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25
Solving Polynomial Systems Using a Branch and Prune Approach
 SIAM Journal on Numerical Analysis
, 1997
"... This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in ..."
Abstract

Cited by 101 (7 self)
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This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called boxconsistency, which approximates the notion of arcconsistency wellknown in artificial intelligence. Boxconsistency is parametrized by an interval extension of the constraint and can be instantiated to produce the HansenSegupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with stateoftheart continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.
A Constraint Satisfaction Approach to a Circuit Design Problem
, 1998
"... A classical circuitdesign problem from Ebers and Moll [6] features a system of nine nonlinear equations in nine variables that is very challenging both for local and global methods. This system was solved globally using an interval method by Ratschek and Rokne [23] in the box [0; 10] 9 . Their ..."
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Cited by 21 (1 self)
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A classical circuitdesign problem from Ebers and Moll [6] features a system of nine nonlinear equations in nine variables that is very challenging both for local and global methods. This system was solved globally using an interval method by Ratschek and Rokne [23] in the box [0; 10] 9 . Their algorithm had enormous costs (i.e., over 14 months using a network of 30 Sun Sparc1 workstations) but they state that "at this time, we know no other method which has been applied to this circuit design problem and which has led to the same guaranteed result of locating exactly one solution in this huge domain, completed with a reliable error estimate." The present paper gives a novel branchandprune algorithm that obtains a unique safe box for the above system within reasonable computation times. The algorithm combines traditional interval techniques with an adaptation of discrete constraintsatisfaction techniques to continuous problems. Of particular interest is the simplicity o...
Genetic Object Recognition Using Combinations of Views
 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 2002
"... We investigate the application of genetic algorithms (GAs) for recognizing real twodimensional (2D) or threedimensional (3D) objects from 2D intensity images, assuming that the viewpoint is arbitrary. Our approach is modelbased (i.e., we assume a predefined set of models), while our recognitio ..."
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Cited by 18 (5 self)
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We investigate the application of genetic algorithms (GAs) for recognizing real twodimensional (2D) or threedimensional (3D) objects from 2D intensity images, assuming that the viewpoint is arbitrary. Our approach is modelbased (i.e., we assume a predefined set of models), while our recognition strategy lies on the recently proposed theory of algebraic functions of views. According to this theory, the variety of 2D views depicting an object can be expressed as a combination of a small number of 2D views of the object. This implies a simple and powerful strategy for object recognition: novel 2D views of an object (2D or 3D) can be recognized by simply matching them to combinations of known 2D views of the object. In other words, objects in a scene are recognized by "predicting" their appearance through the combination of known views of the objects. This is an important idea, which is also supported by psychophysical findings indicating that the human visual system works in a similar way. The main difficulty in implementing this idea is determining the parameters of the combination of views. This problem can be solved either in the space of feature matches among the views ("image space") or the space of parameters ("transformation space"). In general, both of these spaces are very large, making the search very time consuming. In this paper, we propose using GAs to search these spaces efficiently. To improve the efficiency of genetic search in the transformation space, we use singular value decomposition and interval arithmetic to restrict genetic search in the most feasible regions of the transformation space. The effectiveness of the GA approaches is shown on a set of increasingly complex real scenes where exact and nearexact matches are found reliably and q...
2007, â€˜Interval Finite Element as a Basis for Generalized Models of Uncertainty in Engineering Mechanics
 Journal of Reliable Computing
"... Latest scientific and engineering advances have started to recognize the need of defining multiple types of uncertainty. Probabilistic modeling cannot handle situations with incomplete or little information on which to evaluate a probability, or when that information is nonspecific, ambiguous, or co ..."
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Cited by 13 (2 self)
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Latest scientific and engineering advances have started to recognize the need of defining multiple types of uncertainty. Probabilistic modeling cannot handle situations with incomplete or little information on which to evaluate a probability, or when that information is nonspecific, ambiguous, or conflicting [1]. Many generalized models of uncertainty have been developed to treat such situations. Among them, there are five major frameworks that use intervalbased representation of uncertainty, namely: imprecise probabilities, possibility theory, DempsterShafer theory of evidence, Fuzzy set theory, and convex set modeling. Regardless what model is adopted, the proper interval solution will represents the first requirement for any further rigorous formulation. In this work an interval technique is applied to Finite Element Methods. Finite Element Methods (FEM) are an essential and frequently indispensable part of engineering analysis and design. An Interval Finite Element Method (IFEM) is presented that handles stiffness and load uncertainty in the linear static problems of mechanics. Uncertain parameters are introduced in the form of unknown but bounded quantities (intervals). To avoid overestimation, the new formulation is based on an elementbyelement (EBE) technique. Element matrices are
Bounds for linear recurrences with restricted coefficients
 Journal of Inequalities in Pure and Applied Mathematics 4, 2, Article
"... Abstract. This paper provides bounds for secondorder linear recurrences with restricted coefficients. It is determined that whenever the coefficients of the associated monic equation are less than the constant {l/3) 1 1 3, all solutions tend to zero at an exponential rate. This constant is optimal. ..."
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Cited by 13 (6 self)
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Abstract. This paper provides bounds for secondorder linear recurrences with restricted coefficients. It is determined that whenever the coefficients of the associated monic equation are less than the constant {l/3) 1 1 3, all solutions tend to zero at an exponential rate. This constant is optimal. Explicit inequalities are also provided, and some residue class structure is revealed. 1.
