Interval arithmetic

Interval numbers provide a simple means of expressing data uncertainties by the specification of lower and upper bounds. An interval thus represents the range of plausible values for any given quantity. Arithmetic operations on interval numbers also provide a means to propagate uncertainties through a mathematical model. Salient aspects of interval arithmetic are given here further details may also be found in Kaufmann and Gupta (1985). 

Given interval quantities A = J and B = J, the following operations hold  

In Equation C.4, the operation becomes defined if b is negative and b is positive, since that results in the inclusion of zero in the denominator. 

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The crudest form of bounding analysis is just interval arithmetic (Moore 1966 Neumaier 1990). In this approach the uncertainty about each quantity is reduced to a pair of numbers, an upper bound and a lower bound, that circumscribe all the possible values of the quantity. In the analysis, these numbers are combined in such a way to obtain sure bounds on the resulting value. Formally, this is equivalent to a worst case analysis (which tries to do the same thing with only 1 extreme value per quantity). The limitations of such analyses are well known. Both interval arithmetic and any simple worst case analysis... 

The generalization of ordinary arithmetic to closed intervals is known as interval arithmetic. An interval is defined as a closed bounded set of real numbers (Moore, 1979) ... 

Thus, intervals have a dual nature as both a number and a set. The basic interval arithmetic operations are... 

The convergence of the normalization algorithm may be proved if one is able to estimate the perturbation left after a given number of normalization step. This may be done with analytical estimates. Thus, we can apply a formal statement of KAM theorem to a Hamiltonian with a remainder dramatically reduced, thanks to both explicit calculation of the expansion and recursive estimates. A fully rigorous result may be achieved by performing all the calculations using interval arithmetics, so that we have full control on the propagation of roundoff errors. 

Researchers who take this semantic for the set HR see each interval as a wrapper that has information of a real number. From this point of view the multiplication of an interval X HR by itself not always has the same result that the multiplication proposed by Moore. In this approach X is always a normegative interval, since it represents the same real number. This interpretation is usually accepted in the context of interval arithmetic. According to this semantics an interval of real numbers is a real subject to uncertainties, ie... 

Moore, R. E. (1962). Interval arithmetic and automatic error analysis in digital computing. Technical Report 25, Department of Mathematics, Stanford University, Stanford, California. NR-0440211. 

Popova, E. D. (1994). Extended interval arithmetic in ieee floating-point envioronment. Interval Computation pp. 100-129. 

The interval arithmetic approach consists of translating the complete deterministic numerical FE procedure to an interval procedure using the arithmetic operations for addition, subtraction, multiplication and division of interval scalars. The outline of the interval procedure corresponds completely to the deterministic FE analysis. 

The basic formulation of interval arithmetic analysis does not allow to keep track of the relationships between uncertain parameters. The result of this inability is a degree of conservatism that is prohibitively high to be useful in practical applications. Manson (2003) proposes a strategy using affine analysis. The basic idea of affine arithmetic is to keep track of dependency between operands and sub-formulae whilst retaining much of the simplicity of interval arithmetic. As a result, tighter bounds are predicted than with interval arithmetic, especially when multiple iterations are necessary. Manson has applied this approach on systems with two degrees of freedom where all equations can be written explicitly. For more complicated and less explicit operations like FE analysis, the affine analysis approach is still in development. 

Thus, the three step algorithm results in a hybrid procedure in the first step, the /hi)-domain is approximated using a global optimisation approach in the second and third step, the modal and total envelope FRFs are calculated using interval arithmetic. 

Interval analysis was first introduced by Moore (1966). An interval consists of a lower and an upper bound. Interval arithmetic ensures that the interval result of an operation on two intervals contains all achievable real values. This is useful as continuous real variables can be divided into discrete interval sections. If mathematical operations are carried out on these intervals, according to the rules of interval arithmetic, the result will contain the range of all possible values. As a result, the global optimum can be bounded. Hansen (1992) provides a detailed explanation of some interval methods and their application to global optimization. 

The sequential modular approach can be formulated in a way suitable for interval methods. Modules are connected in the same way but must be modified to handle interval arithmetic. A generic module requires point values or intervals for all the input streams and unit parameters and calculates the conditions for the output streams as respectively values or intervals. 

Balanced random interval arithmetic is proposed for improving efficiency in global optimisation extending the ideas of random interval arithmetic where a random combination of standard and inner interval operations is used. The influence of the probability of the standard and inner interval operations to the ranges of functions is experimentally investigated on a manufacturing problem. 

One of the first proponents of interval arithmetic was Moore (1966). Interval arithmetic operates with real intervals x = [xj, 2 ] = x6 9 Xj x < X2, where xi and X2 are real... 

The difficulty is to know the monotonicity of the operands. This requires the computation of the derivatives of each subfunction involved in the expression of the function being studied, which needs a large amount of work. Alt and Lamote (2001) have proposed the idea of random interval arithmetic which is obtained by choosing standard or inner interval operations randomly with the same probability at each step of the computation. It is assumed that the distribution of the centres and radii of the evaluated intervals is normal. The mean values and the standard deviations of the centers and radii of the evaluated intervals computed using random interval arithmetic are used to evaluate an approximate range of the function ... 

Random interval arithmetic assumes that operators in all operations are monotonic. This may be the case when intervals are small and there is only one interval variable. When intervals are wide, as they can be in process engineering problems, operators cannot be assumed to be monotonic. Independent variables cannot be assumed monotonic either. Therefore such random interval arithmetic uses inner interval arithmetic too often and provides results which are too narrow when intervals are wide, so it cannot be applied to global optimization directly. 

Standard interval arithmetic provides guaranteed bounds but they are often too pessimistic. Standard interval arithmetic is used in global optimization providing guaranteed solutions, but there are problems for which the time of optimization is too long. Random interval arithmetic provides bounds closer to the exact range when intervals are small, but it provides too narrow bounds when intervals are wide. 

We would like to have interval methods that are less pessimistic than using standard interval arithmetic and less optimistic than with random interval arithmetic. We expect that the random interval arithmetic will provide wider or narrower bounds depending on... 

Experimental Study of the Balanced Random Interval Arithmetic... 

Figure I. The histograms of the centres and radii of the intervals offunction (10) evaluated using balanced random interval arithmetic with probabilities 0.5, 0.6 and 0.7 and of the function values at uniformly distributed random points.

The ranges of function (10) in 10000 random subregions have been evaluated using balanced random interval arithmetic with different probabilities of standard and inner... 

Alt, R. and Lamotte, J.-L., 2001. Experiments on the evaluation of functional ranges using random interval arithmetic. Mathematics and Computers in Simulation, 56, 17-34. 

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