Interval analysis

Vaidyanathan, R. and El-Halwagi, M. M. (1996). Global optimization of nonconvex MINLP s by interval analysis. In Global Optimization in Engineering Design, (I. E. Grossmann, ed.), pp. 175-194. Kluwer Academic Publishers, Dordrecht, The Netherlands. 

Hansen, Global Optimization Using Interval Analysis (1992)... 

Thus, interval analysis (Moore, 1979 Alefeld and Herzberger, 1983) provides the adequate support and notation formalism to express solu-... 

Moore, R., Methods and Applications of Interval Analysis. SIAM, Philadelphia, 1979. 

Conscious studies using devices for measurement of blood pressure and six chest lead ECG measurements (V2, V4, V6, V10, rV2 and rV4). ECG interval analysis is performed on the V2 lead (RR, PR, QT, QTc intervals, QRS duration). QT dispersion can also be measured. Locomotor activity can be monitored and behavior captured on video using CCTV. 

L. Jaulin, M. Kieffer, O. Didrit, and E. Walter. Applied interval analysis with examples in parameter and state estimation robust control and robotics. Springer-Verlag, Londres, 1991. 

L. Jaulin and E. Walter. Set inversion via interval analysis for nonlinear... 

M. Kieffer, L. Jaulin, and E. Walter. Guaranteed recursive nonlinear state estimation using interval analysis. In IEEE Conf. Decision and Control (CDC), pages 16-18, Florida, USA, December 1998. 

We suggest that what is needed is a bounding approach that marries the advantages of interval analysis with those of probability theory while sidestepping the limitations of both. In the following sections, we describe 2 very different approaches that do just this in different ways. 

Moore RE. 1966. Interval analysis. Englewood Cliffs (NJ) Prentice-HaU. 

An f2 value less than 50 does not necessarily indicate lack of similarity. If the sponsor is of the opinion that the differences observed related to this calculation of f2 are typical for the drug product involved in this SUP AC situation, an appropriate j ustification can be submitted, but only as part of a prior approval supplement. This justification should include additional data to support the claim of similarity, as well as supporting statistical analysis (e.g. 90% confidence interval analysis). If this justification is not found acceptable, the potential effect of the proposed change on the differences in dissolution on bioavailability should be determined. 

E. Hansen. Global optimization using interval analysis. Marcel Dekker Inc., New York, 1992. 

The completely reliable computational technique that we have developed is based on interval analysis. The interval Newton/generalized bisection technique can guarantee the identification of a global optimum of a nonlinear objective function, or can identify all solutions to a set of nonlinear equations. Since the phase equilibrium problem (i.e., particularly the phase stability problem) can be formulated in either fashion, we can guarantee the correct solution to the high-pressure flash calculation. A detailed description of the interval Newton/generalized bisection technique and its application to thermodynamic systems described by cubic equations of state can be found... 

We have applied a global optimization technique, based on interval analysis, to the high-pressure phase equilibrium problem (INTFLASH). It does not require any initial guesses and is guaranteed, both mathematically and computationally, to converge to the correct solution. The interval analysis method and its application to phase equilibria using equation-of-state... 

Hua, J. Z. Brennecke, J. F. Stadtherr, M. A. Reliable Computation of Phase Stability Using Interval Analysis Cubic Equation of State Models. Comput. Chem. Eng. 1998a, 22, 1207-1214. 

This section provides an overview of common methods for quantitative uncertainty analysis of inputs to models and the associated impact on model outputs. Furthermore, consideration is given to methods for analysis of both variability and uncertainty. In practice, commonly used methods for quantification of variability, uncertainty or both are typically based on numerical simulation methods, such as Monte Carlo simulation or Latin hypercube sampling. However, there are other techniques that can be applied to the analysis of uncertainty, some of which are non-probabilistic. Examples of these are interval analysis and fuzzy methods. The latter are briefly reviewed. Since probabilistic methods are commonly used in practice, these methods receive more detailed treatment here. The use of quantitative methods for variability and uncertainty is consistent with, or informed by, the key hallmarks of data... 

Uncertainty expressed in terms of sets of alternatives results from the nonspecificity inherent in each set. Large sets result in less specific predictions (retrodictions, prescriptions, etc.) than their smaller counterparts. One area of mathematics that deals with this kind of uncertainty is interval analysis. ... 

Uncertainties inherent to the risk assessment process can be quantitatively described using, for example, statistical distributions, fuzzy numbers, or intervals. Corresponding methods are available for propagating these kinds of uncertainties through the process of risk estimation, including Monte Carlo simulation, fuzzy arithmetic, and interval analysis. Computationally intensive methods (e.g., the bootstrap) that work directly from the data to characterize and propagate uncertainties can also be applied in ERA. Implementation of these methods for incorporating uncertainty can lead to risk estimates that are consistent with a probabilistic definition of risk. 

The algebraic equations in the chemistry operator co can be much more complicated than the equations usually used for the transport operator 0, but it is still possible to bound the error introduced by approximating co, by using interval analysis. Interval analysis is a branch of mathematics that considers how mathematical operations affect intervals (ranges) [ Tlow, Thigh] rather than single points Y. For error control, one wants to rigorously bound the error... 

Interval analysis is trickier than ordinary mathematics, since in interval analysis some arithmetic operations give outputs which are overestimates of the true range. In other words, if a function f x) takes on values between /min and /max for input values of. v in the interval [.V ow, xhigh], interval analysis may give you an output range f mJ, where/ x>/max and/or f min[Pg.35]

As a result, interval analysis tends to overestimate error bounds. However, there are clever ways to reduce this overestimation, for example by appropriately grouping terms. One of the best and most computationally efficient ways to minimize overestimation of the bounds on the approximation error is to replace terms f Y) in w(Y) by their Taylor models, e.g. the first-order Taylor model is given by Eq. (14) ... 

The vapor-liquid equilibrium was computed from the EOS model using the reliable and robust method of Hua et al 14-16) based on interval analysis. Their method can find the correct thermodynamically stable solution to the vapor-liquid equilibrium problem with mathematical and computational certainty. Additionally, the tangent plane distance method 17,18) was used to test the predicted liquid and vapor phase compositions for global thermodynamic phase stability. 

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