidence that the solution found is the only one in the domain of the nonnegative reals. Our paper reports on experimental computations which used various kinds of interval analytic methods while also sometimes reflecting on Wright-Cutteridge's philosophy and theses. The computations resulted in a guarantee that in the domain of consideration, that is, the interval [0,10] for each of the 9 variables, exactly one solution did exist, which was near the solution known up to now. Finally, our solution could be localized within a parallelpiped with edge lengths between 10−6 and 3.2-10−4.

Experiments using interval analysis for solving a circuit design problem

The Bernstein and B-spline forms are generalized to multivariate polynomials. These forms are combined with a type of Taylor form for multivariate functions to generate realizable forms for multivariate functions.

Interval approximation of higher order to the ranges of functions

We give a short overview of the general ideas involved in solving optimization problems using interval arithmetic. We include a discussion of a few prototype optimization algorithms.

Interval tools for global optimization

A family of centered forms for rational functions dependent on a parameter is investigated. Particular choices of the parameter are investigated in further detail. It is also shown that a centered form of minimum width can be chosen from the family. These centered forms are then generalized to a family of forms of higher order.

J.G. Rokne

Papers

Interval approximation of higher order to the ranges of functions

Interval tools for global optimization

A family of centered forms for rational functions