Interval Subroutine Library Mission

We propose the collection, standardization, and distribution of a full-featured production quality library for reliable scientific computing with routines using interval techniques for use by the wide community of applications developers

Efficient importance sampling in low dimensions using affine arithmetic

Despite the development of sophisticated techniques such as sequential Monte Carlo, importance sampling (IS) remains an important Monte Carlo method for low dimensional target distributions. This paper describes a new technique for constructing proposal distributions for IS, using affine arithmetic. This work builds on the Moore rejection sampler to which we provide a comparison

Treating Non-Smooth Functions as Smooth Functions in Global Optimization and Nonlinear Systems Solvers

this paper. First, required properties of interval extensions for non-smooth functions are discussed. Then, selected formulas from [4] are presented. The formulas for slopes presented here represent improvements (sharper bounds) over those in [4]. A simple, illustrative example is then given. Fourth, a convergence and existence / uniqueness verification theory for interval Newton methods using non-smooth extensions is developed. Finally, formulas and examples are given for which interval Newton methods can find critical points, even if the gradient itself is discontinuous. 2 R. Baker Kearfot

Interval Extensions of Non-Smooth Functions for Global Optimization and Nonlinear Systems Solvers

Most interval branch and bound methods for nonlinear algebraic systems have to date been based on implicit underlying assumptions of continuity of derivatives. In particular, much of the theory of interval Newton methods is based on this assumption. However, derivative continuity is not necessary to obtain effective bounds on the range of such functions. Furthermore, if the first derivatives just have jump discontinuities, then interval extensions can be obtained that are appropriate for interval Newton methods. Thus, problems such as minimax or l 1 approximations can be solved simply, formulated as unconstrained nonlinear optimization problems. In this paper, interval extensions and computation rules are given for the unary operation jxj, the binary operation maxfx; yg and a more general "jump" function Ø(s; x; y). These functions are incorporated into an automatic differentiation and code list interpretation environment. Experimental results are given for nonlinear systems involvin..

On Existence and Uniqueness Verification for Non-Smooth Functions 1 Summary

It is known that interval Newton methods can verify existence and uniqueness of solutions of a nonlinear system of equations near points where the Jacobi matrix of the system is not ill-conditioned. Recently, we have shown how to verify existence and uniqueness, up to multiplicity, for solutions at which the Jacobi matrix is singular. We do this by efficient computation of the topological index over a small box containing the approximate solution. Algorithmically, our techniques mimic the non-singular case (both in algorithmic steps and computational complexity), and can be considered as incomplete Gauss–Seidel sweeps. Since the topological index is defined and computable when the Jacobi matrix is not even defined at the solution, one may speculate that efficient algorithms can be devised for verification in this case, too. In this talk, we discuss, through examples, key techniques underlying our simplification of the calculations that cannot necessarily be used when the function is non-smooth. We also suggest when degree computations involving non-smooth functions may be practical. Our examples also shed light on our published work on verification involving the topological degree.