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Keywords: uncertainty propagation, optimization, stochastic functions, polynomial chaos, Monte Carlo simulation, probability theory.

Summary

It is difficult to find examples of systems to be optimized that do not include some level of uncertainty about the values to assign to some of the parameters or about the actual design of some of the components of the system [1]. Several approaches have been developed to take into account uncertainties in optimization processes. The main techniques are: stochastic programming or robust optimization, fuzzy programming and reliability based design optimization. However, these techniques deal exclusively with approximating some statistical or probabilistic information of the response of the system.

Lopez et al. [2] developed a methodology based on the polynomial chaos expansion (PCE) [3] and stochastic approximation able to fully characterize the probabilistic information of the optimal point of stochastic functions, i.e. it is able to construct the random variable that characterizes this point.

One drawback of this methodology is the stochastic approximation step employed to calculate the coefficients of the PCE [2]. It may require a high computational cost and may have convergence problems in the case of non-convex functions [1].

Thus, the main contribution of this paper is the development a new methodology that avoids this step. Its key concept is the approximation of a random variable when only a sample of such a random variable is available. The first step is the construction of this sample using the MCS. Then, the random variable is represented using the PCE. The deterministic coefficients of the PCE are calculated by matching the moments of the sample to the ones of the PCE. Based on the numerical analysis, the main advantages of the new methodology are:

a): it avoids the stochastic approximation step of the PCE based approach, requiring only deterministic optimization algorithms;
b): the numerical results showed that its computational cost is lower than the one of the PCE based approach and,
c): it provided more accurate results requiring fewer function evaluations.

Even though the function analyzed in this paper was quite simple, the new methodology may be easily extended to more complex function, for example a structure modelled by finite elements. This is a topic of current research by the authors.

References

1: R.J-B. Wets, "Stochastic programming", Handbook for Operations Research and Management Sciences: Vol. 1, Elsevier, Amsterdam, 1989.
2: R.H. Lopez, E.S. Cursi, D. Lemosse, "Approximating the probability density function of the optimal point of optimization problems", Engineering Optimization, 281-303, 2011. doi:10.1080/0305215X.2010.489607
3: N. Wiener, "The homogeneous chaos", Amer. J. Math., 60, 897-936, 1938. doi:10.2307/2371268

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 96 PROCEEDINGS OF THE THIRTEENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping and Y. Tsompanakis Paper 91 A New Approach for the Full Characterization of the Optimal Point of Stochastic Functions R.H. Lopez¹, J.E.S. de Cursi² and A. El-Hami² ¹Department of Physics, Technological and Federal University of Paraná, Curitiba, Brazil ²Mechanical Engineering Department, National Institute of Applied Sciences (INSA), Rouen, France doi:10.4203/ccp.96.91 purchase the full-text of this paper Full Bibliographic Reference for this paper R.H. Lopez, J.E.S. de Cursi, A. El-Hami, "A New Approach for the Full Characterization of the Optimal Point of Stochastic Functions", in B.H.V. Topping, Y. Tsompanakis, (Editors), "Proceedings of the Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 91, 2011. doi:10.4203/ccp.96.91 Keywords: uncertainty propagation, optimization, stochastic functions, polynomial chaos, Monte Carlo simulation, probability theory. Summary It is difficult to find examples of systems to be optimized that do not include some level of uncertainty about the values to assign to some of the parameters or about the actual design of some of the components of the system [1]. Several approaches have been developed to take into account uncertainties in optimization processes. The main techniques are: stochastic programming or robust optimization, fuzzy programming and reliability based design optimization. However, these techniques deal exclusively with approximating some statistical or probabilistic information of the response of the system. Lopez et al. [2] developed a methodology based on the polynomial chaos expansion (PCE) [3] and stochastic approximation able to fully characterize the probabilistic information of the optimal point of stochastic functions, i.e. it is able to construct the random variable that characterizes this point. One drawback of this methodology is the stochastic approximation step employed to calculate the coefficients of the PCE [2]. It may require a high computational cost and may have convergence problems in the case of non-convex functions [1]. Thus, the main contribution of this paper is the development a new methodology that avoids this step. Its key concept is the approximation of a random variable when only a sample of such a random variable is available. The first step is the construction of this sample using the MCS. Then, the random variable is represented using the PCE. The deterministic coefficients of the PCE are calculated by matching the moments of the sample to the ones of the PCE. Based on the numerical analysis, the main advantages of the new methodology are: a) it avoids the stochastic approximation step of the PCE based approach, requiring only deterministic optimization algorithms; b) the numerical results showed that its computational cost is lower than the one of the PCE based approach and, c) it provided more accurate results requiring fewer function evaluations. Even though the function analyzed in this paper was quite simple, the new methodology may be easily extended to more complex function, for example a structure modelled by finite elements. This is a topic of current research by the authors. References 1 R.J-B. Wets, "Stochastic programming", Handbook for Operations Research and Management Sciences: Vol. 1, Elsevier, Amsterdam, 1989. 2 R.H. Lopez, E.S. Cursi, D. Lemosse, "Approximating the probability density function of the optimal point of optimization problems", Engineering Optimization, 281-303, 2011. doi:10.1080/0305215X.2010.489607 3 N. Wiener, "The homogeneous chaos", Amer. J. Math., 60, 897-936, 1938. doi:10.2307/2371268 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £130 +P&P)
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