Review on stochastic approach to round-off error analysis and its applications
Mathematics and Computers in Simulation
A stochastic arithmetic for reliable scientific computation
Mathematics and Computers in Simulation
Validation of results of collocation methods for ODEs with the CADNA library
Applied Numerical Mathematics
The use of the CADNA library for validating the numerical results of the hybrid GMRES algorithm
Applied Numerical Mathematics
Mathematics and Computers in Simulation
An Efficient Stochastic Method for Round-Off Error Analysis
Proceedings of the Symposium on Accurate Scientific Computations
Interval arithmetic and interval analysis: an introduction
Granular computing
Stochastic arithmetic: addition and multiplication by scalars
Applied Numerical Mathematics
Numerical study of algebraic solutions to linear problems involving stochastic parameters
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
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A widely used method to estimate the accuracy of the numerical solution of real life problems is the CESTAC Monte Carlo type method. In this method, a real number is considered as an N-tuple of Gaussian random numbers constructed as Gaussian approximations of the original real number. This N-tuple is called a "discrete stochastic number" and all its components are computed synchronously at the level of each operation so that, in the scope of granular computing, a discrete stochastic number is considered as a granule. In this work, which is part of a more general one, discrete stochastic numbers are modeled by Gaussian functions defined by their mean value and standard deviation and operations on them are those on independent Gaussian variables. These Gaussian functions are called in this context stochastic numbersand operations on them define continuous stochastic arithmetic (CSA). Thus operations on stochastic numbers are used as a model for operations on imprecise numbers. Here we study some new algebraic structures induced by the operations on stochastic numbers in order to provide a good algebraic understanding of the performance of the CESTAC method and we give numerical examples based on the Least squares method which clearly demonstrate the consistency between the CESTAC method and the theory of stochastic numbers.