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A Review of Preconditioners for the Interval GaussSeidel Method
, 1991
"... . Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ae R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in finiteprecision arithmetic. In such methods, the system ..."
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Cited by 50 (16 self)
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. Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ae R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in finiteprecision arithmetic. In such methods, the system F (X) = 0 is transformed into a linear interval system 0 = F (M) +F 0 (X)( ~ X \Gamma M); if interval arithmetic is then used to bound the solutions of this system, the resulting box ~ X contains all roots of the nonlinear system. We may use the interval GaussSeidel method to find these solution bounds. In order to increase the overall efficiency of the interval Newton / generalized bisection algorithm, the linear interval system is multiplied by a preconditioner matrix Y before the interval GaussSeidel method is applied. Here, we review results we have obtained over the past few years concerning computation of such preconditioners. We emphasize importance and connecting relationships,...
Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for Nonlinear Systems
, 1991
"... 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 accumul ..."
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Cited by 20 (9 self)
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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.
An Interval Branch and Bound Algorithm for Bound Constrained Optimization Problems
 JOURNAL OF GLOBAL OPTIMIZATION
, 1992
"... In this paper, we propose modifications to a prototypical branch and bound algorithm for nonlinear optimization so that the algorithm efficiently handles constrained problems with constant bound constraints. The modifications involve treating subregions of the boundary identically to interior region ..."
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Cited by 13 (5 self)
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In this paper, we propose modifications to a prototypical branch and bound algorithm for nonlinear optimization so that the algorithm efficiently handles constrained problems with constant bound constraints. The modifications involve treating subregions of the boundary identically to interior regions during the branch and bound process, but using reduced gradients for the interval Newton method. The modifications also involve preconditioners for the interval GaussSeidel method which are optimal in the sense that their application selectively gives a coordinate bound of minimum width, a coordinate bound whose left endpoint is as large as possible, or a coordinate bound whose right endpoint is as small as possible. We give experimental results on a selection of problems with different properties.
New Interval Methodologies for Reliable Chemical Process Modeling
 COMPUT. CHEM. ENG. 2002
, 2002
"... The use of interval methods, in particular intervalNewton/generalizedbisection techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical proces ..."
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Cited by 10 (8 self)
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The use of interval methods, in particular intervalNewton/generalizedbisection techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical process modeling. The most significant drawback of the currently used interval methods is the potentially high computational cost that must be paid to obtain the mathematical and computational guarantees of certainty. New methodologies are described here for improving the efficiency of the interval approach. In particular, a new hybrid preconditioning strategy, in which a simple pivoting preconditioner is used in combination with the standard inversemidpoint method, is presented, as is a new scheme for selection of the real point used in formulating the intervalNewton equation. These techniques can be implemented with relatively little computational overhead, and lead to a large reduction in the number of subintervals that must be tested during the intervalNewton procedure. Tests on a variety of problems arising in chemical process modeling have shown that the new methodologies lead to substantial reductions in computation time requirements, in many cases by multiple orders of magnitude.
On Interval Weighted Threelayer Neural Networks
 In Proceedings of the 31 Annual Simulation Symposium
, 1998
"... In solving application problems, the data sets used to train a neural network may not be hundred percent precise but within certain ranges. Representing data sets with intervals, we have interval neural networks. By analyzing the mathematical model, we categorize general threelayer neural network t ..."
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Cited by 2 (1 self)
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In solving application problems, the data sets used to train a neural network may not be hundred percent precise but within certain ranges. Representing data sets with intervals, we have interval neural networks. By analyzing the mathematical model, we categorize general threelayer neural network training problems into two types. One of them can be solved by finding numerical solutions of nonlinear systems of equations. The other can be transformed into nonlinear optimization problems. Reliable interval algorithms such as interval Newton/generalized bisection method and interval branchandbound algorithm are applied to obtain optimal weights for interval neural networks. The applicable stateofart interval software packages are reviewed in this paper as well. I. Introduction A. Threelayer Neural Network A threelayer neural network includes I (input), H (hidden), and O (output) layers. Each of these layers contains some nodes, called neurons. There is no any direct links for any...
On Interval Weighted Threelayer Neural Networks
 Proc. of the 31 Annual Simulation Symposium
, 1998
"... In solving application problems, the data sets used to train a neural network may not be hundred percent precise but within certain ranges. Representing data sets with intervals, we have interval neural networks. By analyzing the mathematical model, we categorize general threelayer neural networ ..."
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Cited by 1 (1 self)
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In solving application problems, the data sets used to train a neural network may not be hundred percent precise but within certain ranges. Representing data sets with intervals, we have interval neural networks. By analyzing the mathematical model, we categorize general threelayer neural network training problems into two types. One of them can be solved by flnding numerical solutions of nonlinear systems of equations. The other can be transformed into nonlinear optimization problems. Reliable interval algorithms such as interval Newton/generalized bisection method and interval branchandbound algorithm are applied to obtain optimal weights for interval neural networks. The applicable stateofart interval software packages are reviewed in this paper as well. I.
Parallel Reliable Computing with Interval Arithmetic
"... Reliability of computational results is crucial in computational science and engineering. In this paper, we report some current research results on parallel reliable computing with interval arithmetic. In section 1, a brief introduction to interval arithmetic is provided. In section 2, an interval a ..."
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Cited by 1 (0 self)
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Reliability of computational results is crucial in computational science and engineering. In this paper, we report some current research results on parallel reliable computing with interval arithmetic. In section 1, a brief introduction to interval arithmetic is provided. In section 2, an interval algorithm to reliably solving largescale sparse nonlinear systems of equations is presented. In section 3, polynomial interpolation with interval arithmetic is studied. We conclude this paper with section 4. I. Introduction Interval arithmetic, first introduced by Moore [24] in the 1960's, has become an active research area in scientific computing. Here is the definition of interval arithmetic. Definition 1.1: Let x and y be two real intervals 1 , and op be one of the arithmetic operations +; \Gamma; \Theta, \Xi. Then, x op y = fx op y : x 2 x; y 2 yg, provided that 0 62 y if op represents \Xi. For example, [1; 2] + [\Gamma1; 0] = [0; 2] and [2; 4] \Xi [1; 2] = [1; 4]. Some reasons for...
A Comparison of Some Methods for Bounding Connected and Disconnected Solution Sets of Interval Linear Systems
, 2007
"... Finding bounding sets to solutions to systems of algebraic equations with uncertainties in the coefficients, as well as rapidly but rigorously locating all solutions to nonlinear systems or global optimization problems, involves bounding the solution sets to systems of equations with wide interval c ..."
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Finding bounding sets to solutions to systems of algebraic equations with uncertainties in the coefficients, as well as rapidly but rigorously locating all solutions to nonlinear systems or global optimization problems, involves bounding the solution sets to systems of equations with wide interval coefficients. In many cases, singular systems are admitted within the intervals of uncertainty of the coefficients, leading to unbounded solution sets with more than one disconnected component. This, combined with the fact that computing exact bounds on the solution set is NPhard, limits the range of techniques available for bounding the solution sets for such systems. However, the componentwise nature and other properties make the interval Gaussâ€“Seidel method suited to computing meaningful bounds in a predictable amount of computing time. For this reason, we focus on the interval Gaussâ€“Seidel method. In particular, we study and compare various preconditioning techniques we have developed over the years but not fully investigated, comparing the results. Based on a study of the preconditioners in detail on some simple, speciallydesigned small systems, we propose two heuristic algorithms, then study the behavior of the preconditioners on some larger, randomly generated systems, as well as a small selection of systems from the Matrix Market collection.