Stochastic Arithmetic as a Model of Granular Computing

Numerical simulation is used more and more frequently in the analysis of physical phenomena. A simulation requires several phases. The first phase consists of constructing a physical model based on the results of experimenting with the phenomena. Next, the physical model is approximated by a mathematical model. Generally, the mathematical model contains algebraic expressions, ordinary or partial differential equations, or other mathematical features which are very complex and cannot be solved analytically. Thus, in the third phase the mathematical model must be transformed into a discrete model which can be solved with numerical methods on a computer. In the final phase the discrete model and the associated numerical methods must be translated into a scientific code by the use of a programming language. Unfortunately, when a code is run on a computer all the computations are performed using floating-point (FP) arithmetic which does not deal with real numbers but with ‘machine numbers’ consisting of a finite number of significant figures. Thus the arithmetic of the computer is merely an approximation of the exact arithmetic. It no longer respects the fundamental properties of the latter, so that every result provided by the computer always contains a round-off error, which is sometimes such that the result is false. It is therefore essential to validate all computer-generated results. Furthermore, the data used by the scientific code may contain some uncertainties. It is thus also necessary to estimate the influence of data errors on the results provided by the computer. This chapter is made up in two parts. In the first part, after a brief recalling how round-off error propagation results from FP arithmetic, the CESTAC method (Controle et Estimation STochastique des Arrondis de Calcul) is summarized. This method is a probabilistic approach to the analysis of round-off error propagation and to the analysis of the influence that uncertainties in data have on computed results. It is presented from both a theoretical and a practical point of view. The CESTAC method gives rise to stochastic arithmetic which is presented as a model of granular computing in a similar fashion to interval arithmetic and interval analysis [1]. Theoretically, in stochastic arithmetic the granules are Gaussian random variables and the tools working on these granules are the operators working on Gaussian random variables. In practice, stochastic arithmetic is discretized and is termed discrete stochastic arithmetic (DSA). In this case granules of DSA are the samples provided by the CADNA (Control of Accuracy and