Abstract
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 numbers and 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.
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Alt, R., Lamotte, J.-L., Markov, S.: Numerical Study of Algebraic Solutions to Linear Problems Involving Stochastic Parameters. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2005. LNCS, vol. 3743, pp. 273–280. Springer, Heidelberg (2006)
Alt, R., Lamotte, J.-L., Markov, S.: Abstract structures in stochastic arithmetic. In: Bouchon-Meunier, B., Yager, R.R. (eds.) Proc. 11th Conference on Information Processing and Management of Uncertainties in Knowledge-based Systems (IPMU 2006), EDK, Paris, pp. 794–801 (2006)
Alt, R., Markov, S.: On Algebraic Properties of Stochastic Arithmetic. Comparison to Interval Arithmetic. In: Kraemer, W., Gudenberg, J.W.v. (eds.) Scientific Computing, Validated Numerics, Interval Methods, pp. 331–342. Kluwer Academic Publishers, Dordrecht (2001)
Alt, R., Vignes, J.: Validation of Results of Collocation Methods for ODEs with the CADNA Library. Appl. Numer. Math. 20, 1–21 (1996)
Chesneaux, J.M., Vignes, J.: Les fondements de l’arithmétique stochastique. C.R. Acad. Sci., Paris, Sér. I, Math. 315, 1435–1440 (1992)
Delay, F., Lamotte, J.-L.: Numerical simulations of geological reservoirs: improving their conditioning through the use of entropy. Mathematics and Computers in Simulation 52, 311–320 (2000)
NTLAB—INTerval LABoratory V. 5.2., www.ti3.tu-harburg.de/~rump/intlab/
Lamotte, J.-L., Epelboin, Y.: Study of the numerical stability of a X-RAY diffraction model. In: Computational Engineering in Systems Applications, CESA 1998 IMACS Multiconference, Nabeul-Hammamet, Tunisia, vol. 1, pp. 916–919 (1998)
Markov, S.: Least squares approximations under interval input data, Contributions to Computer Arithmetic and Self-Validating Numerical Methods. In: Ullrich, C.P. (ed.) IMACS Annals on computing and applied mathematics, Baltzer, vol. 7, pp. 133–147 (1990)
Markov, S., Alt, R.: Stochastic arithmetic: Addition and Multiplication by Scalars. Appl. Numer. Math. 50, 475–488 (2004)
Markov, S., Alt, R., Lamotte, J.-L.: Stochastic Arithmetic: S-spaces and Some Applications. Numer. Algorithms 37(1–4), 275–284 (2004)
Rokne, J.G.: Interval arithmetic and interval analysis: An introduction, Granular computing: An emerging paradigm, Physica-Verlag GmbH, 1–22 (2001)
Scott, N.S., et al.: Numerical ‘health check’ for scientific codes: The CADNA approach. Comput. Physics communications 176(8), 507–521 (2007)
Toutounian, F.: The use of the CADNA library for validating the numerical results of the hybrid GMRES algorithm. Appl. Numer. Math. 23, 275–289 (1997)
Toutounian, F.: The stable A T A-orthogonal s-step Orthomin(k) algorithm with the CADNA library. Numer. Algo. 17, 105–119 (1998)
Vignes, J., Alt, R.: An Efficient Stochastic Method for Round-Off Error Analysis. In: Miranker, W.L., Toupin, R.A. (eds.) Accurate Scientific Computations. LNCS, vol. 235, pp. 183–205. Springer, Heidelberg (1986)
Vignes, J.: Review on Stochastic Approach to Round-Off Error Analysis and its Applications. Math. and Comp. in Sim. 30(6), 481–491 (1988)
Vignes, J.: A Stochastic Arithmetic for Reliable Scientific Computation. Math. and Comp. in Sim. 35, 233–261 (1993)
Yao, Y.Y.: Granular Computing: basic issues and possible solutions. In: Wang, P.P. (ed.) Proc. 5th Joint Conference on Information Sciences, Atlantic City, N. J., USA, February 27– March 3, Assoc. for Intelligent Machinery, vol. I, pp. 186–189 (2000)
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Alt, R., Lamotte, JL., Markov, S. (2008). Numerical Study of Algebraic Problems Using Stochastic Arithmetic. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2007. Lecture Notes in Computer Science, vol 4818. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78827-0_12
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DOI: https://doi.org/10.1007/978-3-540-78827-0_12
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