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Polynomial chaos for the approximation of uncertainties: Chances and limits

Published online by Cambridge University Press:  01 April 2008

F. AUGUSTIN ,
A. GILG ,
M. PAFFRATH ,
P. RENTROP  and
U. WEVER
F. AUGUSTIN
Affiliation:
Technical University of Munich, Department of Mathematics, 85748 Garching, Germany email: augustin@ma.tum.de; toth-pinter@ma.tum.de
A. GILG
Affiliation:
Siemens AG, Corporate Technology, Otto-Hahn-Ring 6, 81730 Munich, Germany email: albert.gilg@siemens.com; meinhard.paffrath@siemens.com; utz.wever@siemens.com
M. PAFFRATH
Affiliation:
Siemens AG, Corporate Technology, Otto-Hahn-Ring 6, 81730 Munich, Germany email: albert.gilg@siemens.com; meinhard.paffrath@siemens.com; utz.wever@siemens.com
P. RENTROP
Affiliation:
Technical University of Munich, Department of Mathematics, 85748 Garching, Germany email: augustin@ma.tum.de; toth-pinter@ma.tum.de
U. WEVER
Affiliation:
Siemens AG, Corporate Technology, Otto-Hahn-Ring 6, 81730 Munich, Germany email: albert.gilg@siemens.com; meinhard.paffrath@siemens.com; utz.wever@siemens.com
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Abstract

In technical applications, uncertainties are a topic of increasing interest. During the last years the Polynomial Chaos of Wiener (Amer. J. Math. 60(4), 897–936, 1938) was revealed to be a cheap alternative to Monte Carlo simulations. In this paper we apply Polynomial Chaos to stationary and transient problems, both from academics and from industry. For each of the applications, chances and limits of Polynomial Chaos are discussed. The presented problems show the need for new theoretical results.

Type
A Survey in Mathematics for Industry
Information
European Journal of Applied Mathematics , Volume 19 , Issue 2 , April 2008 , pp. 149 - 190
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Alpert, B. K. (1993) A class of bases in L 2 for the sparse representation of integral operators. SIAM J. Math. Anal. 24, 246–262.CrossRefGoogle Scholar
[2]Augustin, F. (2007) Zur Numerik des Polynomiellen Chaos bei Differentialgleichungen, Diplomarbeit an der Technischen Universität München, Fachbereich Mathematik.Google Scholar
[3]Breitung, K. W. (1994) Asymptotic Approximations for Probability Integrals, Lecture Notes in Mathematics, Vol. 1592, Springer, Berlin-Heidelberg.CrossRefGoogle Scholar
[4]Bronstein, I. N., Semendjajew, K.A Musiol, G. & Mühlig, H. (2001) Taschenbuch der Mathematik, 5th ed., Verlag Harri Deutsch, Frankfurt.Google Scholar
[5]Cameron, R. & Martin, W. (1947) The orthogonal development of nonlinear functionals in series of Fourier–Hermite functionals. Ann. of Math. 48 (2), 385–392.CrossRefGoogle Scholar
[6]Davis, P. J. & Rabinowitz, P. (1975) Methods of Numerical Integration, Academic Press, New York.Google Scholar
[7]Denk, G. & Schäffler, S. (1997) Adams methods for the efficient solution of stochastic differential equations with additive noise. Computing. 59 (2), 153–161.CrossRefGoogle Scholar
[8]Fox, B. L. (1999) Strategies for Quasi-Monte Carlo, Kluwer Academic Publishers Group, Dordrecht, the Netherlands.CrossRefGoogle Scholar
[9]Gautschi, W. (1982) On generating orthogonal polynomials. SIAM J. Sci. Statist. Comput. 3, 289–317.CrossRefGoogle Scholar
[10]Georgii, H.-O. (2002) Stochastik. Einführung in die Wahrscheinlichkeitstheorie und Statistik, deGruyter Lehrbuch.Google Scholar
[11]Ghanem, R. (1999) Ingredients for a general purpose stochastic finite elements implementation. Comp. Methods Appl. Mech. Engrg. 168, 19–34.CrossRefGoogle Scholar
