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Contractor Programming
 Artificial Intelligence
"... Abstract. This paper describes a solver programming method, called contractor programming, that copes with two issues related to constraint processing over the reals. First, continuous constraints involve an inevitable step of solver design. Existing softwares provide an insufficient answer by restr ..."
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Abstract. This paper describes a solver programming method, called contractor programming, that copes with two issues related to constraint processing over the reals. First, continuous constraints involve an inevitable step of solver design. Existing softwares provide an insufficient answer by restricting users to choose among a list of fixed strategies. Our first contribution is to give more freedom in solver design by introducing programming concepts where only configuration parameters were previously available. Programming consists in applying operators (intersection, composition, etc.) on algorithms called contractors that are somehow similar to propagators. Second, many problems with real variables cannot be cast as the search for vectors simultaneously satisfying the set of constraints, but a large variety of different outputs may be demanded from a set of constraints (e.g., a paving with boxes inside and outside of the solution set). These outputs can actually be viewed as the result of different contractors working concurrently on the same search space, with a bisection procedure intervening in case of deadlock. Such algorithms (which are not strictly speaking solvers) will be made easy to build thanks to a new branch & prune system, called paver. Thus, this paper gives a way to deal harmoniously with a larger set of problems while giving a fine control on the solving mechanisms. The contractor formalism and the paver system are the two contributions. The approach is motivated and justified through different cases of study. An implementation of this framework named Quimper is also presented. 1
Motivations for an arbitrary precision interval arithmetic and the MPFI library
 Reliable Computing
, 2002
"... Nowadays, computations involve more and more operations and consequently errors. The limits of applicability of some numerical algorithms are now reached: for instance the theoretical stability of a dense matrix factorization (LU or QR) is ensured under the assumption that n 3 u < 1, where n is t ..."
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Cited by 39 (6 self)
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Nowadays, computations involve more and more operations and consequently errors. The limits of applicability of some numerical algorithms are now reached: for instance the theoretical stability of a dense matrix factorization (LU or QR) is ensured under the assumption that n 3 u < 1, where n is the dimension of the matrix and u = 1 + − 1, with 1 + the smallest floatingpoint larger than 1; this means that n must be less than 200,000, which is almost reached by modern simulations. The numerical quality of solvers is now an issue, and not only their mathematical quality. Let us cite studies performed by the CEA (French Nuclear Agency) on the simulation of nuclear plant accidents and also softwares controlling and possibly correcting numerical programs, such as Cadna [10] or Cena [20]. Another approach consists in computing with certified enclosures, namely interval arithmetic [21, 2, 18]. The fundamental principle of this arithmetic consists in replacing every number by an interval enclosing it. For instance, π cannot be exactly represented using a binary or decimal arithmetic, but it
Solving the Forward Kinematics of a GoughType Parallel Manipulator with Interval Analysis
, 2004
"... We consider in this paper a Goughtype parallel robot and we present an efficient algorithm based on interval analysis that allows us to solve the forward kinematics, i.e., to determine all the possible poses of the platform for given joint coordinates. This algorithm is numerically robust as numeri ..."
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We consider in this paper a Goughtype parallel robot and we present an efficient algorithm based on interval analysis that allows us to solve the forward kinematics, i.e., to determine all the possible poses of the platform for given joint coordinates. This algorithm is numerically robust as numerical roundoff errors are taken into account; the provided solutions are either exact in the sense that it will be possible to refine them up to an arbitrary accuracy or they are flagged only as a "possible" solution as either the numerical accuracy of the computation does not allow us to guarantee them or the robot is in a singular configuration. It allows us to take into account physical and technological constraints on the robot (for example, limited motion of the passive joints). Another advantage is that, assuming realistic constraints on the velocity of the robot, it is competitive in term of computation time with a realtime algorithm such as the Newton scheme, while being safer.
Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty
 11733, SAND20070939. hal00839639, version 1  28 Jun 2013
"... Sandia is a multiprogram laboratory operated by Sandia Corporation, ..."
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Sandia is a multiprogram laboratory operated by Sandia Corporation,
Reachability Analysis of Nonlinear Systems with Uncertain Parameters using Conservative Linearization
"... Abstract — Given an initial set of a nonlinear system with uncertain parameters and inputs, the set of states that can possibly be reached is computed. The approach is based on local linearizations of the nonlinear system, while linearization errors are considered by Lagrange remainders. These error ..."
