Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems

Abstract

Abstract--Zusammenfassung Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for NonlinearSystems. Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of the original nonlinear system sometimes leads to nonconvergent iteration.In this paper, we examine interval iterations based on an expanded system obtained from the intermediate quantities in the original system. In this system, there is no overestimation in entries of the interval Jacobi matrix, and nonlinearities can be taken into account to obtain sharp bounds. We present an example in detail, algorithms, and detailed experimental results obtained from applying our algorithms to the example. AMS Subject Classifications: Primary: 65H10; secondary: 65G10Key words: Nonlinear algebraic systems, interval arithmetic, automatic differentiation. Verbesserung der Intervalliteration fiir niehtlineare Gleichungssysteme durch Zerlegung arithmetischerAusdriicke. Intervalliterationen k6nnen in Verbindung mit anderen Verfahren verwendet werden, um alle Ltsungen eines nichlinearen Gleichungsystems in einem gegebenen Gebiet mit Sicherheit abzuschgtzen, und auch um Approximationen der Ltsungen solcher Systeme zu veriflzieren. Die Abschiitzungen in den Veffahren sind jedoch manchmal nieht hinreichend genau, da (Jbersehfitzungen in der Berechnung uud in dem Gebraueh der Invervall-Jacobi Matrix auflreten.In der vorliegenden Arbeit wcrden Intervalliterationen auf einem erweiterten Gleichungssystem behandelt. In diesem System gibt es keine Oberschiitzungen der Einzelkomponenten der Intervall-Jacobi Matrix, und fiir die Nichtlinearitfiten ktnnen Absch~tzungen angegeben werden. Anhand eines Beispiels wird die Wirkungsweise der behandelten Algorithmen demonstriert.

New interval methodologies for reliable chemical process modeling

“…Methods that seek to make improvements in solving the linear interval system defined by Eq. (1), the interval-Newton equation (e.g., Kearfott, 1990;Hu, 1990;Kearfott et al, 1991;Gan et al, 1994;Hansen, 1997); or some combination of the above (e.g., Madan, 1990;Dinkel et al, 1991;Kearfott, 1991;Kearfott, 1997). A comprehensive review or these and other techniques is beyond the scope of this paper.…”

Efficient interval partitioning for constrained global optimization

“…Such binary trees enable interval propagation over all subexpressions of the constraints and the objective function [BMV94]. Interval propagation and function trees are used by [Kea91] in improving interval Newton approach by decomposition and variable expansion, by [SP99] in automated problem reformulation, by [Sah03] and by [TS04] where feasibility based range reduction is achieved by tightening variable bounds.…”

Advances in Interval Methods for Deterministic Global Optimization in Chemical Engineering

“…Their goal was to develop an approach that would significantly reduce the number of subintervals that must be tested in the interval-Newton algorithm, but that could be implemented with little computational overhead, so that large savings in computation time could be realized. In a pivoting preconditioner [12], only one element of the preconditioning row y i , called the pivot element, is nonzero, and it is assigned a value of one. In this case, Eq.…”

“…One way to improve the efficiency of the interval-Newton approach for solving a nonlinear equation system is to more tightly bound the solution set of the linear interval equation system that is at the core of this approach. In this paper, we review recent preconditioning techniques [5,11,12], for this purpose, and a new bounding strategy based on the use of linear programming (LP) techniques is presented. Using this approach it is possible to exactly (within round out) determine the desired bounds on the solution set of the linear interval system.…”