[12]Ghanem, R., Masri, S., Pellissetti, M. & Wolfe, R. (2005) Identification and prediction of stochastic dynamical systems in a polynomial chaos basis. Comput. Methods Appl. Mech. Engrg. 194, 1641–1654.CrossRefGoogle Scholar
[13]Ghanem, R. & Spanos, P. D. (1991) Stochastic Finite Elements – A Spectral Approach, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[14]Hairer, E.Nφrsett, S. P. & Wanner, G. (1993) Solving Ordinary Differential Equations 1, 2nd ed., Springer-Verlag, Berlin.Google Scholar
[15]Hairer, E. & Wanner, G. (1996) Solving Ordinary Differential Equations 2, 2nd ed., Springer-Verlag, Berlin.CrossRefGoogle Scholar
[16]Herzog, M., Gilg, A., Paffrath, M., Rentrop, P. & Wever, U. (2007) Intrusive versus non-intrusive methods for stochastic finite elements, In: Breitner, M.Denk, G. & Rentrop, P. (editors), From Nano to Space, Applied Mathematics inspired by R. Bulirsch, Springer, Berlin-Heidelberg.Google Scholar
[17]Hohenbichler, M., Gollwitzer, S., Kruse, W. & Rackwitz, R. (1987) New light on first- and second-order reliability methods. Struct. Saf. 4 (4), 267–284.CrossRefGoogle Scholar
[18]Hohenbichler, M. & Rackwitz, R. (1983) First-order concepts in system reliability. Struct. Saf. 1 (3), 177–188.CrossRefGoogle Scholar
[19]Janson, S. (1997) Gaussian–Hilbert Spaces, Cambridge University Press.CrossRefGoogle Scholar
[20]Karniadakis, G. E., Su, C. H., Xiu, D., Lucor, D., Schwab, C. & Todor, R. A. (2005). Generalized Polynomial Chaos Solution for Differential Equations with Random Inputs, ETH Zürich. (Research Report No. 2005-01).Google Scholar
[21]Kiureghian, A. D. (2000) The geometry of random vibrations and solutions by FORM and SORM. Probab. Eng. Mech. 15 (1), 81–90.CrossRefGoogle Scholar
[22]Kuske, R. & Keller, J. B. (1997) Large deviation theory for stochastic difference equations. EJAM. 8 (6), 567–580.Google Scholar
[23]LeMaitre, O. P.Knio, O. M.Najm, H. N. & Ghanem, R. G. (2004) Uncertainty propagation using Wiener–Haar expansions. J. Comput. Phys. 197, 28–57.CrossRefGoogle Scholar
[24]LeMaitre, O. P.Najm, H. N.Ghanem, R. G. & Knio, O. M. (2004) Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197, 502–531.CrossRefGoogle Scholar
[25]Loève, M. (1978) Probability Theory 2, 4th ed., Springer-Verlag, Berlin.Google Scholar
[26]Loh, W.-L. (1996) On latin hypercube sampling. Ann. Stat. 24 (5), 2058–2080.CrossRefGoogle Scholar
[27]Lorenz, E. N. (1963) Deterministic nonperiodic flow. J. Athmosph. Sci. 20, 130–141.2.0.CO;2>CrossRefGoogle Scholar
[28]Lucor, D. & Karniadakis, G. E. (2004) Adaptive generalized polynomial chaos for nonlinear random oscillators. SIAM J. Sci. Comput. 26 (2), 720–735.CrossRefGoogle Scholar
[29]Lucor, D. & Karniadakis, G. E. (2004) Noisy inflows cause a shedding-mode switching in flow past an oscillating cylinder. Phys. Rev. Lett. 92, Id 154501.CrossRefGoogle Scholar
[30]Lucor, D., Su, C. H. & Karniadakis, G. E. Karniadakis (2004) Generalized polynomial chaos and random oscillators. Int. J. Meth. Engng. 60, 571–596.CrossRefGoogle Scholar
[31]Madras, N. N. (2002) Lectures on Monte Carlo Methods, American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
[32]Meintrup, D. & Schäffler, S. (2005) Stochastik, Theorie und Anwendungen, Springer-Verlag, Berlin.Google Scholar
[33]Melchers, R. E. (1999) Structural Reliability Analysis and Prediction, John Wiley & Sons, Chichester.Google Scholar