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Abstract — Given an initial set of a nonlinear system with uncertain parameters and inputs, the set of states that can possibly be reached is computed. The approach is based on local linearizations of the nonlinear system, while linearization errors are considered by Lagrange remainders. These errors are added as uncertain inputs, such that the reachable set of the locally linearized system encloses the one of the original system. The linearization error is controlled by splitting of reachable sets. Reachable sets are represented by zonotopes, allowing an efficient computation in relatively highdimensional space. I.
Validated solutions of initial value problems for parametric ODEs
 Appl Num Math
"... Accepted for publication in Applied Numerical Mathematics In initial value problems for ODEs with intervalvalued parameters and/or initial values, it is desirable in many applications to be able to determine a validated enclosure of all possible solutions to the ODE system. Much work has been done ..."
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Accepted for publication in Applied Numerical Mathematics In initial value problems for ODEs with intervalvalued parameters and/or initial values, it is desirable in many applications to be able to determine a validated enclosure of all possible solutions to the ODE system. Much work has been done for the case in which initial values are given by intervals, and there are available software packages that deal with this case. However, less work has been done on the case in which parameters are given by intervals. We describe here a new method for obtaining validated solutions of initial value problems for ODEs with intervalvalued parameters. The method also accounts for intervalvalued initial values. The effectiveness of the method is demonstrated using several numerical examples involving parametric uncertainties.
Granular Computing: An Emerging Paradigm
, 2001
"... We provide an overview of Granular Computing a rapidly growing area of information processing aimed at the construction of intelligent systems. We highlight the main features of Granular Computing, elaborate on the underlying formalisms of information granulation and discuss ways of their developme ..."
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Cited by 28 (0 self)
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We provide an overview of Granular Computing a rapidly growing area of information processing aimed at the construction of intelligent systems. We highlight the main features of Granular Computing, elaborate on the underlying formalisms of information granulation and discuss ways of their development. We also discuss the concept of granular modeling and present the issues of communication between formal frameworks of Granular Computing. © 2007 World Academic Press, UK. All rights reserved.
Guaranteed proofs using interval arithmetic
 Proceedings of the 17th Symposium on Computer Arithmetic, Cape Cod
, 2005
"... This paper presents a set of tools for mechanical reasoning of numerical bounds using interval arithmetic. The tools implement two techniques for reducing decorrelation: interval splitting and Taylor’s series expansions. Although the tools are designed for the proof assistant system PVS, expertise o ..."
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This paper presents a set of tools for mechanical reasoning of numerical bounds using interval arithmetic. The tools implement two techniques for reducing decorrelation: interval splitting and Taylor’s series expansions. Although the tools are designed for the proof assistant system PVS, expertise on PVS is not required. The ultimate goal of the tools is to provide guaranteed proofs of numerical properties with a minimal humantheorem prover interaction. 1
Novel Approaches to Numerical Software with Result Verification
 NUMERICAL SOFTWARE WITH RESULT VERIFICATION, INTERNATIONAL DAGSTUHL SEMINAR, DAGSTUHL
, 2003
"... Traditional design of numerical software with result verification is based on the assumption that we know the algorithm ¦¨§� © ©���� £��������� � that transforms input © ©�� into �� � £��������� � ©���� the output, and we £��������� � know the intervals of possible values of the inputs. Many real ..."
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Cited by 27 (19 self)
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Traditional design of numerical software with result verification is based on the assumption that we know the algorithm ¦¨§� © ©���� £��������� � that transforms input © ©�� into �� � £��������� � ©���� the output, and we £��������� � know the intervals of possible values of the inputs. Many reallife problems go beyond this paradigm. In some cases, we do not have an algorithm ¦, we only know some relation (constraints) between ©� � and. In other cases, in addition to knowing the intervals, we may know some relations between; we may have some information about the probabilities of different values of © � , and we may know the exact values of some of the inputs (e.g., we may know that © £ ���¨�� �). In this paper, we describe the approaches for solving these reallife problems. In Section 2, we describe interval consistency techniques related to handling constraints; in Section 3, we describe techniques that take probabilistic information into consideration, and in Section 4, we overview techniques for processing exact real numbers.
3D Deformable Image Registration: A Topology Preservation Scheme Based on Hierarchical Deformation Models and Interval Analysis Optimization
 IEEE Transactions on Image Processing
, 2005
"... c ○ 2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other ..."
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Cited by 26 (2 self)
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c ○ 2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.