Interval Linear Constraint Solving Using the Preconditioned Interval GaussSeidel Method
 IN PROCEEDINGS OF THE TWELFTH INTERNATIONAL CONFERENCE ON LOGIC PROGRAMMING, LOGIC PROGRAMMING
, 1994
"... We propose the use of the preconditioned interval GaussSeidel method as the backbone of an efficient linear equality solver in a CLP(Interval) language. The method, as originally designed, works only on linear systems with square coefficient matrices. Even imposing such a restriction, a naive incor ..."
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Cited by 12 (1 self)
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We propose the use of the preconditioned interval GaussSeidel method as the backbone of an efficient linear equality solver in a CLP(Interval) language. The method, as originally designed, works only on linear systems with square coefficient matrices. Even imposing such a restriction, a naive incorporation of the traditional preconditioning algorithm in a CLP language incurs a high worstcase time complexity of O(n^4), where n is the number of variables in the linear system. In this paper, we generalize the algorithm for general linear systems with m constraints and n variables, and give a novel incremental adaptation of preconditioning of O(n 2 (n + m)) complexity. The efficiency of the incremental preconditioned interval GaussSeidel method is demonstrated using largescale linear systems.
Reliable Computation of Phase Stability and Equilibrium from the SAFT Equation of State
 Industrial & Engineering Chemistry Research
, 2001
"... In recent years, molecularlybased equations of state, as typified by the SAFT (statistical associating fluid theory) approach, have become increasingly popular tools for the modeling of phase behavior. However, whether using this, or even much simpler models, the reliable calculation of phase behav ..."
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Cited by 9 (6 self)
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In recent years, molecularlybased equations of state, as typified by the SAFT (statistical associating fluid theory) approach, have become increasingly popular tools for the modeling of phase behavior. However, whether using this, or even much simpler models, the reliable calculation of phase behavior from a given model can be a very challenging computational problem. A new methodology is described that is the first completely reliable technique for computing phase stability and equilibrium from the SAFT model. The method is based on interval analysis, in particular an interval Newton/generalized bisection algorithm, which provides a mathematical and computational guarantee of reliability, and is demonstrated using nonassociating, selfassociating, and crossassociating systems. New techniques are presented that can also be exploited when conventional pointvalued solution methods are used. These include the use of a volumebased problem formulation, in which the core thermodynamic function for phase equilibrium at constant temperature and pressure is the Helmholtz energy, and an approach for dealing with the internal iteration needed when there are association effects. This provides for direct, as opposed to iterative, determination of the derivatives of the internal variables. 1
Indexing Based on Algebraic Functions of Views
 COMPUTER VISION AND IMAGE UNDERSTANDING
, 1998
"... this paper, we propose the use of algebraic functions of views for indexingbased object recognition. During indexing, we consider groups of model points and we represent all the views (i.e., images) that they can produce in a hash table. The images that a group of model points can produce are compu ..."
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Cited by 6 (5 self)
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this paper, we propose the use of algebraic functions of views for indexingbased object recognition. During indexing, we consider groups of model points and we represent all the views (i.e., images) that they can produce in a hash table. The images that a group of model points can produce are computed by combining a small number of reference views which contain the group using algebraic functions of views. Fundamental to this procedure is a methodology, based on Singular Value Decomposition and Interval Arithmetic, for estimating the allowable ranges of values that the parameters of algebraic functions can assume. During recognition, scene groups are used to retrieve from the hash table the most feasible model groups that might have produced the scene groups. The use of algebraic functions of views for indexingbased recognition offers a number of advantages. First of all, the hash table can be built using a small number of reference views per object. This is in contrast to current approaches which build the hash table using either a large number of reference views or 3D models. Most importantly, recognition does not rely on the similarity between reference views and novel views; all that is required for the novel views is to contain common groups of points with a small number of reference views. Second, verification becomes simpler. This is because candidate models can now be backprojected onto the scene by applying a linear transformationona small number of reference views of the candidate model. Finally, the proposed approach is more general and extendible. This is because algebraic functions of views have been shown to exist over a wide range of transformations and projections. The recognition performance of the proposed approach is demonstrated using both artific...
Solving interval constraints by linearization in computeraided design. Reliable Computing
, 2006
"... Abstract. Current parametric CAD systems require geometric parameters to have fixed values. Specifying fixed parameter values implicitly adds rigid constraints on the geometry, which have the potential to introduce conflicts during the design process. This paper presents a soft constraint representa ..."
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Cited by 6 (5 self)
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Abstract. Current parametric CAD systems require geometric parameters to have fixed values. Specifying fixed parameter values implicitly adds rigid constraints on the geometry, which have the potential to introduce conflicts during the design process. This paper presents a soft constraint representation scheme based on nominal interval. Interval geometric parameters capture inexactness of conceptual and embodiment design, uncertainty in detail design, as well as boundary information for design optimization. To accommodate underconstrained and overconstrained design problems, a doubleloop GaussSeidel method is developed to solve linear constraints. A symbolic preconditioning procedure transforms nonlinear equations to separable form. Inequalities are also transformed and integrated with equalities. Nonlinear constraints can be bounded by piecewise linear enclosures and solved by linear methods iteratively. A sensitivity analysis method that differentiates active and inactive constraints is presented for design refinement. 1.
Solving Polynomial Systems Using a Branch and Prune Approach
 SIAM Journal on Numerical Analysis
, 1997
"... This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in ..."
Abstract

Cited by 2 (0 self)
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This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called boxconsistency, which approximates the notion of arcconsistency wellknown in artificial intelligence. Boxconsistency is parametrized by an interval extension of the constraint and can be instantiated to produce HansenSegupta narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with stateoftheart continuatio...