[34]Novak, E. & Ritter, K. (1996) High-dimensional integration of smooth functions over cubes. Numer. Math. 75 (1), 79–97.CrossRefGoogle Scholar
[35]Oran, E. S. & Boris, J. B. (1987) Numerical Simulation of Reacting Flows, Elsevier Science Ltd., New York.Google Scholar
[36]Orszag, S. A. & Bisonette, L. R. (1967) Dynamical properties of truncated Wiener–Hermite expansions. Phys. Fluids. 10, 2603–261.CrossRefGoogle Scholar
[37]Paffrath, M. & Wever, U. (2005) The Probabilistic Optimizer RODEO, Internal Report Siemens AG.Google Scholar
[38]Paffrath, M. & Wever, U. (2007) Adaptive polynomial chaos expansion for failure detection. J. Comput. Phys. 226, 263–281,.CrossRefGoogle Scholar
[39]Petras, K. Asymptotically minimal Smolyak cubature, Preprint, available at http://www-public.tu-bs.de:8080/~petras/software.html.Google Scholar
[40]Petras, K. (2001) Fast calculation of coefficients in the Smolyak algorithm. Numerical Algorithms 26 (2), 93–109.CrossRefGoogle Scholar
[41]Pradtlwarter, H. J. & Schueller, G. I. (1997) On advanced Monte Carlo simulation procedures in stochastic structural dynamics. Int. J. Non-Linear Mech. 32 (4), 735–744.CrossRefGoogle Scholar
[42]Pulch, R. & Emmerich van, C. (Feb. 2007) Polynomial chaos for simulating random volatilities, Bergische Universität Wuppertal, Fachbereich Mathematik und Naturwissenschaften.Google Scholar
[43]Rackwitz, R. (2001) Reliability analysis – A review and some perspectives. Struct. Saf. 23 (4), 365–395.CrossRefGoogle Scholar
[44]Rice, S. O. (1980) Distribution of quadratic forms in normal random variables evaluation by numerical integration. SIAM J. Sci. Comput. 1 (4), 438–448.CrossRefGoogle Scholar
[45]Rosenblatt, M. (1952) Remarks on a multivariate transformation. Ann. Math. Stat. 23 (3), 470–472.CrossRefGoogle Scholar
[46]Schäffler, S. (1995) Unconstrained global optimization using stochastic integral equations. Optimization. 35, 43–60.CrossRefGoogle Scholar
[47]Schäffler, S., Schultz, R. & Weinzierl, K. (2002) A Stochastic method for the solution of unconstrained vector optimization problems. J.Optim. Theory Appl. 114, 209–222.CrossRefGoogle Scholar
[48]Schevenels, M., Lombaert, G. & Degrande, G. (2004) Application of the stochastic finite element method for Gaussian and non-Gaussian systems. In: Proceedings of ISMA, K. U. Leuven Department of Mechanical Engineering, PMA.Google Scholar
[49]Swift, R. J. (2002) A stochastic predator–prey model. Irish Math. Soc. Bull. (48), 57–63.CrossRefGoogle Scholar
[50]Taylor, R. L. FEAP Version 7.5 theory manual. http://www.ce.berkeley.edu/rlt/feap/theory.pdf.Google Scholar
[51]Taylor, R. L. (1988) FEAP: A Finite Element Analysis Program. Department of Civil and Environmental Engineering, University of California at Berkley. Research report.Google Scholar
[52]Volterra, V. (1931) Variations and fluctations of the number of individuals in animal species living together. In: Animal Ecology, McGraw-Hill.Google Scholar
[53]Wan, X. & Karniadakis, G. E. (2005) An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209, 617–642.CrossRefGoogle Scholar
[54]Wan, X. & Karniadakis, G. E. (2006) Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (3), 901–928.CrossRefGoogle Scholar
[55]Werner, D. (2005) Functional Analysis, 5th ed., Springer Verlag, Berlin-Heidelberg.Google Scholar
[56]Wiener, N. (1938) The homogeneous chaos. Amer. J. Math. 60 (4), 897–936.CrossRefGoogle Scholar
[57]Xiu, D. & Karniadakis, G. E. (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. Siam J. Sci. Comput. 24 (2), 619–644.CrossRefGoogle Scholar