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Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation
, 1999
"... Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated (also called interval) methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique s ..."
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Cited by 32 (8 self)
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Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated (also called interval) methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. To date, the only effective approach for computing guaranteed enclosures of the solution of an IVP for an ODE has been interval methods based on Taylor series. This thesis derives a new approach, an interval HermiteObreschkoff (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same order and stepsize, our IHO scheme has a smaller truncation error and better...
Analytic Constraint Solving and Interval Arithmetic
 In POPL’00 ACM SIGPLANSIGACT Symposium on Principles of Programming Languages
, 1999
"... In this paper we describe the syntax, semantics, and implementation of the constraint logic programming language CLP(F) and we prove that the implementation is sound. This language is an example of a new approach to scientific programming which we call analytic constraint logic programming (ACLP). T ..."
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Cited by 18 (7 self)
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In this paper we describe the syntax, semantics, and implementation of the constraint logic programming language CLP(F) and we prove that the implementation is sound. This language is an example of a new approach to scientific programming which we call analytic constraint logic programming (ACLP). The idea behind ACLP is that it provides an intervalbased constraint language in which higher order mathematical objects (e.g. ODEs, PDEs, function transforms, etc.) can be used to define scientifically interesting constraints on real numbers. All real numbers are associated to intervals (initially [\Gamma1; 1]), and the goal of an ACLP constraint solver is to narrow those intervals without removing any solutions to the specified ACLP constraints. After describing the syntax and semantics of the constraint language for CLP(F) and giving several examples, we show how to convert these analytic constraints into second order interval arithmetic constraints. We then present an algorithm for solvi...
editors. Applications of Interval Computations
 Applied Optimization
, 1996
"... compute derivatives of interval functions fast ..."
An Interval HermiteObreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation
 Developments in Reliable Computing
, 1998
"... To date, the only effective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary differential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval HermiteObreschkoff (IHO) method, for co ..."
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Cited by 14 (3 self)
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To date, the only effective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary differential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval HermiteObreschkoff (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same stepsize and order, our IHO scheme has a smaller truncation error, better stability, and requires fewer Taylor coefficients and highorder Jacobians. The stability properties of the ITS and IHO methods are investigated. We show as an important byproduct of this analysis that the stability of an interval method is determined not only by the stability function of the underlying formula, as in a standard method for an IVP for an ODE, but also by the associated formula for the truncation error.
Applications of interval computations to earthquakeresistant engineering: how to compute derivatives of interval functions fast
 Reliable Computing
, 1995
"... Abstract. One of the main sources of destruction during earthquake is resonance. Therefore, the following idea has been proposed. We design special control linkages between floors that are normally unattached to the building but can be attached if necessary. They are so designed that adding them cha ..."
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Cited by 3 (2 self)
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Abstract. One of the main sources of destruction during earthquake is resonance. Therefore, the following idea has been proposed. We design special control linkages between floors that are normally unattached to the building but can be attached if necessary. They are so designed that adding them changes the building’s characteristic frequency. We continuously monitor displacements within the structure, and when they exceed specified limits, the linkages are engaged in a way to control structural motion. This idea can also be applied to avoid vibrational destruction of large aerospace structures. To check for a resonance, one must know not only the displacements x(t), but also the rates ˙x(t) with which they change. So, we must estimate the velocity from the approximately known function values. Since we consider a small time interval, the function x(t) can be well approximated by its first Taylor expansion terms. Therefore, we consider two cases: when x(t) is linear, and when it is quadratic. In mathematical terms, for some n, we know n + 1 real numbers t0, t1,..., tn (called moments of time), n numbers x1,..., xn (called measurement results), and n numbers εi
From Interval Analysis to Taylor Models An Overview
"... Interval arithmetic has been widely used in enclosure methods for almost 40 years. Today, it is a well established tool for the calculation of rigorous error bounds for many problems in numerical analysis. Despite its overall success, interval arithmetic suffers from two drawbacks: the dependency pr ..."
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Cited by 2 (0 self)
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Interval arithmetic has been widely used in enclosure methods for almost 40 years. Today, it is a well established tool for the calculation of rigorous error bounds for many problems in numerical analysis. Despite its overall success, interval arithmetic suffers from two drawbacks: the dependency problem and the socalled wrapping effect, which both may overestimate the true error of some computation. To reduce overestimation, Taylor models have been developed as a symbiosis of a computer algebra method and interval arithmetic by M. Berz and his group since the 1990s. Software implementations of Taylor models have been applied to a variety of problems, such as global optimization problems, validated multidimensional integration, or the solution of ODEs and DAEs. The validated solution of ODEs is used for exemplifying the distinction of interval methods and Taylor model methods. 1 Interval Computations
Interval Computations as an Important Part of Granular Computing: An Introduction
"... This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing. ..."
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This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing.
On the Blunting Method in the Verified Integration of ODEs
, 2009
"... Verified methods for the integration of initial value problems (IVPs) for ODEs aim at computing guaranteed error bounds for the flow of an ODE and maintaining a low level of overestimation at the same time. This paper is concerned with one of the sources of overestimation: a matrixvector product des ..."
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Verified methods for the integration of initial value problems (IVPs) for ODEs aim at computing guaranteed error bounds for the flow of an ODE and maintaining a low level of overestimation at the same time. This paper is concerned with one of the sources of overestimation: a matrixvector product describing a parallelepiped in the phase space. We analyze the socalled blunting method developed by Berz and Makino, which consists of a special choice of the matrix in this product. For the linear model problem u ′ = Au, u(0) = u0 ∈ u0, where u ∈ R m, A ∈ R m×m, m ≥ 2, and u0 is a given interval vector, we compare the convergence behavior of the blunting method with that of the wellknown QR interval method. For both methods, the amount of overestimation of the flow of the initial set depends on the spectral radius of some welldefined matrix. We show that under certain conditions, the spectral radii of the matrices that describe the excess propagation in the QR method and in the blunting method, respectively, have the same limits, so that the excess propagation in both methods is similar.
Challenges in Constraintbased Analysis of Hybrid Systems ⋆
"... Abstract. In the analysis of hybrid discretecontinuous systems, rich arithmetic constraint formulae with complex Boolean structure arise naturally. The iSAT algorithm, a solver for such formulae, is aimed at bounded model checking of hybrid systems. In this paper, we identify challenges emerging fr ..."
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Abstract. In the analysis of hybrid discretecontinuous systems, rich arithmetic constraint formulae with complex Boolean structure arise naturally. The iSAT algorithm, a solver for such formulae, is aimed at bounded model checking of hybrid systems. In this paper, we identify challenges emerging from planned and ongoing work to enhance the iSAT algorithm: First, we propose an extension of iSAT to directly handle ordinary differential equations as constraints. Second, we outline the recently introduced generalization of the iSAT algorithm to deal with probabilistic hybrid systems and some open research issues in that context. Third, we present ideas on how to move from bounded to unbounded model checking by using the concept of interpolation. Finally, we discuss the adaption of some parallelization techniques to the iSAT case, which will hopefully lead to performance gains in the future. By presenting these open research questions, this paper aims at fostering discussions on these extensions of constraint solving.