Stochastic arithmetic and verification of mathematical models

Stochastic arithmetic enables one to estimate round-off error propagation using a probabilistic approach. With Stochastic arithmetic, the numerical quality of any simulation program can be controlled. Furthermore by detecting all the instabilities which may occur at run time, a numerical debugging of the user code can be performed. Stochastic arithmetic can be used to dynamically control approximation methods. Such methods provide a result which is affected by a truncation error inherent to the algorithm used and a round-off error due to the finite precision of the computer arithmetic. If the discretization step decreases, the truncation error also decreases, but the round-off error increases. Therefore it can be difficult to control these two errors simultaneously. In order to obtain with an approximation method a result for which the global error (consisting of both the truncation error and the round-off error) is minimal, a strategy, based on a converging sequence computation, has been proposed. Computation is carried out until the difference between two successive iterates has no exact significant digit. Then it is possible to determine which digits of the result obtained are in common with the exact solution. This strategy can apply to the computation of integrals using the trapezoidal rule, Simpson's rule, Romberg's method or the Gauss—Legendre method.