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25
Interval Analysis on Directed Acyclic Graphs for Global Optimization
 J. Global Optimization
, 2004
"... A directed acyclic graph (DAG) representation of optimization problems represents each variable, each operation, and each constraint in the problem formulation by a node of the DAG, with edges representing the ow of the computation. ..."
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Cited by 40 (8 self)
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A directed acyclic graph (DAG) representation of optimization problems represents each variable, each operation, and each constraint in the problem formulation by a node of the DAG, with edges representing the ow of the computation.
Globally convergent autocalibration using interval analysis
 PAMI
, 2004
"... Università degli studi di Verona We address the problem of autocalibration of a moving camera with unknown constant intrinsic parameters. Existing autocalibration techniques use numerical optimization algorithms whose convergence to the correct result cannot be guaranteed, in general. To address thi ..."
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Cited by 25 (8 self)
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Università degli studi di Verona We address the problem of autocalibration of a moving camera with unknown constant intrinsic parameters. Existing autocalibration techniques use numerical optimization algorithms whose convergence to the correct result cannot be guaranteed, in general. To address this problem, we have developed a method where an interval branchandbound method is employed for numerical minimization. Thanks to the properties of Interval Analysis this method converges to the global solution with mathematical certainty and arbitrary accuracy, and the only input information it requires from the user are a set of point correspondences and a search box. The cost function is based on the HuangFaugeras constraint of the fundamental matrix, and a closed form expression for its Jacobian and Hessian matrices is derived through matrix differential calculus. A recently proposed interval extension based on Bernstein polynomial forms has been investigated to speed up the search for the solution. Finally, experimental results on synthetic and real images are presented.
On Taylor model based integration of ODEs
 SIAM J. Numer. Anal
"... Abstract. Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a ..."
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Cited by 13 (0 self)
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Abstract. Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a solution. Overestimation, however, is a potential drawback of verified methods. For some problems, the computed error bounds become overly pessimistic, or the integration even breaks down. The dependency problem and the wrapping effect are particular sources of overestimations in interval computations. Berz and his coworkers have developed Taylor model methods, which extend interval arithmetic with symbolic computations. The latter is an effective tool for reducing both the dependency problem and the wrapping effect. By construction, Taylor model methods appear particularly suitable for integrating nonlinear ODEs. We analyze Taylor model based integration of ODEs and compare Taylor model methods with traditional enclosure methods for IVPs for ODEs. AMS subject classifications. 65G40, 65L05, 65L70. Key words. Taylor model methods, verified integration, ODEs, IVPs.
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.
Deterministic global optimization for parameter estimation of dynamic systems
 Industrial and Engineering Chemistry Research
, 2006
"... A method is presented for deterministic global optimization in the estimation of parameters in models of dynamic systems. The method can be implemented as an ɛglobal algorithm, or, by use of the intervalNewton method, as an exact algorithm. In the latter case, the method provides a mathematically ..."
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Cited by 6 (4 self)
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A method is presented for deterministic global optimization in the estimation of parameters in models of dynamic systems. The method can be implemented as an ɛglobal algorithm, or, by use of the intervalNewton method, as an exact algorithm. In the latter case, the method provides a mathematically guaranteed and computationally validated global optimum in the goodness of fit function. A key feature of the method is the use of a new validated solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of dynamic systems with intervalvalued parameters, as well as on the first and secondorder sensitivities of the state variables with respect to the parameters. The computational efficiency of the method is demonstrated using several benchmark problems.
IntervalType and Affine ArithmeticType Techniques for Handling Uncertainty in Expert Systems
 Journal of Computational and Applied Mathematics
"... Expert knowledge consists of statements Sj (facts and rules). The expert’s degree of confidence in each statement Sj can be described as a (subjective) probability (some probabilities are known to be independent). Examples: if we are interested in oil, we should look at seismic data (confidence 90%) ..."
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Cited by 5 (5 self)
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Expert knowledge consists of statements Sj (facts and rules). The expert’s degree of confidence in each statement Sj can be described as a (subjective) probability (some probabilities are known to be independent). Examples: if we are interested in oil, we should look at seismic data (confidence 90%); a bank A trusts a client B, so if we trust A, we should trust B too (confidence 99%). If a query Q is deducible from facts and rules, what is our confidence p(Q) in Q? We can describe Q as a propositional formula F in terms of Sj; computing p(Q) exactly is NPhard, so heuristics are needed. Traditionally, expert systems use technique similar to straightforward interval computations: we parse F and replace each computation step with corresponding probability operation. Problem: at each step, we ignore 1 the dependence between the intermediate results Fj; hence intervals are too wide. Example: the estimate for P (A ∨ ¬A) is not 1. Solution: similarly to affine arithmetic, besides P (Fj), we also compute P (Fj & Fi) (or P (Fj1 &... & Fj k)), and on each step, use all combinations of l such probabilities to get new estimates. Results: e.g., P (A ∨ ¬A) is estimated as 1. 1 Formulation of the Problem Expert knowledge usually consists of statements Sj: facts and rules. The main objective is, given a query Q, to check whether Q follows from the expert knowledge. For example, in the knowledge base
Guaranteed state and parameter estimation for nonlinear continuoustime systems with boundederror measurements
 Industrial and Engineering Chemistry Research
, 2007
"... A strategy for state and parameter estimation in nonlinear, continuoustime systems is presented. The method provides guaranteed enclosures of all state and parameter values that are consistent with boundederror output measurements. Key features of the method are the use of a new validated solver f ..."
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Cited by 4 (3 self)
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A strategy for state and parameter estimation in nonlinear, continuoustime systems is presented. The method provides guaranteed enclosures of all state and parameter values that are consistent with boundederror output measurements. Key features of the method are the use of a new validated solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of nonlinear dynamic systems with intervalvalued parameters and initial states, and the use of a constraint propagation strategy on the Taylor models used to represent the solutions of the dynamic system. Numerical experiments demonstrate the use and computational efficiency of the method.
Interval Computations and IntervalRelated Statistical Techniques: Tools for Estimating Uncertainty of the Results of Data Processing and Indirect Measurements
"... In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on ..."
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Cited by 4 (1 self)
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In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on [−∆, ∆], and to use the corresponding statistical techniques. In some situations, however, this approach underestimates the error of indirect measurements. It is therefore desirable to directly process this interval uncertainty. Such “interval computations” methods have been developed since the 1950s. In this chapter, we provide a brief overview of related algorithms, results, and remaining open problems.
Towards More Efficient Interval Analysis: Corner Forms and a Remainder Interval Newton Method
, 2005
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Theoretical Explanation of Bernstein Polynomials’ Efficiency: They Are Optimal Combination of Optimal EndpointRelated Functions
, 2011
"... In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval [x, x] as linear combinations of Bernstein polynomials (x − x) k · (x − x) n−k. In this paper, we provide a theoretical explanation for this empirical success: namely, we show t ..."
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Cited by 1 (1 self)
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In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval [x, x] as linear combinations of Bernstein polynomials (x − x) k · (x − x) n−k. In this paper, we provide a theoretical explanation for this empirical success: namely, we show that under reasonable optimality criteria, Bernstein polynomials can be uniquely determined from the requirement that they are optimal combinations of optimal polynomials corresponding to the interval’s endpoints.