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82
An Introduction to Affine Arithmetic
, 2003
"... Affine arithmetic (AA) is a model for selfvalidated computation which, like standard interval arithmetic (IA), produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as all internal truncation and roundoff errors. Unlike standard I ..."
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Cited by 16 (0 self)
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Affine arithmetic (AA) is a model for selfvalidated computation which, like standard interval arithmetic (IA), produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as all internal truncation and roundoff errors. Unlike standard IA, the quantity representations used by AA are firstorder approximations, whose error is generally quadratic in the width of input intervals. In many practical applications, the higher asymptotic accuracy of AA more than compensates for the increased cost of its operations.
Static analysis of the accuracy in control systems: Principles and experiments
 In FMICS, volume 4916 of Lecture
"... Abstract. Finite precision computations can severely affect the accuracy of computed solutions. We present a complete survey of a static analysis based on abstract interpretation, and a prototype implementing this analysis for C codes, for studying the propagation of rounding errors occurring at eve ..."
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Abstract. Finite precision computations can severely affect the accuracy of computed solutions. We present a complete survey of a static analysis based on abstract interpretation, and a prototype implementing this analysis for C codes, for studying the propagation of rounding errors occurring at every intermediary step in floatingpoint computations. In the first part of this paper we briefly present all the domains and techniques used in the implemented analyzer, called FLUCTUAT. We describe in the second part, the experiments made on real industrial codes, at Institut de Radioprotection et de Sûreté Nucléaire and at HispanoSuiza, respectively coming from the nuclear industry and from aeronautics industry. This paper aims at filling in the gaps between some theoretical aspects of the static analysis of floatingpoint computations that have been described in [13, 14, 21], and the necessary choices of algorithms and implementation, in accordance with practical motivations drawn from real industrial cases.
Reliable TwoDimensional Graphing Methods for Mathematical Formulae with Two Free Variables
, 2001
"... present s a series of new algorit hms for reliably graphingt wodimensional implicit equat ions and inequalit ies. A clear st andard for int erpret ingt he graphs generat ed byt wodimensional graphing soft ware is int roduced and used t o evaluat et he present ed algorit hms. The first approach pr ..."
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Cited by 15 (0 self)
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present s a series of new algorit hms for reliably graphingt wodimensional implicit equat ions and inequalit ies. A clear st andard for int erpret ingt he graphs generat ed byt wodimensional graphing soft ware is int roduced and used t o evaluat et he present ed algorit hms. The first approach present ed uses a st andard int erval arit hmet ic library. This approach is shownt o be fault y; an analysis oft he failure reveals a limit at ion of st andard int erval arit hmet ic. Subsequent algorit hms are developed in parallel wit h improvement s and ext#E sions t# t# e int erval ari t#met# c used byt he graphing algorit hms. Graphs exhibit ing a variet y of mat hemat ical and art ist ic phenomena are shownt o be graphed correct ly byt he present ed algorit hms. A brief comparison of t he final algorit hm present edt o ot her graphing algorit hms is included.
Extentions of Affine Arithmetic: Application to Unconstrained Global Optimization
 Journal of Universal Computer Science
"... Abstract: Global optimization methods in connection with interval arithmetic permit to determine an accurate enclosure of the global optimum, and of all the corresponding optimizers. One of the main features of these algorithms consists in the construction of an interval function which produces an e ..."
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Abstract: Global optimization methods in connection with interval arithmetic permit to determine an accurate enclosure of the global optimum, and of all the corresponding optimizers. One of the main features of these algorithms consists in the construction of an interval function which produces an enclosure of the range of the studied function over a box (right parallelepiped). We use here affine arithmetic in global optimization algorithms, in order to elaborate new inclusion functions. These techniques are implemented and then discussed. Three new affine and quadratic forms are introduced. On some polynomial examples, we show that these new tools often yield more efficient lower bounds (and upper bounds) compared to several wellknown classical inclusion functions. The three new methods, presented in this paper, are integrated into various Branch and Bound algorithms. This leads to improve the convergence of these algorithms by attenuating some negative effects due to the use of interval analysis and standard affine arithmetic.
