Symbolic Computing in Probabilistic and Stochastic Analysis

Open access

Abstract

The main aim is to present recent developments in applications of symbolic computing in probabilistic and stochastic analysis, and this is done using the example of the well-known MAPLE system. The key theoretical methods discussed are (i) analytical derivations, (ii) the classical Monte-Carlo simulation approach, (iii) the stochastic perturbation technique, as well as (iv) some semi-analytical approaches. It is demonstrated in particular how to engage the basic symbolic tools implemented in any system to derive the basic equations for the stochastic perturbation technique and how to make an efficient implementation of the semi-analytical methods using an automatic differentiation and integration provided by the computer algebra program itself. The second important illustration is probabilistic extension of the finite element and finite difference methods coded in MAPLE, showing how to solve boundary value problems with random parameters in the environment of symbolic computing. The response function method belongs to the third group, where interference of classical deterministic software with the non-linear fitting numerical techniques available in various symbolic environments is displayed. We recover in this context the probabilistic structural response in engineering systems and show how to solve partial differential equations including Gaussian randomness in their coefficients.

Binder, K. and Heermann, D. (1997). Monte Carlo Simulation in Statistical Physics, Springer Verlag, Berlin/Heidelberg.

Bjorck, A. (1996). Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA.

Brandt, S. (1999). Data Analysis Statistical and Computational Methods for Scientists and Engineers, Springer-Verlag, New York, NY.

Burczyński, T. (1995). Boundary element method in stochastic shape design sensitivity analysis and identification of uncertain elastic solids, Engineering Analysis with Boundary Elements 15(2): 151–160.

Chakraverty, S. (2014). Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems, IGI Global, Hershey.

Falsone, G. (2005). An extension of the Kazakov relationship for non-Gaussian random variables and its use in the non-linear stochastic dynamics, Probabilistic Engineering Mechanics 20(1): 45–56.

Feller, W. (1965). An Introduction to Probability Theory and Its Applications, Wiley, New York, NY.

Grigoriu, M. (2000). Stochastic mechanics, International Journal of Solids and Structures 37(1–2): 228–248.

Hurtado, J. and Barbat, A. (1998). Monte-Carlo techniques in computational stochastic mechanics, Archives of Computer Methods in Engineering 37(1): 3–30.

Kamiński, M. (2005). Computational Mechanics of Composite Material, Springer-Verlag, London/New York, NY.

Kamiński, M. (2013). The Stochastic Perturbation Method for Computational Mechanics, Wiley, Chichester.

Kleiber, M. and Hien, T. (1992). The Stochastic Finite Element Method, Wiley, Chichester.

Kwiatkowska, M., Norman, G., Sproston, J. and Wang, F. (2007). Symbolic model checking for probabilistic timed automata, Information and Computation 205(7): 1027–1077.

Kwiatkowska, M., Parker, D., Zhang, Y. and Mehmood, R. (2004). Dual-processor parallelisation of symbolic probabilistic model checking, 12th International Symposium Modeling, Analysis and simulation of Computer and Telecommunication Systems MASCOTS’04, Volendaam, The Netherlands, pp. 123–130.

López, N., Nunez, M. and Rodriguez, I. (2006). Specification, testing and implementation relations for symbolic-probabilistic system, Theoretical Computer Science 353(1–3): 228–248.

Melchers, R. (2002). Structural Reliability Analysis and Prediction, Wiley, Chichester.

Moller, B. and Beer, M. (2004). Fuzzy Randomness. Uncertainty in Civil Engineering and Computational Mechanics, Springer-Verlag, Berlin/Heidelberg.

Nayfeh, A.H. (2000). Perturbation Method, Wiley-VCH Verlag GmbH, Weinheim.

Peng, X., Geng, L., Liyan, W., Liu, G. and Lam, K. (1998). A stochastic finite element method for fatigue reliability analysis of gear teeth subjected to bending, Computational Mechanics 21(3): 253–261.

Sakata, S., Ashida, F., Kojima, T. and Zako, M. (2008). Three-dimensional stochastic analysis using a perturbation-based homogenization method for elastic properties of composite material considering microscopic uncertainty, International Journal of Solids and Structures 45(3–4): 894–907.

Schueller, G. (2007). On the treatment of uncertainties in structural mechanics and analysis, Computers and Structures 85(5–6): 235–243.

Shachter, R.D., D’Ambrosio, B. and Del Favero, B. (1990). Symbolic probabilistic inference in belief networks, Proceedings of the 8th National Conference on Artificial Intelligence AAAI-90, Boston, MA, USA, pp. 126–131.

Shannon, C. (1948). A mathematical theory of communication, The Bell System Technical Journal 27(4): 623–656.

Sobczyk, K. and Spencer, B. (1992). Random Fatigue: From Data to Theory, Academic Press, Boston, MA.

Spanos, P. and Ghanem, R. (1991). Stochastic Finite Elements. A Spectral Approach, Springer-Verlag, Berlin/Heidelberg.

To, C. and Kiu, M. (1994). Random responses of discretized beams and plates by the stochastic central difference method with time co-ordinate transformation, Computers and Structures 53(3): 727–738.

Van Noortwijk, J. and Frangopol, D. (2004). Two probabilistic life-cycle maintenance models for deteriorating civil infrastructures, Probabilistic Engineering Mechanics 19(4): 345–359.

Wiggins, J. (1987). Option values under stochastic volatility. Theory and empirical evidence, Journal of Financial Economics 19(2): 351–372.

Zienkiewicz, O. and Taylor, R. (2005). Finite Element Method for Solid and Structural Mechanics, Elsevier, Amsterdam.

International Journal of Applied Mathematics and Computer Science

Journal of the University of Zielona Góra

Journal Information


IMPACT FACTOR 2017: 1.694
5-year IMPACT FACTOR: 1.712

CiteScore 2017: 2.20

SCImago Journal Rank (SJR) 2017: 0.729
Source Normalized Impact per Paper (SNIP) 2017: 1.604

Mathematical Citation Quotient (MCQ) 2017: 0.13

Cited By

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 114 114 18
PDF Downloads 32 32 5