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51
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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Cited by 10 (6 self)
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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.
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 object is desc ..."
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Cited by 9 (7 self)
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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 9 (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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Cited by 9 (1 self)
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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.
Fast Ray Tracing of Arbitrary Implicit Surfaces with Interval and Affine Arithmetic
"... Existing techniques for rendering arbitraryform implicit surfaces are limited, either in performance, correctness or flexibility. Ray tracing algorithms employing interval arithmetic (IA) or affine arithmetic (AA) for rootfinding are robust and general in the class of surfaces they support, but tr ..."
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Cited by 9 (4 self)
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Existing techniques for rendering arbitraryform implicit surfaces are limited, either in performance, correctness or flexibility. Ray tracing algorithms employing interval arithmetic (IA) or affine arithmetic (AA) for rootfinding are robust and general in the class of surfaces they support, but traditionally slow. Nonetheless, implemented efficiently using a stackdriven iterative algorithm and SIMD vector instructions, these methods can achieve interactive performance for common algebraic surfaces on the CPU. A similar algorithm can also be implemented stacklessly, allowing for efficient ray tracing on the GPU. This paper presents these algorithms, as well as an inclusionpreserving reduced affine arithmetic (RAA) for faster raysurface intersection. Shader metaprogramming allows for immediate and automatic generation of symbolic expressions and their interval or affine extensions. Moreover, we are able to render even complex forms robustly, in realtime at high resolution.
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 8 (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.
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.
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 6 (0 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.
Combining Multiple Inclusion Representations in Numerical Constraint Propagation
 Publications Three Representative Papers
"... Abstract — This paper proposes a novel generic scheme enabling the combination of multiple inclusion representations to propagate numerical constraints. The scheme allows bringing into the constraint propagation framework the strength of inclusion techniques coming from different areas such as inter ..."
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Cited by 4 (4 self)
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Abstract — This paper proposes a novel generic scheme enabling the combination of multiple inclusion representations to propagate numerical constraints. The scheme allows bringing into the constraint propagation framework the strength of inclusion techniques coming from different areas such as interval arithmetic, affine arithmetic and mathematical programming. The scheme is based on the DAG representation of the constraint system. This enables devising finegrained combination strategies involving any factorable constraint system. The paper presents several possible combination strategies for creating practical instances of the generic scheme. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic and linear programming illustrate the flexibility and efficiency of the approach. I.
Ray casting implicit procedural noises with reduced affine arithmetic
 Dept. of Comp. Science, The University of Sheffield
, 2005
"... A method for ray casting implicit surfaces, defined with procedural noise models, is presented. The method is robust in that it is able to guarantee correct intersections at all image pixels and for all types of implicit surfaces. This robustness comes from the use of an affine arithmetic representa ..."
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Cited by 3 (3 self)
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A method for ray casting implicit surfaces, defined with procedural noise models, is presented. The method is robust in that it is able to guarantee correct intersections at all image pixels and for all types of implicit surfaces. This robustness comes from the use of an affine arithmetic representation for the quantity that expresses the variation of the implicit function along a ray. Affine arithmetic provides a bounding interval estimate which is tighter than the interval estimates returned by conventional interval arithmetic. Our ray casting method is also efficient due to a proposed modification in the data structure used to hold affine arithmetic quantities. This modified data structure ultimately leads to a reduced affine arithmetic model. We show that such a reduced affine arithmetic model is able to retain all the tight estimation capabilities of standard affine arithmetic, in the context of ray casting implicit procedural noises, while being faster to compute and more efficient to store. We also show that, without this reduced model, affine arithmetic would not have any advantage over the more conventional interval arithmetic for ray casting the class of implicit procedural surfaces that we are interested in visualizing.