Jean Luc Lamotte

Stochastic arithmetic: s-spaces and some applications

It has been recently shown that computation with stochastic numbers as regard to addition and multiplication by scalars can be reduced to computation in familiar vector spaces. In this work we show how this can be used for the algebraic solution of linear systems of equations with stochastic right-hand sides. On several examples we compare the algebraic solution with the simulated solution using the CADNA package.

On the numerical solution to linear problems using stochastic arithmetic

It has been recently shown that computation with stochastic numbers as regard to addition and multiplication by scalars can be reduced to computation in familiar vector spaces. This result allows us to solve certain practical problems with stochastic numbers and to compare algebraically obtained results with practical applications of stochastic numbers, such as the ones provided by the CESTAC method. Such comparisons give additional information related to the stochastic behavior of random roundings in the course of numerical computations. A number of original numerical experiments are presented that agree with the expected theoretical results.

Numerical study of algebraic solutions to linear problems involving stochastic parameters

We formulate certain numerical problems with stochastic numbers and compare algebraically obtained results with experimental results provided by the CESTAC method. Such comparisons give additional information related to the stochastic behavior of random roundings in the course of numerical computations. The good coincidence between theoretical and experimental results confirms the adequacy of our algebraic model and its possible application in the numerical practice. n nAMS Subject Classification: 65C99, 65G99, 93L03.

Kinetic and FT-IR study for the mechanism of addition of hydrogen sulfide to methyl acrylate over solid basic catalysts

Abstract Addition of H2S to methyl acrylate was performed in a batch reactor at 413xa0K, 400xa0rpm and at different concentrations in methyl acrylate and H2S in the presence of a commercially available mixed magnesium–aluminum basic oxide, namely the KW 2200 catalyst. Under chemical regime, the rate law was shown to follow a “poisoned” Eley–Rideal mechanism in which H2S reacts in the adsorbed state, whereas methyl acrylate, also capable of being adsorbed on the catalyst, would react in a physisorbed state. FT-IR study of methyl acrylate and H2S adsorption performed on MgO are in agreement with these kinetic results. Co-adsorption experiments showed that both methyl acrylate and H2S were strongly adsorbed on MgO. By varying the order of introduction of the two reactants, i.e. adsorption of H2S followed by methyl acrylate addition or methyl acrylate adsorption followed by H2S addition, it was shown that chemisorbed H2S species like HS− species which result from H2S dissociative adsorption were the active species and that strongly adsorbed methyl acrylate was not involved in the reaction.

Numerical Study of Algebraic Problems Using Stochastic Arithmetic

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.

Error-Free Transformation in Rounding Mode toward Zero

In this paper, we provide new error-free transformations for the sum and the product of two floating-point numbers. These error-free transformations are well suited for the CELL processor. We prove that these transformations are error-free, and we perform numerical experiments on the CELL processor comparing these new error-free transformations with the classic ones.

An (almost) direct deployment of the Fast Multipole Method on the Cell processor

This paper presents the first deployment of the Fast Multipole Method on the Cell processor (PowerXCellxa08i). We rely on the matrix formulation with BLAS routines of the FMB code (Fast Multipole with BLAS) in order to directly and efficiently offload the most time consuming operators of both far field and near field computations on the Cell heterogeneous cores. We detail the difficulties that had to be solved first, and we finally obtain a deployment in single and double precisions, which scales linearly on several Cell blades and which is able to handle both uniform and non-uniform distributions of particles. We also present our performance results and comparisons with multicore CPUs, as well as the limitations of our deployment on the Cell processor.

Testing Stochastic Arithmetic and CESTAC Method Via Polynomial Computation

The CESTAC method and its implementation known as CADNA software have been created to estimate the accuracy of the solution of real life problems when these solutions are obtained from numerical methods implemented on a computer. The method takes into account uncertainties on data and round-off errors. On another hand a theoretical model for this method in which operands are gaussian variables called stochastic numbers has been developed. In this paper numerical examples based on the Lagrange polynomial interpolation and polynomial computation have been constructed in order to demonstrate the consistency between the CESTAC method and the theory of stochastic numbers. Comparisons with the interval approach are visualized.

Parallel Integration Across Time of Initial Value Problems Using PVM

This paper shows the efficiency of PVM in the solution of initial value problems on distributed architectures. The method used here is a collocation method showing a large possibility of parallelism across time. The solution of the linear or non linear system, which is a part of the method is obtained with Picards iterations using divided differences. This solution is also obtained in parallel as well as the values of the approximated solution at different times. Numerical examples are considered. The tests have been performed on a network of workstations and on the Connection Machine CM5.