Sampling Implicit Objects With PhysicallyBased Particle Systems
 Computers & Graphics
, 1996
"... . After reviewing three classical sampling methods for implicit objects, we describe a new sampling method that is not based on scanning the ambient space. In this method, samples are "randomly" generated using physicallybased particle systems. Introduction In computer graphics, an obje ..."
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. After reviewing three classical sampling methods for implicit objects, we describe a new sampling method that is not based on scanning the ambient space. In this method, samples are "randomly" generated using physicallybased particle systems. Introduction In computer graphics, an object is described either by a set of sample points or by an analytic scheme that uses mathematical equations to define its geometry and topology. Descriptions based on samples occur in areas such as medical images and terrain models. Analytical descriptions are usually found in applications of geometric modeling, such computeraided design and manufacture. When an object is described by samples, a reconstruction scheme is needed to recover its geometry and topology from the samples. This problem, called structuring, consists of providing a combinatorial structure to the samples in order to (ideally) recover the exact topology of the object and an approximation of its geometry. When the object is describe...
Bounded clustering  finding good bounds on clustered light transport
 in: Proc. Pacific Graphics '98, IEEE Computer
, 1998
"... Clustering is a very e cient technique to applynite element methods to the computation of radiosity solutions of complex scenes. Both computation time and memory consumption can be reduced dramatically by grouping the primitives of the input scene into a hierarchy of clusters and allowing for light ..."
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Cited by 10 (3 self)
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Clustering is a very e cient technique to applynite element methods to the computation of radiosity solutions of complex scenes. Both computation time and memory consumption can be reduced dramatically by grouping the primitives of the input scene into a hierarchy of clusters and allowing for light exchange between all levels of this hierarchy. However, problems can arise due to clustering, when gross approximations about a cluster's content result in unsatisfactory solutions or unnecessary computations. In the clustering approach for di use global illumination described in this paper, light exchange between two objects  patches or clusters  is bounded by using geometrical and shading information provided by every object through a uniform interface. With this uniform view of various kinds of objects, comparable and reliable error bounds on the light exchange can be computed, which then guide a standard hierarchical radiosity algorithm. 1.
Modified Affine Arithmetic Is More Accurate than Centered Interval Arithmetic or Affine Arithmetic
 Martin (Eds.), Lecture Notes in Computer Science 2768, Mathematics of Surfaces, SpringerVerlag
, 2003
"... In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a boxshaped region: (i) using interval arithmetic in centered form is always more accurate than standard a#ne arithmetic, and (ii) modified a#ne arithmetic is always more accurate than i ..."
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In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a boxshaped region: (i) using interval arithmetic in centered form is always more accurate than standard a#ne arithmetic, and (ii) modified a#ne arithmetic is always more accurate than interval arithmetic in centered form. Test results show that modified a#ne arithmetic is not only more accurate but also much faster than standard a#ne arithmetic. We thus suggest that modified a#ne arithmetic is the method of choice for evaluating algebraic functions, such as implicit surfaces, over a box.
Taylor Forms  Use and Limits
 Reliable Computing
, 2002
"... This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 19 ..."
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Cited by 8 (2 self)
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This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001, although the details given are not sufficient to check the validity of their claims.
Approximating Parametric Curves with Strip Trees using Affine Arithmetic
"... We show how to use affine arithmetic to represent a parametric curve with a strip tree. The required bounding rectangles for pieces of the curve are computed by exploiting the linear correlation information given by affine arithmetic. As an application, we show how to compute approximate distance ..."
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Cited by 7 (2 self)
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We show how to use affine arithmetic to represent a parametric curve with a strip tree. The required bounding rectangles for pieces of the curve are computed by exploiting the linear correlation information given by affine arithmetic. As an application, we show how to compute approximate distance fields for parametric curves.